1 Introduction↩︎

We are concerned with the boundary behaviour of convex variational problems with linear growth and Dirichlet boundary conditions, that is we consider integrands \(f:\overline{\Omega}\times \mathbb{R}^{n\times d}\rightarrow \mathbb{R}\), where \(\Omega\subset \mathbb{R}^{d}\) is some bounded domain with a sufficiently regular boundary, of which we assume that they are convex in the second variable and there are constants \(C_0\) and \(C_0'\) such that \[\begin{align} C_0|\xi|-C_0'\leq f(y,\xi)\leq C_0'(1+|\xi|) \end{align}\] for all \(y\) and \(\xi\).

For these, we study the minimization problem (potentially also with lower order terms to be specified later on) \[\begin{align} \min_{u\in W_{u_0}^{1,1}(\Omega, \mathbb{R}^m)} \int_\Omega f(x,\mathrm{D}u(x))\,\mathrm{d}x\label{org32prob} \end{align}\tag{1}\] for some boundary datum \(u_0\). It is classical that this problem will in general not have minimizers due to the failure of weak^\(\ast\)-closedness of \(W_{u_0}^{1,1}\) and elementary counterexamples exist¹. Instead, to obtain a solution, one has to replace the functional with its lower semicontinuous hull, yielding the problem \[\begin{align} \MoveEqLeft\min_{u\in BV(\Omega, \mathbb{R}^n)} \int_\Omega f(x,\mathrm{D}^au(x))\,\mathrm{d}x+\int_\Omega f^\infty(x,\frac{\mathrm{d}\mathrm{D}^s u}{\mathrm{d}|\mathrm{D}^s u|}(x))\,\mathrm{d}|\mathrm{D}^s u|(x)\\ &+\int_{\partial\Omega} f^\infty(x,(u_0-u)(x)\otimes \nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x),\label{rel32prob} \end{align}\tag{2}\] where \(\nu_x\) is the (outer) normal of \(\partial\Omega\) at \(x\) and \[\begin{align} f^\infty(x,\xi)=\lim_{x'\rightarrow x, t\rightarrow +\infty} \frac{1}{t}f(x',t\xi)\label{rec32func} \end{align}\tag{3}\] is the recession function (see Section 2 for the other definitions), to which the direct method in the calculus of variations is applicable, and a (not necessarily unique) minimizer always exists.

It is a natural question under which circumstances these modifications are actually necessary. The issue of the prevalence of the space \(BV\) is reasonably well understood, one can show that under extra ellipticity conditions on the integrand, interior Sobolev or even Hölder regularity does hold, see e.g. [1]–[5], while without ellipticity, easy counterexamples exist.

This work is concerned with the other modification, that is the replacement of the Dirichlet boundary condition with the penalty term \(\int_{\partial\Omega} f^\infty(x,(u_0-u)(x)\otimes \nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x)\), whose necessity is a much less explored question and only understood in special cases and only in the scalar case \(n=1\).

Whether or not the boundary datum is attained largely hinges on curvature conditions on the boundary.

The two most classical cases in which the boundary behaviour is understood are that of the non-parametric Plateau problem, which consists of finding a function whose graph is a minimal surface with a given boundary, and which corresponds to \(n=1\) and \(f(y,\xi)=\sqrt{1+|\xi|^2}\) and the least gradient problem, which is the Dirichlet problem for the \(1\)-Laplacian, corresponding to \(n=1\) and \(f(y,\xi)=|\xi|\).

For the Plateau problem, the first existence results for \(d=2\) were obtained by Bernstein over a century ago [6], while Serrin and Jenkins [7] showed in 1968 that for \(d\geq 2\), in domains whose boundary has non-negative mean curvature, a \(C^2\) solution exists for any \(C^2\) boundary datum, while for every domain with boundary parts of negative mean curvature counterexamples exist.

These classical results can also be shown to work with merely continuous (or a.e.continuous) boundary data [8], [9] and can also be extended to the prescribed mean curvature equation [10] (that is the problem 1 /2 with \(f(y,\xi)=\sqrt{1+|\xi|^2}\) and an additional term \(\int_\Omega Hu\,\mathrm{d}x\), corresponding to a right-hand side term in the Euler-Lagrange equation) or anisotropic versions [11]. We refer the reader e.g.to the textbook [12] for further information. On the other hand, for boundary data which is merely \(L^1\), Baldo and Modica constructed an example in [13] where the boundary data is not attained in a trace sense and in which the solution is also not unique.

For the least gradient problem, which is relevant e.g.for the computational methods for minimal surfaces [14], optimal design [15], or image denoising and inpainting [16], [17], existence of solutions for continuous boundary data in domains whose boundary has positive mean curvature was first shown by Sternberg, Williams and Ziemer in 1992 in [18], who also established that under these conditions solutions are unique. Again, positive mean curvature is a necessary condition here (the function in the footnote 1 earlier is a counterexample for negative curvature). Spradlin and Tamasan showed in [19] that having some kind of regularity assumption on the boundary data is also necessary, as there exists \(L^1\)-boundary data for which there is no minimizer attaining the boundary data in the trace sense. Górny, on the other hand, showed in [20] that continuity of the boundary datum can be weakened to continuity a.e.

For discontinuous boundary data, uniqueness of minimizers is in general wrong (see e.g.[21]), though quite a lot of structure on the set of minimizers is enforced [22].

The generalisation to the anisotropic least gradient problem, in which \(f\) is an \(x\)-dependent norm, was, motivated by applications in conductivity imaging, considered by Jerrard, Moradifam, and Nachman in [23], where it is shown that solutions exist for continuous boundary data if the mean curvature condition is suitably adapted to \(f\) and are unique under additional regularity assumptions on \(f\).

We refer to [24]–[29] for some further recent developments and to the textbook [30] for general background reading.

A few works on the boundary behaviour of other integrands with linear growth do exist, for instance in [31] Beck, Bulíček and Maringová showed that under stronger ellipticity conditions on the integrand, which do not hold in the aformentioned special cases, solutions attaining the boundary values do exist even without a curvature condition on the boundary, see also [32] for an earlier similar result. In [33], plasticity-type problems were considered.

The purpose of this work is to prove existence results in a more general setting, which covers both the minimal surface equation and the least gradient problem, as well as other integrands with linear growth, and extends the aforementioned results to boundary values that have (fractional) Sobolev or \(BV\) regularity instead of continuity and to non-scalar-valued problems. To the best of the author’s knowledge, these are the first results for boundary attainment both for fully discontinuous data and for systems.

1.1 Precise Set-up and main results↩︎

The scalar case \(n=1\)

Our approach is the following: We consider \(u\) which are minimizers of the problem \[\begin{align} \MoveEqLeft\min_{\substack{u\in BV(\Omega),\\ \int_\Omega \lambda |u|^2\,\mathrm{d}x<\infty}}\int_\Omega f(x,\mathrm{D}^au(x))\,\mathrm{d}x+\int_\Omega f^\infty(x,\frac{\mathrm{d}\mathrm{D}^s u}{\mathrm{d}|\mathrm{D}^s u|}(x))\,\mathrm{d}|\mathrm{D}^s u|(x)\\ &+\int_{\partial\Omega}f^\infty(x,(u_0-u)\otimes \nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x)+\int_\Omega gu+\frac{\lambda(x)}{2}|u-h|^2\,\mathrm{d}x\label{gen32rel32prob} \end{align}\tag{4}\] (definitions can be found in Section 2), which is the relaxation of the problem \[\begin{align} \MoveEqLeft\min_{\substack{u\in W_{u_0}^{1,1}(\Omega),\\ \int_\Omega \lambda |u|^2\,\mathrm{d}x<\infty}}\int_\Omega f(x,\mathrm{D}u(x))\,\mathrm{d}x+\int_\Omega gu+\frac{\lambda(x)}{2}|u-h|^2\,\mathrm{d}x.\label{gen32org32prob0} \end{align}\tag{5}\] A more detailed account of the relation between these problems is given in Section 3.

As already mentioned, the relaxed problem 4 is amenable to the direct method, presuming \(g\) is sufficiently small and \(\lambda\geq 0\) (see Proposition 8 for details), and therefore the existence of minimizers is quite straightforward, though they might not be unique.

Our results will then show that any minimizer of the problem 4 must have trace \(u_0\) under suitable assumptions.

Regarding the domain, we assume the following:

\(\Omega\subset \mathbb{R}^d\) is open, bounded and has Lipschitz boundary. We let \(U\subset\partial\Omega\) be an open subset of the boundary, of which we assume that it is \(C^2\) and that its (relative) boundary in \(\partial\Omega\) is \(C^2\).

The assumption about the relative boundary is not restrictive because \(u=u_0\) is a purely local property and therefore, if \(\partial U\) is not regular, one can cover \(U\) with more regular subsets and apply the theorem to each of them.

Our assumptions on the integrand \(f\) in the scalar setting \(n=1\) are the following:

\(f:\overline{\Omega}\times \mathbb{R}^d\rightarrow \mathbb{R}\) is such that:

• \(f(x,\cdot)\) is convex for each \(x\in \overline{\Omega}\).

• There is a constant \(C_1>0\), not depending on \(x\) or \(\xi\), such that for all \(x,\xi\), it holds that \[\begin{align} \frac{1}{C_1}|\xi|-C_1\leq f(x,\xi)\leq C_1(|\xi|+1).\label{bd32gf1} \end{align}\tag{6}\]

• There is some \(R>0\), such that for each \(x\in\overline{\Omega}\), the function \(f(x,\cdot)\) is in \(C^2(\mathbb{R}^{d}\backslash B_R(0))\) and it holds that \[\begin{align} \sup_{\xi:\, \xi\geq |a|}|\xi|\int_{|\xi|}^\infty \langle \mathrm{D}_\xi^2f(x,s\frac{\xi}{|\xi|})\frac{\xi}{|\xi|},\frac{\xi}{|\xi|} \rangle\,\mathrm{d}s\xrightarrow{|a|\rightarrow +\infty} 0\label{conv32df} \end{align}\tag{7}\] uniformly in \(x\) in a neighborhood of \(U\).

• \(f\) is continuous in the joint variable \((x,\xi)\in \overline{\Omega}\times \mathbb{R}^d\).

• The limit 3 in the definition of \(f^\infty\) exists for all \(x\in \overline{\Omega}\) and all \(\xi\in \mathbb{R}^d\).

• Regarding the regularity of the recession function, defined in 3 , we assume that²

  1. \(f^\infty\in C_x^1C_\xi^1(\overline{\Omega},B_2(0)\backslash B_{\frac{1}{2}}(0))\)

  2. \(f^\infty \in C_\xi^2(B_2(0)\backslash B_{\frac{1}{2}}(0))\) and it holds that \(\sup_{x\in \overline{\Omega}} \left\lVert f(x,\cdot) \right\rVert_{C^2(B_2(0)\backslash B_{\frac{1}{2}}(0))}<\infty\).

This includes the cases \(f(x,\xi)=|\xi|, \sqrt{1+|\xi|^2}\) as well as \(x\)-dependent versions thereof (presuming the \(x\)-dependence is sufficiently regular), and the function \(\min(|\xi|^2,|\xi|)\) used for Hencky plasticity as well as many others.

Obviously, the mean curvature condition needs to be adapted to \(f\). We let \(\mathfrak{d}:\Omega\rightarrow \mathbb{R}_{>0}\) denote the distance to the boundary of \(\partial\Omega\), which is a \(C^2\) function in a neighborhood of each \(x\in U\) by the implicit function theorem. We then set \[\begin{align} H_{\partial\Omega,f}(x):=\min\left(-\operatorname{div}(\mathrm{D}_\xi f^\infty(x,\mathrm{D}\mathfrak{d}(x))),\,\operatorname{div}(\mathrm{D}_\xi f^\infty(x,-\mathrm{D}\mathfrak{d}(x)))\right)\label{def32hf} \end{align}\tag{8}\] for \(x\in U\). If \(f^\infty\) is even, both terms in the minimum agree, and this is essentially the same definition as used by Jerrard, Moradifam, and Nachman for the anisotropic least gradient problem in [23]. In this case, this generalizes the mean curvature (normalized to be \(d-1\) for the unit sphere) as it is the first variation of the perimeter-type functional \(\int_{\partial\Omega} f^\infty(x,\nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x)\). In particular, if \(f^\infty=|\cdot|\), then this is the classical mean curvature.

On the other hand, if \(f\) is not even, these terms might not be the same, and it is not difficult to construct one-dimensional examples showing that one needs to take both terms into account. Let us also note that, as a consequence of [A1] and [A2], \(H_{\partial\Omega,f}\) is continuous on \(U\).

Regarding \(g\) and the curvature we assume that: \(g\in L^{\max(2,d)}(\Omega)\) is bounded in a neighborhood of \(U\) and there is a \(c>0\) such that it holds that³ \[\begin{align} \mathop{\mathrm{essinf}}_{\Omega\ni y\rightarrow x} H_{\partial\Omega,f}(x)-|g(y)|>c \end{align}\] for all \(x\in U\).

In particular this implies that \[\begin{align} \inf_{x\in U}H_{\partial\Omega,f}(x)>0. \end{align}\] Let us also point out that \(g\in L^d(\Omega)\) ensures that the integral \(\int gu\,\mathrm{d}x\) in the functional exists because \(BV\hookrightarrow L^{\frac{d}{d-1}}\).

We make the following assumptions on \(\lambda\) and \(h\): \(\lambda\in L^\infty(\Omega)\) and \(\lambda\geq 0\) and \(h\in L^2(\Omega)\).

At least one, but not necessarily more, of the following three statements holds: \[\begin{align} &\lambda=0 \quad \text{in a neighborhood of U.}\\ &h\in BV \quad \text{in a neighborhood of U.}\\ & h\in C^0\quad \text{in a neighborhood of U.} \end{align}\] In the second or third case, we also require \[\begin{align} h=u_0\text{ on U.} \end{align}\]

We remark that if the third condition holds, then \(u_0\) must be automatically continuous.

Regarding the regularity of the boundary datum, we consider the following cases: At least one (but not necessarily more) of the following three conditions holds:

• \(u_0\in BV(U)\cap L^1(\partial\Omega)\)

• \(u_0\in W^{\alpha,p}(U)\cap L^1(\partial\Omega)\) for some \(\alpha\in (0,1)\) and \(p\in (2,\infty)\) with \(\alpha p\geq 2\)

• \(u_0\in C^0(U)\cap L^1(\partial\Omega)\) and (\(f=f^\infty\) and (\(\lambda=0\) or \(h\in C^0\) in a neighborhood of \(U\)))

The definitions of the spaces are also recalled in Sections 2.2 and 2.4. Let us also stress that, unlike previous works, this does not require \(u_0\) (or any of its representatives) to be continuous at any point.

We also point out that the case \(\lambda=g=h=0\) is of course included here, and in this case the assumptions [A3]-[A5] just reduce to \(\inf H_{\partial\Omega,f}>0\).

The extra assumption that \(u\in L^2\) is a technical limitation due to the fact that the version of the Euler-Lagrange equation which is used in the proof (see Lemma 3) is in full generality only available for \(u\in L^2\). We expect that using the ideas from the \(L^1\)-theory of the parabolic versions of these problems [34], [35], it is probably possible to remove this assumption, at least in some cases. In fact, in the simplest special case of the least gradient problem, we will show in Corollary 5 that this can be removed with an easy truncation argument.

Nevertheless, it is usually not difficult to verify that \(u\in L^2\), it is for instance always true if there is a maximum principle (see [36] for a detailed discussion of when this is true), if \(\lambda> 0\), or if \(d=2\) it follows from the Sobolev inequality since \(u\in BV(\Omega)\hookrightarrow L^2(\Omega)\) if \(d=2\).

The vectorial case \(n\geq 1\) In the vectorial case, we do not include lower order terms and assume that \(f\) is autonomous, i.e.we consider minimizers \(u\) of the following problem: \[\begin{align} \MoveEqLeft\min_{u\in BV(\Omega, \mathbb{R}^n)} \int_\Omega f(\mathrm{D}^au(x))\,\mathrm{d}x+\int_\Omega f^\infty(\frac{\mathrm{d}\mathrm{D}^s u}{\mathrm{d}|\mathrm{D}^s u|}(x))\,\mathrm{d}|\mathrm{D}^s u|(x)\\ &+\int_{\partial\Omega} f^\infty((u_0-u)(x)\otimes \nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x),\label{vproblem} \end{align}\tag{9}\] which is the relaxed version of the problem \[\begin{align} \MoveEqLeft\min_{u\in W_{u_0}^{1,1}(\Omega, \mathbb{R}^n)} \int_\Omega f(\mathrm{D}u(x))\,\mathrm{d}x.\label{og32vproblem} \end{align}\tag{10}\] Again, 9 is the natural extension of Problem 10 and has (not necessarily unique) minimizers by the direct method (cf.Propositions 7 and 8).

Regarding the domain, we shall assume that: \(\Omega\subset \mathbb{R}^d\) is a bounded domain with Lipschitz boundary. \(U\subset \partial\Omega\) is a subset of the boundary which can be written as a graph of a uniformly convex \(C^2\)-function and whose relative boundary in \(\partial\Omega\) is \(C^2\).

Again, the assumption that \(U\) is a graph with a regular boundary is not restrictive, as the theorem’s statement is purely local.

Regarding the integrand \(f\) we assume:

• \(f:\mathbb{R}^{n\times d}\rightarrow \mathbb{R}\) is convex.

• There is a constant \(C_1>0\), such that \[\begin{align} \frac{1}{C_1}|\xi|-C_1\leq f(\xi)\leq C_1(|\xi|+1).\label{bd32gf2} \end{align}\tag{11}\]

• There is an \(R>0\), such that \(f\in C^2(\mathbb{R}^{n \times d}\backslash B_R(0))\cap C^1(\mathbb{R}^{n\times d}\backslash \{0\})\), and it holds that \[\begin{align} \sup_{\xi:\, \xi\geq |a|}|\xi|\int_{|\xi|}^\infty \langle \mathrm{D}_\xi^2f(s\frac{\xi}{|\xi|})\frac{\xi}{|\xi|},\frac{\xi}{|\xi|} \rangle\,\mathrm{d}s\xrightarrow{|a|\rightarrow +\infty} 0.\label{almost32hom} \end{align}\tag{12}\]

• \(f^\infty\in C^2(\mathbb{R}^{n\times d}\backslash\{0\})\).

• There are positively \(0\)-homogeneous and odd functions \(\mu_1\) and \(\mu_2\) such that \[\begin{align} \mathrm{D}_\xi f^\infty(v_1\otimes v_2)=\mu_1(v_1)\otimes \mu_2(v_2)\quad \text{for all v_1,v_2\neq 0.}\label{r132split} \end{align}\tag{13}\]

• There is a \(C_2>0\) such that for all \(\xi\neq 0\) it holds that\[\begin{align} \mathrm{D}_\xi^2((f^\infty)^2)(\xi)\geq C_2 \mathrm{Id}\quad \text{in the sense of positive definiteness}.\label{uni32conv} \end{align}\tag{14}\]

• If for some \(\xi\neq 0\), there is a \(\xi'\) with \(\mathrm{D}_\xi f^\infty(\xi')=\mathrm{D}_\xi f(\xi)\) then \(\mathrm{D}_\xi f^\infty(\xi')=\mathrm{D}_\xi f^\infty(\xi)\).

We remark that 13 implies that \(f^\infty\) is even.

Regarding the boundary data we assume that: \[\begin{align} u_0\in L^1(\partial\Omega,\mathbb{R}^n)\cap BV(U,\mathbb{R}^n)\cap L^2(U,\mathbb{R}^n).\label{re32u02} \end{align}\tag{15}\]

Some model integrands for which these assumptions hold are for instance \(f(\xi)=|\xi|, \sqrt{1+|\xi|^2}\) or \(\sqrt{\sum_{i=1}^n A\xi_{i,\bullet}\cdot \xi_{i,\bullet}}\) for any symmetric positive definite \(A\in \mathbb{R}^{d\times d}\). Let us stress that this does not contain the minimal surface problem in higher codimension as it is not convex (see e.g.[37]).

Our main theorem in this case is the following.

This follows from the theorem with the same argument as for Corollary 1.

On the other hand, we also have a smooth counterexample showing the necessity of 13 .

Of course, this data also fulfills [B1] and [B3]. This example illustrates how the situation here is quite different from the scalar setting, as for every scalar, even, uniformly convex and autonomous integrand, the ball would fulfill the mean curvature condition ⁴.

Roughly speaking, the mechanism for smooth non-attainment here is that there is an extension of \(u_0\) such that a part of \(\partial\Omega\) is a minimal surface with respect to the anisotropic scalar integrand \(f(u_0(y)\otimes \cdot)\).

1.2 Further discussion↩︎

Uniqueness Our results also imply new uniqueness results for minimizers in a lot of cases.

There are many existing theorems about the regularity of these kinds of problems (including the vectorial case), showing that under certain ellipticity assumptions, which hold e.g.for the Plateau problem, any minimizer must automatically be in \(W^{1,1}\), see e.g.[1]–[4]. On the other hand, for the least gradient problem, strict convexity does not hold (and neither does interior regularity), and uniqueness might fail in this setting, see e.g.[21]. A classical counterexample due to Santi [40] (see also [36]) also shows that in almost nowhere mean-convex domains, uniqueness might fail even though interior regularity holds.

We also refer to [36] for a more detailed discussion of continuity in the boundary datum and similar issues.

The Dirichlet problem for the Rudin-Osher-Fatemi functional

Let us comment on the special case in which \(n=1\) and \(\lambda=1\) and \(f(x,\xi)=|\xi|\), in which the functional reduces (up to a constant) to the famous Rudin-Osher-Fatemi functional [16] given by \[\begin{align} \int_\Omega |\mathrm{D} u|+\int_\Omega\frac{1}{2}(u-\bar{h})^2\,\mathrm{d}x\quad \text{(with \bar{h}=h+g)}\label{ROF} \end{align}\tag{16}\] equipped with Dirichlet boundary values \(u_0\). The question when this Dirichlet problem (usually the functional is considered with a Neumann boundary condition) is solvable was raised by Brezis in [41] and was previously open for \(d>1\).

In this case, our results give the following answer to this question:

Similar statements can also be made for \(u_0\in BV(\partial\Omega)\) or \(u_0\in W^{\alpha,p}(\partial\Omega)\). Of course, the result also holds for other values of \(\lambda\) if \(\min H_{\partial\Omega}-\lambda|u-\bar{h}|>0\).

Let us also mention that if \(d<8\), then \(u\) inherits the interior regularity of \(\bar{h}\) in the sense that if \(\bar{h}\in C^{\beta}(\Omega)\), then \(u\in C^\beta(\Omega)\) for \(\beta\in (0,1]\) as shown in [42].

The bound of \(H_{\partial\Omega}\) on \(|u_0-\bar{h}|\) is also sharp if \(d\geq 3\), for \(|u_0-\bar{h}|>H_{\partial\Omega}\) a counterexample is given by \[\begin{align} u_0=0;\quad \bar{h}=(1+t)\operatorname{div}\frac{x}{|x|};\quad u=t\operatorname{div}\frac{x}{|x|}\quad \text{ with \Omega=B_1(0)} \end{align}\] for any \(t\in \mathbb{R}_{>0}\), since \(\operatorname{div}\frac{x}{|x|}=\frac{d-1}{|x|}\). To see that this is indeed a counterexample, one notes that on the one hand minimizers of both the original problem and the relaxed version 4 must be unique due to the strictly convex \((u-\bar{h})^2\)-term. On the other hand, one can check by direct calculation that this \(u\) is indeed a minimizer of the relaxed problem by using the subdifferential characterisation in the Lemmata 3 and 5 with \(z=-\frac{x}{|x|}\).

We also stress that the necessity of non-negative mean curvature persists here, even if \(\bar{h}=u_0\) on the boundary. A counterexample is for instance given by \(\Omega=B_1(0)\backslash B_{\frac{1}{2}}(0)\subset \mathbb{R}^2\) with the data \[\begin{align} u=0\quad \text{and}\quad \bar{h}=u_0=\frac{4}{3|x|}-\frac{4}{3}, \end{align}\] which does not attain the boundary data at \(|x|=\frac{1}{2}\). To verify that this is indeed a counterexample, one can again use the uniqueness and Lemma 3, together with \(z=\frac{2}{3}x-\frac{4}{3}\frac{x}{|x|}\).

On the other hand, optimality of this result is certainly an open-ended question, and if one takes more information on the structure of the data into account, then better results for \(d=1\) are available, see e.g.[41] and [43].

The trace space of functions of least gradient Let us consider the special case of the least gradient problem, that is 4 with \(g=\lambda=h=0\) and \(f(x,\xi)=|\xi|\). In [30] Górny and Mazón raised the question of whether for every \(u_0\in BV(\partial\Omega)\) there is a minimizer \(u\) of 4 for which \(u=u_0\) in the trace sense (which was already shown in the special case \(d=2\) by Górny in [44]).

Our result answers this question positively if \(\partial\Omega\) is \(C^2\) and has positive mean curvature (if the mean curvature is negative, there are of course smooth counterexamples). In particular, in this context, it is also quite easy to remove the assumption that \(u\in L^2\) in the theorem.

The (short) proof is given in Section 6.

Comments on the optimality of the assumptions

The size bound on \(g\): We already gave a (somewhat implicit) example of the sharpness of the size bound on \(g\) in [A3] for \(d\geq 3\) in the counterexample for the ROF functional in Subsection [sec:rofsec] above.

If one allows other integrands, then there are also easier counterexamples in one dimension, for instance one can consider \(\Omega=(0,1)\) and \(f(x,\xi)=a(x)|\xi|\) for \(a\in C^2\), then the generalized mean curvature at \(1\) is \(a'(1)\) and at \(0\) it is \(-a'(0)\). If both values are positive, then one can then check, using Lemma 3 with \(z=a(x)\) that \(u=0\) is a minimizer for \(g=-a'(x)\) and \(\lambda=0\) and boundary datum \(u_0(1)=1\) and \(u_0(0)=-1\).

The regularity of \(u_0\): Whether or not the spaces \(BV\) and \(W^{\alpha,p}\) (with \(\alpha p\geq 2\)) are optimal is not clear, the construction of a counterexample given by Spradlin and Tamasan in [19] seems to be limited to spaces \(W^{\alpha,1}\) with \(\alpha <\frac{2}{3}\).

It would be an interesting question whether or not the required regularity of the boundary datum improves for elliptic integrands (such as e.g.\(\sqrt{1+|\xi|^2}\)).

The structure of the lower order terms: There are plenty of works considering similar problems with different lower order terms, for instance, non-linear eigenvalue problems [45], saddle point problems [46] or parabolic versions [47]–[50].

Some of these fall under our framework by using that their Euler-Lagrange equations have the same structure as the functional here (see Lemma 5) if one considers \(g\) as the contribution of the additional nonlinearities and that fulfilling the Euler-Lagrange equation is equivalent to being a minimizer (cf.Lemma 5) (this e.g.works with the saddle points in [46] if the prefactor of the lower order nonlinearity is small enough). In full generality, results analogous to ours are simply not going to be true, e.g.for the Cheeger problem [51], there are easy examples (such as balls) where the situation seems to be completely different.

Regarding the parabolic setting, we expect that the technique here should work under similar assumptions and should produce a bound of the type \(\partial_t|u-u_0|\leq -H_{\partial\Omega,f}\) on \(\partial\Omega\) if \(u\neq u_0\).

Finally, we remark that of course not every assumption here has been optimized towards maximal generality, and we expect that some of them can be weakened (e.g.the regularity of \(h\)) without too much effort.

1.3 Organisation of the paper↩︎

In Section 2, we introduce the necessary preliminaries about function spaces and convex analysis. In Section 3, we recall the general machinery for linear growth functionals in the calculus of variations and state the characterisations of their subdifferentials, which the proofs of the Theorems 1 and 3 in the subsequent Section 4 use. Most of the proofs of the technical lemmata used there are postponed to Section 5. The short proof of Corollary 5 is given in Section 6. The counterexample in Theorem 4 is constructed in Section 7. Finally, the proof of a lemma from convex analysis, which we believe to be well known but could not find in a satisfying citable form, is given in the Appendix 8.

2 Preliminaries and notation↩︎

2.1 Notational conventions↩︎

We shall write \(A\gtrsim B\) if there is some constant \(C\), depending on \(\Omega,n,d, f\) or \(\kappa\), but not \(x\) or \(\epsilon\), such that \(A\geq CB\), sometimes, we shall also explicitly write the constant, in this case it is allowed to change its value from line to line. If we specifically wish to highlight the dependence of the implicit constant on a certain parameter, we shall denote that by a subscript, as in e.g.\(\gtrsim_{u_0}\). Sometimes we also write \(C(t)\) to highlight the dependence on \(t\) (or some other parameter). If such an inequality holds in both directions, we shall use the symbol \(\approx\) for that.

The divergences of matrices are always taken row-wise. \(S^{d-1}\) is the \(d-1\)-dimensional unit sphere. \(\mathbb{1}\) denotes indicator functions. \(\mathcal{L}^d\) and \(\mathcal{H}^d\) are the \(d\)-dimensional Lebesgue/Hausdorff measure. The spaces of smooth and compactly supported smooth test functions and the space of distributions are denoted by \(\mathcal{D}\) and \(\mathcal{D}'\). We denote the truncation by \[\begin{align} T_b(a)=\min(b,\max(a,-b)) \quad \text{for a\in \mathbb{R} and b\in \mathbb{R}_{\geq 0}}.\label{def32trunc} \end{align}\tag{17}\]

2.2 The space \(BV\)↩︎

Let \(O\subset \mathbb{R}^d\) be a bounded set with Lipschitz boundary. We denote the outer normal at \(x\in \partial O\) with \(\nu_x\) (sometimes also written as just \(\nu\)) and use the same notation for the normal on \(\partial\Omega\).

The space \(BV(O,\mathbb{R}^n)\) of functions of bounded variation is defined as the the space of functions \(w\in L^1(O,\mathbb{R}^n)\) for which the distributional derivative \(\mathrm{D}w\in \mathcal{D}'(O,\mathbb{R}^n)\) is represented by integration against a bounded, signed, \(\mathbb{R}^{n\times d}\)-valued Radon measure, which is also denoted by \(\mathrm{D} w\), i.e.the space of functions for which there is a measure \(\mathrm{D}w\) with \[\begin{align} -\int_O w\operatorname{div}\phi \,\mathrm{d}x=\int_O \phi \mathrm{d}\mathrm{D}w \quad \forall\phi\in \mathcal{D}(O). \end{align}\] The space \(BV(O,\mathbb{R}^n)\) is a Banach space when endowed with the norm \[\begin{align} \left\lVert w \right\rVert_{BV(O,\mathbb{R}^n)} := \left\lVert w \right\rVert_{L^1(O,\mathbb{R}^n)} + |\mathrm{D}w|(O), \end{align}\] where \(|\mathrm{D}w|\) denotes the total variation of \(\mathrm{D}w\).

We say \(w_m\rightarrow w\) strictly if \(w_m\rightarrow w\) in \(L^1(O,\mathbb{R}^n)\) and \(|\mathrm{D}w_m|(O)\rightarrow |\mathrm{D}w|(O)\). One can show that \(C^\infty(\overline{O})\) is dense in \(BV(O,\mathbb{R}^n)\) with respect to strict convergence.

We shall write \(\mathrm{D}^a w\) and \(\mathrm{D}^sw\) for the absolutely continuous and singular parts of the measure, i.e.\[\begin{align} \mathrm{D}^a w(x)=\frac{\mathrm{d}\mathrm{D}^a w}{\mathrm{d}\mathcal{L}^d}(x) \quad \text{and}\quad \mathrm{D}^s w=\mathrm{D}w-\mathrm{D}^a w\mathcal{L}^d. \end{align}\] It can be shown that there is a linear, bounded trace \(w\rightarrow w|_{\partial O}\in L^1(O,\mathbb{R}^n)\) on \(BV(O,\mathbb{R}^n)\), which is continuous under strict convergence and agrees with the classical trace on \(C^1\). We will omit the “\(|_{\partial O}\)” for the rest of the paper.

Furthermore, one has the same Poincaré and Sobolev inequalities as for \(W^{1,1}\), i.e.for every \(q\in [1,\frac{d}{d-1}]\), there is a \(C(O,q)>0\), such that it holds that \[\begin{align} \left\lVert w \right\rVert_{L^q(O)}\leq C(O,q)\left(|\mathrm{D}w|(O)+\int_{\partial O}|w|\,\mathrm{d}\mathcal{H}^{d-1}\right)\quad \forall w\in BV(O),\label{poincare} \end{align}\tag{18}\] furthermore \(BV(O)\) embeds into \(L^q(O)\) and the embedding is compact for \(q\in [1,\frac{d}{d-1})\).

Whenever \(\mathfrak{g}:\mathbb{R}^n\rightarrow \mathbb{R}^m\) is Lipschitz and \(w\in BV(O,\mathbb{R}^n)\), then \(\mathfrak{g}\circ w\in BV(O,\mathbb{R}^m)\). A particularly important case for us is that if \(n=m=1\), then it holds that \(\max(a,w)\in BV(O)\) for any \(w\in BV(O)\) and \(a\in \mathbb{R}\), in particular it also holds that \[\begin{align} |\mathrm{D}\max(a,w)|(\mathcal{U})\leq|\mathrm{D}w|(\mathcal{U})\label{measure32trunc} \end{align}\tag{19}\] for every measurable \(\mathcal{U}\subset O\) and of course this also holds for minima instead of maxima and any combination of the two (such as \(T_b\)).

If \(M\subset \mathbb{R}^{d}\) is a compact embedded \(C^1\)-manifold of dimension \(d'<d\) (the only relevant case for us are subsets of \(\partial\Omega\)), then we define \(BV(M,\mathbb{R}^n)\) by first defining the norm \[\begin{align} \left\lVert w \right\rVert_{BV(M,\mathbb{R}^n)}:=\left\lVert w \right\rVert_{L^1(M,\mathbb{R}^n)}+\left\lVert \nabla_\tau w \right\rVert_{L^1(M,\mathbb{R}^{n\times d'})}, \end{align}\] first for \(w\in C^1(M,\mathbb{R}^n)\) where \(\nabla_\tau\) is the tangential gradient, taken with respect to any orthonormal basis of the tangent space. We then define the norm for \(w\notin C^1(M,\mathbb{R}^n)\) by \[\begin{align} \left\lVert w \right\rVert_{BV(M,\mathbb{R}^n)}=\inf_{C^1\ni w_m\rightarrow w\in L^1}\left\lVert w_m \right\rVert_{BV(M,\mathbb{R}^n)}. \end{align}\] It can be shown that \(\nabla_\tau w_m\) converges weakly^\(*\) to a measure which is independent of the minimizing sequence [52]. This definition is also equivalent to using charts or distributional derivatives.

The reader may e.g.consult the textbook [53] for further reading.

The Anzellotti Pairing In order to develop a satisfying theory for the Euler-Lagrange equation, it is necessary to have some notion of product between \(\mathrm{D}w\) and discontinuous functions, which is done via partial integration. We refer to Anzelotti’s original paper [54] as a source⁵. We define \[\begin{align} X_p(O,\mathbb{R}^{n\times d}):=\big\{z\in L^\infty(O,\mathbb{R}^{n\times d})\,\big|\, \operatorname{div}z\in L^p(O,\mathbb{R}^n)\big\}. \end{align}\] Here, the divergence is the distributional one and taken row-wise. One can then define a distribution \((z,\mathrm{D}w)\) for \(z\in X_p(O,\mathbb{R}^{n\times d})\) and \(w\in (BV\cap L^q)(O,\mathbb{R}^n)\) with \(\frac{1}{p}+\frac{1}{q}=1\) by \[\begin{align} (z,\mathrm{D}w)(\phi)= - \int_O \phi w\cdot\operatorname{div}(z)\,\mathrm{d}x- \int_O w \otimes\nabla \phi :z\,\mathrm{d}x\quad \text{for all } \phi \in \mathcal{D}(O). \end{align}\] It can be shown that this defines a bounded Radon measure with \(|(z,\mathrm{D}w)|<<|\mathrm{D}w|\) and that it holds that \[\begin{align} |(z,\mathrm{D}w)|(\mathcal{U})\leq \left\lVert z \right\rVert_{L^\infty}|\mathrm{D}w|(\mathcal{U})\label{anz32bas32est} \end{align}\tag{20}\] for every measureable \(\mathcal{U}\subset O\). It further generalizes the pointwise product in the sense that \[\begin{align} \frac{\mathrm{d}(z,\mathrm{D}w)}{\mathrm{d}\mathcal{L}^d}=\langle z,\frac{\mathrm{d}\mathrm{D}^aw}{\mathrm{d}\mathcal{L}^d} \rangle\quad \text{\mathcal{L}^d-a.e.}, \end{align}\] in particular, for \(w\in W^{1,1}(O,\mathbb{R}^n)\) it is the pointwise product. Regarding the singular part of the measure, one can show that if \(z(x)\in \mathcal{C}\) for some closed convex set \(\mathcal{C}\) in some open \(\mathcal{O}\subset O\), then it holds that \[\begin{align} \frac{\mathrm{d}(z,\mathrm{D}w)}{\mathrm{d}|\mathrm{D}^sw|}(x)\in \mathcal{C}\cdot \frac{\mathrm{d}\mathrm{D}^sw}{\mathrm{d}|\mathrm{D}^sw|}(x)\quad \text{for |\mathrm{D}^sw|-a.e.\;x\in \mathcal{O}}.\label{anz32rep} \end{align}\tag{21}\] In fact, more precise expressions for the density as suitable weak limits exist [39], [55].

Furthermore \(z\in X_p(O,\mathbb{R}^{n\times d})\) has a normal trace \([z,\nu]\in L^\infty(\partial O,\mathbb{R}^n)\) on \(\partial O\), which is linear in \(z\) and fulfills \(\left\lVert [z,\nu] \right\rVert_{L^\infty(\partial O,\mathbb{R}^n)}\leq \left\lVert z \right\rVert_{L^\infty(O,\mathbb{R}^{n\times d})}\). For \(z\in C^1\) it agrees a.e.with the pointwise normal trace, \(z\nu_x=\sum_{i=1}^d z_{mi}(x)\nu_x^i\), where \(\nu_x^i\) are the components of \(\nu_x\). Similarly to 21 , we have that whenever \(z(x)\in \mathcal{C}\) for some closed convex set \(\mathcal{C}\) in some open \(\mathcal{O}\subset O\), then it holds that \[\begin{align} [z,\nu](x)\in \mathcal{C} \nu_x\quad \text{for \mathcal{H}^{d-1}-a.e.\;x\in \partial(\mathcal{O}\cap O)}.\label{nt32rep} \end{align}\tag{22}\] Furthermore, one has the following Gauss-Green type formula for \(w\in (BV\cap L^q)(O,\mathbb{R}^n)\) (for \(\frac{1}{p}+\frac{1}{q}=1\)): \[\begin{align} \int_O \langle w,\operatorname{div}z \rangle\,\mathrm{d}y+\int_O (z,\mathrm{D}w)=\int_{\partial O}\langle w,[z,\nu] \rangle\,\mathrm{d}\mathcal{H}^{d-1}.\label{gg32form} \end{align}\tag{23}\] We will also use the following chain rule-type estimate for the pairing.

2.3 Elements of convex analysis↩︎

Let \(Y\) be a (real) separable Hilbert space, and let \(\mathcal{G}:Y\rightarrow \mathbb{R}\cup\{+\infty\}\) be convex and lower semicontinuous (with the usual conventions for \(+\infty\)), then we say that \(v\in Y\) is in the subdifferential of \(\mathcal{G}\) at \(w\in Y\), written \(v\in \partial\mathcal{G}(w)\), if \(\mathcal{G}(w)<\infty\) and \[\begin{align} \langle w'-w,v \rangle\leq \mathcal{G}(w')-\mathcal{G}(w) \quad \forall w'\in Y. \end{align}\] If \(\mathcal{G}\) is Gateaux-differentiable at \(w\), then \(\partial\mathcal{G}(w)=\{\mathrm{D}\mathcal{G}(w)\}\).

One can show that if \(Y\) is finite-dimensional and \(\mathcal{G}\) is finite everywhere, then \(\partial\mathcal{G}(w)\) is nonempty for all \(w\). We also note that some \(w\in Y\) is a minimizer of \(\mathcal{G}\) if and only if \(0\in \partial\mathcal{G}(w)\).

We set \[\begin{align} \mathop{\mathrm{Ran}}(\partial\mathcal{G})=\{v\,\big|\, v\in \partial\mathcal{G}(w)\text{ for some w\in Y}\}. \end{align}\] For functions also depending on other parameters, we denote the dependence on the parameter by a subscript, e.g.for the subdifferential of \(f(x,\xi)\) with respect to \(\xi\), we write \(\partial_\xi f(x,\xi)\).

We now consider integrands \(\bar{f}:\overline{\Omega}\times \mathbb{R}^{n\times d}\) which are convex in the second variable, continuous in the joint variable, and fulfill a growth bound of the form \[\begin{align} C_3^{-1}|\xi|-C_3\leq \bar{f}(x,\xi)\leq C_3(|\xi|-1),\label{gb32barf} \end{align}\tag{24}\] with some \(C_3>0\) not depending on \(x\) or \(\xi\).

We define the recession function \(\bar{f}^\infty\) as \[\begin{align} \bar{f}^\infty(x,\xi)=\lim_{x'\rightarrow x,\, t\rightarrow +\infty}\frac{1}{t}\bar{f}(x,t\xi), \end{align}\] and it is easy to show that, if this limit exists, then 24 implies \[\begin{align} C_3^{-1}|\xi|\leq\bar{f}^\infty(x,\xi)\leq C_3|\xi|\label{gb32rec} \end{align}\tag{25}\] with the same constant \(C_3\).

In particular, if [A2] resp.[B2] hold, then it holds that \[\begin{align} C_1^{-1}|\xi|\leq f^\infty(x,\xi)\leq C_1|\xi|,\label{gc32finf} \end{align}\tag{26}\] with the same \(C_1\) as in these assumptions.

The recession function is positively \(1\)-homogeneous in the second variable, and if \(\bar{f}\) is positively \(1\)-homogeneous in the second variable, then \(\bar{f}^\infty=\bar{f}\).

Regarding the subdifferentials of \(\bar{f}\) and \(\bar{f}^\infty\), we have the following relation \[\begin{align} \overline{\mathop{\mathrm{Ran}}{\partial_\xi\bar{f}(x,\cdot)}}=\mathop{\mathrm{Ran}}{\partial_\xi\bar{f}^\infty(x,\cdot)}\subset \overline{B_{C_3}(0)}.\label{ball32subdiff} \end{align}\tag{27}\] This also implies Lipschitz continuity of \(\bar{f}\), more precisely \[\begin{align} |\bar{f}(x,\xi_1)-\bar{f}(x,\xi_2)|\leq C_3|\xi_1-\xi_2|\label{lipsch} \end{align}\tag{28}\] with the same \(C_3\).

The Legendre transform of \(\bar{f}\) is defined as \[\begin{align} \bar{f}^*(x,\xi^*)=\sup_{\xi\in \mathbb{R}^{n\times d}}\langle \xi^*,\xi \rangle-\bar{f}(x,\xi). \end{align}\] It can be shown that \(\bar{f}^*(x,\cdot)=+\infty\) outside of \(\overline{\mathop{\mathrm{Ran}}{\partial_\xi\bar{f}(x,\cdot)}}\) and that \[\begin{align} \bar{f}(x,\xi)=\sup_{\xi^*\in \mathbb{R}^{n\times d}}\langle \xi^*,\xi \rangle-\bar{f}^*(x,\xi^*), \end{align}\] where the supremum is attained at \(\xi^*\) if and only if \(\xi^*\in \partial_\xi f(x,\xi)\).

If \(\bar{f}^\infty\) is differentiable in \(\xi\) at \((x,\xi)\), we furthermore have \[\begin{align} \bar{f}^\infty(x,\xi)=\sup_{\xi^*\in\mathop{\mathrm{Ran}}(\partial_\xi\bar{f}^\infty(x,\cdot))} \langle \xi,\xi^* \rangle=\langle \xi,\mathrm{D}_\xi \bar{f}^\infty(x,\xi) \rangle,\label{doal} \end{align}\tag{29}\] and it holds that \(\partial_\xi\bar{f}^\infty(x,0)=\mathop{\mathrm{Ran}}(\partial_\xi\bar{f}^\infty(x,\cdot))\). We also note that our assumptions imply a quantitative version of this for \(f^\infty\):

We believe this Lemma to be quite well-known, but were not able to find this exact version in the literature; therefore, a proof can be found in the Appendix 8.

We also note for further reference that, if [A2] resp.[B2] hold, then \[\begin{align} \lim_{t\rightarrow +\infty} \mathrm{D}_\xi f(x,t\xi)\cdot\frac{\xi}{|\xi|}=\mathrm{D}_\xi f^\infty(x,\xi)\cdot\frac{\xi}{|\xi|}=f^\infty(x,\frac{\xi}{|\xi|})\qquad \forall \xi\neq 0.\label{conv32grad} \end{align}\tag{30}\] Indeed this follows from the fact that the expression on the left-hand side is bounded from below by \(\frac{1}{2t|\xi|}(f(x,t\xi)-f(x,\frac{1}{2}t\xi))\) and from above by \(\frac{1}{t|\xi|}(f(x,2t\xi)-f(x,t\xi))\) by convexity, which both converge to \(f^\infty(x,\frac{\xi}{|\xi|})\) by definition.

The reader may refer e.g.to the books [56]–[58] for further background reading.

2.4 Fractional Sobolev spaces↩︎

Let \(\mathcal{U}\subset \partial\Omega\) be an open subset of the boundary which is \(C^1\) and whose (relative) boundary is \(C^1\), let \(\alpha\in (0,1)\) and \(p\in (1,\infty)\), then we define \[\begin{align} \left\lVert w \right\rVert_{W^{\alpha,p}(\mathcal{U})}:=\left\lVert w \right\rVert_{L^p(\mathcal{U})}+\left(\int_{\mathcal{U}^2} \frac{|w(x)-w(y)|^p}{|x-y|^{d-1+\alpha p}}\,\mathrm{d}x\,\mathrm{d}y\right)^\frac{1}{p} \end{align}\] and define the space \(W^{\alpha,p}(\mathcal{U})\) as the subset of \(L^p(\mathcal{U})\) for which this norm is finite. This definition is equivalent to defining the space via a diffeomorphism to a subset of the \(\mathbb{R}^{d-1}\) and using the same norm there. We refer e.g.to the monograph [59] for background reading.

The main property which we will require is the following one: If \(w\in W^{\alpha,p}(\mathcal{U})\), then there are functions \(r_\delta\in L^p(\mathcal{U})\), such that for every a.e.\(x,y\in \mathcal{U}\) with \(|x-y|\leq \delta\) it holds that \[\begin{align} |w(x)-w(y)|\leq |x-y|^\alpha(r_\delta(x)+r_\delta(y))\label{cald32grad} \end{align}\tag{31}\] and that furthermore \[\begin{align} \lim_{\delta\searrow 0} \left\lVert r_\delta \right\rVert_{L^p(\mathcal{U})}=0.\label{r32lim} \end{align}\tag{32}\] A proof of this can be found e.g.in [60], where it is shown for open subsets of the \(\mathbb{R}^d\), for subsets of \(\partial\Omega\) it follows by parametrizing the boundary.

Let us also stress that we never use any kind of fractional derivative here, and \(\mathrm{D}^a\) and \(\mathrm{D}^s\) never mean anything other than the absolutely continuous and singular part of the gradient.

3 Existence of minimizers and characterisation of the subdifferential↩︎

In this section, we will recall some of the classical existence theory for variational problems with linear growth and characterisations of their subdifferential. Most of this is standard and well known to the expert, we nevertheless provide some of the proofs to show how to deal with the lower-order terms.

For the most part, we will treat the scalar and vector-valued setting simultaneously here, and statements where neither of the two is specified apply to both. In particular, we consider \(f\) as a function on \(\overline{\Omega}\times \mathbb{R}^{n \times d}\) here of which we assume that either \(n=1\) or that it is independent of the first variable.

The expression \(\int f(x,\mathrm{D}w)\,\mathrm{d}x\) is not defined in a classical sense if \(w\notin W^{1,1}\). Instead one defines \[\begin{align} \mathcal{F}_{u_0}(w):=&\int_{\Omega} f(x,\mathrm{D}^aw(x))\,\mathrm{d}x+\int_{\Omega}f^\infty(x,\frac{\mathrm{d}\mathrm{D}^sw}{|\mathrm{d}\mathrm{D}^sw|}(x))\,\mathrm{d}|\mathrm{D}^sw|(x)\label{def32fu0}\\ &+\int_{\partial\Omega} f^\infty(x,(u_0-w)(x)\otimes \nu_x)\,\mathrm{d}\mathcal{H}^{d-1}(x) \end{align}\tag{33}\] for \(u_0\in L^1(\partial\Omega,\mathbb{R}^n)\) and \(w\in BV(\Omega,\mathbb{R}^n)\). Clearly, this functional is \(\int_\Omega f(x,\mathrm{D}w)\,\mathrm{d}x\) when restricted to \(W_{u_0}^{1,1}(\Omega,\mathbb{R}^n)\).

This functional is coercive on \(BV(\Omega,\mathbb{R}^n)\) in the sense that \[\begin{align} \mathcal{F}_{u_0}(w)\geq C_1^{-1}\left(|\mathrm{D}w|(\Omega)+\int_{\partial\Omega}|w|\,\mathrm{d}\mathcal{H}^{d-1}\right)-C_1(\mathcal{L}^{d}(\Omega)+\left\lVert u_0 \right\rVert_{L^1(\partial\Omega)})\label{f32coer} \end{align}\tag{34}\] with the constant \(C_1\) from the assumptions 6 resp.@eq:bd32gf2 , as one directly sees from using 25 and the triangle inequality.

This is a classical result from [62], except for the point about \(L^q\)/\(L^\infty\)-convergence in b), which one can achieve with the standard construction, but which regrettably does not seem to be stated anywhere in the literature. We therefore give a short sketch here of why the construction in [1] still works with \(L^q\)-convergence.

This implies the following two Propositions:

3.1 Characterisation of the subdifferential↩︎

The functional \(\mathcal{F}_{u_0}\) is in general not differentiable (e.g.when is \(f\) not differentiable), instead the natural object to characterise minimizers is the subdifferential as defined in Subsection 2.3.

To define \(\mathcal{F}_{u_0}\) on a Hilbert space, we will consider \(\mathcal{F}_{u_0}\) as a functional on \(L^2(\Omega,\mathbb{R}^n)\) by extending it as \(+\infty\) to \(L^2(\Omega,\mathbb{R}^n)\backslash BV(\Omega,\mathbb{R}^n)\), which is easily seen to be convex and lower semicontinuous, as a consequence of the coercivity bound 34 . By an abuse of notation, we still denote this by \(\mathcal{F}_{u_0}\). Furthermore, the infimum of the functional remains the same thanks to part b) of Proposition 6 above.

We remark that, thanks to the uniform continuity of \(\mathrm{D}_\xi f^\infty\) and the formula 22 , for such a \(z\) it also holds that \[\begin{align} [z,\nu_x](x)\in \big\{\langle z_0,\nu_x \rangle\,\big|\, z_0\in \overline{\mathop{\mathrm{Ran}}(\partial_\xi f(x,\cdot))}\big\}\label{z32tr32pw} \end{align}\tag{36}\] pointwise a.e.on \(\partial\Omega\) in both the scalar and the vector setting (where one interprets the product as a matrix-vector product in the vectorial setting).

Thanks to ?? , we also see that ?? (resp.@eq:cv4 ) imply that \[\begin{align} [z,\nu_x]=\mathrm{D}_\xi f^\infty(x,(u_0-w)(x)\otimes \nu_x)\nu_x \quad \text{ if w(x)\neq u_0(x), up to a \mathcal{H}^{d-1}-zero set}\label{good32bd32cond} \end{align}\tag{37}\] where the product on the right is understood as a matrix-vector product and the gradient is understood as a \(n\times d\) matrix.

For future reference, let us also note that ?? resp.@eq:cv4 together with 27 also imply that \[\begin{align} \left\lVert z \right\rVert_{L^\infty(\Omega,\mathbb{R}^{n\times d})}\leq C_1\lesssim 1 \label{uni32bd32z} \end{align}\tag{38}\] with the \(C_1\) from [A2] resp.[B2].

There are many versions and special cases of this characterisation in the literature, the closest probably being [49], which additionally assumes that \(f^\infty\) is even and \(n=1\) and [50], which considers \(f\) which are rough in \(x\) and only depend on a part of the gradient, which leads to the result being written in rather implicit way.

We can now formulate a suitable form of the Euler-Lagrange equations for 4 and 9 .

4 Proof of the main theorems 1 and 3↩︎

4.1 A few words about the strategy↩︎

Unlike other works, our approach is not based on barriers. Instead, we will transform \(\Omega\) to a ball in the scalar case and use the subdifferential characterisation to show integral estimates in small spherical caps (called \(D_x^\epsilon\) below) for \(u-\hat{u}_x\), where \(\hat{u}_x\) is a suitable extension of \(u_0\). The most important point in the \(BV\)-case is that if the boundary condition 37 for \(z\) holds on most of some spherical cap \(C_x^\epsilon=\partial\Omega\cap \partial D_x^\epsilon\), then, as a consequence of the uniform convexity of the set \(\mathop{\mathrm{Ran}}(\partial_\xi f^\infty)\) of allowed values for \(z\), one actually controls \(z\) in \(D_x^\epsilon\) too as shown in Lemma 11, which allows for sufficiently strong estimates.

The fact that \(u_0\in BV\) enters through estimates on the gradient for the extension \(\hat{u}_x\) (see Lemma 9), the crucial point about it is that we can keep control of the modulus of integrability of the absolutely continuous part of the gradient of the extension (see Lemma 10).

The proof then proceeds by using the Morse covering theorem to cover the bad set \(\{u\neq u_0\}\) with caps \(C_x^\epsilon\) for which one has this control over \(z\) and on which \(|\mathrm{D}^su_0|\) is small, and recovering the trace from the weighted integrals over the \(D_x^\epsilon\), as these become infinitesimally close to the boundary in the limit.

In the \(W^{\alpha,p}\)-case the basic scheme is the same, though instead of an exact extension of \(u_0\), we use local averages, which makes a few gradients disappear, but comes at the cost of having to control how well \(u_0\) is approximated by its local averages, which is the part that needs the fractional Sobolev regularity.

The \(C^0\) case is easier and mostly the \(p\rightarrow \infty\)-limit of the estimates in the \(W^{\alpha,p}\)-case.

The vectorial case \(n\geq 1\) uses the same scheme as the \(BV\)-case, where there are extra limitations on the structure of \(f\) coming from the fact that splitting into the positive and negative parts, as it is done in the scalar case, does not work well anymore, and one hence has to work in a way that handles both parts at once. In particular, we work with the original \(\Omega\), which is the part that requires uniform convexity of the boundary.

4.2 Preparations for the proof of Theorem 1↩︎

We will assume in the entire proof that the Assumptions [A1]-[A6] hold and all of the lemmata use some of them, though we will not explicitly state this.

Transformation of the domain

It will be much easier to carry out the proof if \(U=\partial\Omega\) is a subset of a sphere. We will therefore use a diffeomorphism and show that the assumptions are stable under this, as the following Lemma shows. Furthermore, if this sphere has very small curvature, we can also obtain a more useful version of the curvature condition.

The lemma is proven in Section 5.

Now it suffices to show that \(u=u_0\) in every such \(U'\) as in b). By the Lemma, it is not restrictive to assume that every point in b) was already true for \(U'\) and that \(\Phi\) is the identity (otherwise we use the diffeomorphism which the lemma yields, rename everything, and use that by a) all the assumptions remain true). We also consider \(\kappa\) to be fixed of order \(1\) and allow the implicit constants to depend on it.

Since the statement of the theorem is local, it is not restrictive to only show that \(u-u_0\) vanishes on \(U''\) for every (relatively) open set \(U''\subset U'\) with positive distance to the boundary of \(U'\). It is also not restrictive to assume that both \(U''\) and \(U'\) have a sufficiently regular (relative) boundary.

Let \(x\in U''\) and \(\epsilon>0\), we set \[\begin{align} &C_x^\epsilon=B_\epsilon(x)\cap\partial\Omega\label {def C1}\\ &D_x^\epsilon=\left\{y\in \Omega\cap B_\epsilon(x)\,\big|\,\langle y-x,\nu_x \rangle\geq-\frac{1}{2}\kappa\epsilon^2\right\}\tag{42}\\ &E_x^\epsilon=\partial D_x^\epsilon\backslash C_x^\epsilon=\left\{y\in \overline{\Omega}\cap \overline{B_\epsilon(x)}\,\big|\,\langle y-x,\nu_x \rangle=-\frac{1}{2}\kappa\epsilon^2\right\}.\tag{43} \end{align}\] We only consider \(\epsilon\) such that \(\epsilon<\mathop{\mathrm{dist}}(U'',\partial\Omega\backslash U')\), so that \(C_x^\epsilon\subset U'\) and there is no intersection between \(D_x^\epsilon\) and \(\partial\Omega\backslash U'\). We furthermore also only consider \(\epsilon\) smaller than the \(\epsilon_0\) in part b) of Lemma 7, so that the statement of the Lemma is applicable in \(D_x^\epsilon\) with a \(c_0\) uniform in \(x\in U''\) and small \(\epsilon\).

An easy calculation shows that \(E_x^\epsilon\) is a piece of a plane with normal \(\nu_x\) and that the boundary of the spherical cap \(D_x^\epsilon\) is precisely \(E_x^\epsilon\cup C_x^\epsilon\). See also Figure 1 for a sketch.

We furthermore have the following elementary geometric properties:

The Lemma is proven in Section 5.2.

4.3 Proof of Theorem 1 in the case \(u_0\in BV\)↩︎

We will only show that \(u\leq u_0\) holds \(\mathcal{H}^{d-1}\)-a.e.on \(U''\), the other inequality follows in the same way by appropriately changing the signs.

We will use a comparision vector field \(\hat{z}_{x}\), which we define as \[\begin{align} \hat{z}_{x}(y)=\mathrm{D}_\xi f^\infty(y,- \nu_x).\label{def32hatz} \end{align}\tag{44}\] It follows from the Assumption [A2] that \[\hat{z}_{x}\in C^1(\Omega)\subset X_2(\Omega)\label{reg32hatz}\tag{45}\] and by 27 , we also have \[\begin{align} \left\lVert \hat{z}_{x} \right\rVert_{L^\infty(C_x^\epsilon)}\leq C_1\lesssim1.\label{bound32hatz} \end{align}\tag{46}\] We will then define the function \(\hat{u}_{x}\) as the extension of \(u_0\) along the characteristic curves of \(\hat{z}_{x}\), more precisely:

The Lemma is proven in Section 5.3.

Crucially, we can also lift de-La-Vallée-Poussin-type estimates for the absolutely continuous part of \(\mathrm{D}u_0\) to the absolutely continuous part of \(\mathrm{D}\hat{u}_{x}\).

This lemma is also proven in Section 5.3.

We now take \(z\in X_2(\Omega,\mathbb{R}^d)\) such that the conditions ?? ?? in the characterisation of the subdifferential hold with \(v=-\lambda(u-h)-g\) and \(w=u\), this is possible because of the characterisation of minimizers in the Lemmata 3 and 5.

We use the truncations \(T_b\), defined in 17 , where \(b\in \mathbb{R}_{\geq 0}\) is some arbitrary number. The space \(BV\) is closed under the application of \(T_b\), by e.g.the chain rule as described in Section 2.2. Therefore, we can use the Gauss-Green formula 23 for the domain \(D_x^\epsilon\) and \((T_b(u)-T_b(\hat{u}_{x}))_{+}\), where we denote the positive part by \(a_+=\max(a,0)\), as well as \((z-\hat{z}_{x})\), to the effect that \[\begin{align}\label{part32int} \MoveEqLeft\int_{C_x^\epsilon} [z-\hat{z}_{x},\nu_y](T_b(u)-T_b(u_0))_{+}\,\mathrm{d}\mathcal{H}^{d-1}(y)+\int_{E_x^\epsilon} [z-\hat{z}_{x},-\nu_x](T_b(u)-T_b(\hat{u}_{x}))_{+}\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &=\int_{D_x^\epsilon}(\lambda (u-h) +g-\operatorname{div}\hat{z}_{x}) (T_b(u)-T_b(\hat{u}_{x}))_+\,\mathrm{d}y+\int_{D_x^\epsilon}(z-\hat{z},\mathrm{D}(T_b(u)-T_b(\hat{u}_{x}))_{+}), \end{align}\tag{47}\] where we have used that the outer normal on \(E_x^\epsilon\) is \(-\nu_x\). We claim that the left-hand side is non-positive: Thanks to ?? , we have \[\begin{align} [z,\nu_y]=-f^\infty(y,-\nu_y)\quad \text{on C_x^\epsilon} \label{znu321} \end{align}\tag{48}\] whenever \(u-u_0>0\) at \(y\), due to the positive \(1\)-homogeneity of \(f^\infty\). By the regularity and definition of \(\hat{z}_{x}\) and 29 , we have \[\begin{align} [\hat{z}_{x},\nu_y] =\langle \mathrm{D}_\xi f^\infty(y,-\nu_x),\nu_y \rangle\geq -f^\infty(y,-\nu_y).\label{znu322} \end{align}\tag{49}\] Since the integral over \(C_x^\epsilon\) is automatically zero at the points where \(u\leq u_0\), this shows that \[\begin{align} \int_{C_x^\epsilon} [z-\hat{z}_{x},\nu_y](T_b(u)-T_b(u_0))_{+}\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq 0. \end{align}\] Regarding the other integral, we have on \(E_x^\epsilon\) \[\begin{align} [\hat{z}_x,-\nu_x](y)=\langle \mathrm{D}_\xi f^\infty(y,-\nu_x),-\nu_x \rangle= f^\infty(y,-\nu_x), \end{align}\] by 29 as well as the regularity and definition of \(\hat{z}_{x}\). Since we also have \[\begin{align} [z,-\nu_x]\in \left\{z_0\cdot(- \nu_x)\,\big|\, z_0\in \mathop{\mathrm{Ran}}(\partial_\xi f^\infty(y,\cdot)\right\} \end{align}\] thanks to 21 , ?? and the uniform continuity of \(\mathrm{D}f^\infty\), we see from 29 that \[\begin{align} [z-\hat{z}_{x},-\nu_x]\leq 0, \end{align}\] yielding \[\begin{align} \int_{E_x^\epsilon} [z-\hat{z}_{x},-\nu_x](T_b(u)-T_b(\hat{u}_{x}))_{+}\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq 0. \end{align}\] We hence obtain that \[\begin{align} \label{part32int2} \int_{D_x^\epsilon}(\lambda (u-h) +g-\operatorname{div}\hat{z}_{x}) (T_b(u)-T_b(\hat{u}_{x}))_+\,\mathrm{d}y\leq -\int_{D_x^\epsilon}(z-\hat{z},\mathrm{D}(T_b(u)-T_b(\hat{u}_{x}))_{+}). \end{align}\tag{50}\] We can further estimate the right-hand side, using Lemma 1 and the triangle inequality as \[\begin{align} \label{anz32tr} \MoveEqLeft-\int_{D_x^\epsilon}(z-\hat{z},\mathrm{D}(T_b(u)-T_b(\hat{u}_{x}))_{+})\leq \int_{D_x^\epsilon}|(z-\hat{z},\mathrm{D}T_b(\hat{u}_{x}))|+|(z-\hat{z},\mathrm{D}u)_-|, \end{align}\tag{51}\] where the lower “\(-\)” denotes the negative part of the measure.

We would like to show that the right-hand side of 50 becomes suitably small if \(z\) equals \(\mathrm{D}_\xi f^\infty(y,- \nu_y)\) on most of \(C_x^\epsilon\), which will follow from the next Lemma.

The lemma is proven in Sections 5.4 and 5.5.

We fix an arbitrary (small) \(\delta>0\) now. By Radon-Nikodym [64]⁶, applied to the indicator function \(\mathbb{1}_{u>u_0}\) and the measure \(|\mathrm{D}^su_0|\) with respect to the measure \(\mathcal{H}^{d-1}\) on \(U'\), we know that for \(\mathcal{H}^{d-1}\)-a.e.\(x\in U''\cap \{u> u_0\}\) there exist an \(\epsilon_{0,x}>0\), which can be chosen arbitrarily small, such that for all \(\epsilon\leq \epsilon_{0,x}\) we have \[\begin{align} &\mathcal{H}^{d-1}(\{y\in C_x^{\epsilon}\,\big|\, u(y)\leq u_0(y)\})\leq \delta\mathcal{H}^{d-1}(C_x^{\epsilon})\tag{52}\\ &|\mathrm{D}^su_0|(C_x^{\epsilon})\leq \delta\mathcal{H}^{d-1}(C_x^{\epsilon}).\tag{53} \end{align}\] In particular, by the Morse covering theorem [64]⁶, there exist a (countable) family of \(C_{x_j}^{\epsilon_j}\) (which depends on \(\delta\)), such that \[\begin{align} \mathcal{H}^{d-1}\left((U''\cap \{u>u_0\})\backslash \bigcup_j C_{x_j}^{\epsilon_j}\right)=0 \end{align}\] and such that 52 and 53 hold for each of these spherical caps and such that the elements of this family are pairwise disjoint. Furthermore, we may pick \(\max_j \epsilon_j\) arbitrarily small.

By the boundary condition ?? , and its equivalent form 37 , we see that 52 implies ?? , and in particular, on each such element of the covering, the assertion of the conditional statement of Lemma 11 a) holds.

For further reference, we note that, as a consequence of the disjointness and of Lemma 8, it holds that \[\begin{align} \sum_j \epsilon_j^{d-1}\approx\sum_j \mathcal{H}^{d-1}(C_{x_j}^{\epsilon_j}) \leq \mathcal{H}^{d-1}(U')\lesssim 1.\label{sum32eps} \end{align}\tag{54}\] Now if we divide by \(\epsilon_j^2\), and sum 50 together with 51 over the covering, it holds that

\[\begin{align}\label{eff32est1} \MoveEqLeft\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}(\lambda(u-h) +g-\operatorname{div}\hat{z}_{x_j}) (T_b(u)-T_b(\hat{u}_{x_j}))_+\,\mathrm{d}y\\ &\leq \sum_j\epsilon_j^{-2}\left( \int_{D_{x_j}^{\epsilon_j}} |(z-\hat{z}_{x_j},\mathrm{D}T_b(\hat{u}_{x_j}))|+\int_{D_{x_j}^{\epsilon_j}} |(z-\hat{z}_{x_j},\mathrm{D}u)_-|\right). \end{align}\tag{55}\] The rest of the proof in this case now consists of proving the following two claims.

We start with proving the first claim. It holds that \[\begin{align} \MoveEqLeft\sum_j\epsilon_j^{-2}\left( \int_{D_{x_j}^{\epsilon_j}} |(z-\hat{z}_{x_j},\mathrm{D}T_b(\hat{u}_{x_j}))|+\int_{D_{x_j}^{\epsilon_j}} |(z-\hat{z}_{x_j},\mathrm{D}u)_-|\right)\nonumber\\ &\lesssim\sum_j\epsilon_{j}^{-2}\bigg(\int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|(y)|\mathrm{D}^aT_b(\hat{u}_{x_j})|(y)\,\mathrm{d}y+\left(\left\lVert z \right\rVert_{L^\infty}+\left\lVert \hat{z}_{x_j} \right\rVert_{L^\infty}\right)|\mathrm{D}^sT_b(\hat{u}_{x_j})|(D_{x_j}^{\epsilon_j})\nonumber\\ &\quad+\int_{D_{x_j}^{\epsilon_j}}k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\bigg)\nonumber\\ &\lesssim\sum_j\epsilon_{j}^{-2}\left(\int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|(y)|\mathrm{D}^a\hat{u}_{x_j}|(y)\,\mathrm{d}y+|\mathrm{D}^s\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j})+\int_{D_{x_j}^{\epsilon_j}}k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\right),\label{c1} \end{align}\tag{56}\] where in the first step we have split the first Anzelotti pairing into the absolutely continuous and singular part, which we estimated with its total variation and we have used ?? for the second pairing, while the last step follows directly from 19 , as well as the bounds 46 and 38 on \(z\) and \(\hat{z}_{x}\).

It follows directly from ?? , 53 and 54 that \[\begin{align} \sum_j\epsilon_{j}^{-2}|\mathrm{D}^s\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j})\lesssim \delta \sum_j \mathcal{H}^{d-1}(C_{x_j}^{\epsilon_j})\lesssim \delta\rightarrow 0. \label{p1} \end{align}\tag{57}\] Regarding the last summand in 56 , we split into the parts where \(|z-\hat{z}_{x_j}|\geq \sigma\) and where \(|z-\hat{z}_{x_j}|\leq \sigma\) for some real \(\sigma\) to be chosen later, to see that \[\begin{align} \MoveEqLeft\sum_j\epsilon_{j}^{-2}\int_{D_{x_j}^{\epsilon_j}}k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\leq k(\sigma)\sum_j\epsilon_{j}^{-2}\mathcal{L}^d(D_{x_j}^{\epsilon_j})+\left\lVert k \right\rVert_{\sup}\sum_j\epsilon_{j}^{-2}\mathcal{L}^d(D_{x_j}^{\epsilon_j}\cap\{|z-\hat{z}_{x_j}|\geq \sigma\})\nonumber\\ &\lesssim k(\sigma)\sum_j \epsilon_{j}^{d-1}+\sum_j \frac{1}{\sigma^2}\left(\int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y+\epsilon_j^{d-1}(\epsilon_j+\delta)\right),\label{est32kpart} \end{align}\tag{58}\] here the last step used ?? for the first summand and Lemma 11 a), as well as ?? and Chebyshevs’ inequality for the second summand.

By Lemma 8 c) and the assumption on the \(C_{x_j}^{\epsilon_j}\), the \(D_{x_j}^{\epsilon_j}\) do not intersect and are also contained in a neighborhood of size \(\approx\max_j\epsilon_j^2\) of the boundary of \(\Omega\) by definition. Therefore, since \(\operatorname{div}z=-\lambda(u-h)-g\) is integrable by the assumptions on the functions, we have by e.g.dominated convergence \[\begin{align} \sum_j\int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y\xrightarrow{\max_j\epsilon_j}0.\label{dom32conv} \end{align}\tag{59}\] We can furthermore let \(\sigma=\sigma(\delta,\max_j\epsilon_j)\geq \max(\delta^\frac{1}{3},(\max_j \epsilon_j)^\frac{1}{3})\) go to \(0\) so slowly that \[\begin{align} \frac{1}{\sigma^2}\left(\sum_j\int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y+\epsilon_j^{d-1}(\epsilon_j+\delta)\right)\xrightarrow{\max_j\epsilon_j,\delta\rightarrow 0}0,\label{dom32conv2} \end{align}\tag{60}\] since the second part of the sum goes to \(0\) by 54 . Hence, using 54 for the first summand in 58 , we see that \[\begin{align} \label{p2} \sum_j\epsilon_{j}^{-2}\int_{D_{x_j}^{\epsilon_j}}k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\lesssim k(\sigma)+\frac{1}{\sigma^2}\left(\sum_j\int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y+\epsilon_j^{d-1}(\epsilon_j+\delta)\right)\xrightarrow{\max_j\epsilon_j,\delta\rightarrow 0} 0. \end{align}\tag{61}\] The first part of the sum in 56 uses a similar argument, we split \(|\mathrm{D}^a \hat{u}_{x_j}|\) into the small and big parts to see that \[\begin{align} \MoveEqLeft\sum_j\epsilon_{j}^{-2}\int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|(y)|\mathrm{D}^a\hat{u}_{x_j}|(y)\,\mathrm{d}y\\ &\lesssim \sigma'\sum_j\epsilon_{j}^{-2} \int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|\,\mathrm{d}y +\sum_j \epsilon_j^{-2}(\left\lVert z \right\rVert_{L^\infty}+\left\lVert \hat{z}_{x_j} \right\rVert_{L^\infty})\int_{D_{x_j}^{\epsilon_j}}\mathbb{1}_{|\mathrm{D}^a\hat{u}_{x_j}|\geq \sigma'}|\mathrm{D}^a\hat{u}_{x_j}|(y)\,\mathrm{d}y\label{1st32part} \end{align}\tag{62}\] with a real number \(\sigma'>0\) to be fixed later. For the first sum, we can use Cauchy-Schwarz on \(\cup_j D_{x_j}^{\epsilon_j}\) together with ?? and ?? to see that \[\begin{align} \sum_j\epsilon_{j}^{-2} \int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|\,\mathrm{d}y\leq \left(\sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}1\,\mathrm{d}y\right)^\frac{1}{2}\left(\sum_j\epsilon_{j}^{-2} \int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|^2\,\mathrm{d}y\right)^\frac{1}{2}\nonumber\\ \lesssim \left(\sum_j \epsilon_j^{d-1}\right)^\frac{1}{2}\left(\sum_j\epsilon_{j}^{d-1}(\epsilon_j+\delta)+ \int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y\right)^\frac{1}{2}.\label{551} \end{align}\tag{63}\] By 54 and 59 , this goes to \(0\).

The other summand in 62 on the other hand is estimated, using the bounds 46 and 38 on \(z\) and \(\hat{z}_{x}\), as well as Lemma 10, as

\[\begin{align} \MoveEqLeft\sum_j \epsilon_j^{-2}(\left\lVert z \right\rVert_{L^\infty}+\left\lVert \hat{z}_{x_j} \right\rVert_{L^\infty})\int_{D_{x_j}^{\epsilon_j}}\mathbb{1}_{|\mathrm{D}^a\hat{u}_{x_j}|\geq \sigma'}|\mathrm{D}^a\hat{u}_{x_j}|(y)\,\mathrm{d}y\lesssim \frac{\sigma'}{i(\sigma')}\sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}i(|\mathrm{D}^a\hat{u}_{x_j}|(y))\,\mathrm{d}y\nonumber\\ &\lesssim_{u_0} \frac{\sigma'}{i(\sigma')}\int_{U'} i(|\mathrm{D}^a u_0|(y))\,\mathrm{d}\mathcal{H}^{d-1}(y)\xrightarrow{\sigma'\rightarrow +\infty}0,\label{last32part} \end{align}\tag{64}\] where we used ?? and the disjointness of the sets in the last step.

Therefore, by 63 and 64 , we see that, if we let \(\sigma'\rightarrow \infty\) sufficiently slow, that \[\begin{align} \sum_j\epsilon_{j}^{-2}\int_{D_{x_j}^{\epsilon_j}}|z-\hat{z}_{x_j}|(y)|\mathrm{D}^a\hat{u}_{x_j}|(y)\,\mathrm{d}y\xrightarrow{\max_j\epsilon_j,\delta\rightarrow 0}0.\label{p3} \end{align}\tag{65}\] Combining 56 , 57 , 61 and 65 , we see the Claim 1, and hence that \[\begin{align} \limsup_{\max_j\epsilon_j,\delta\rightarrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}(\lambda(u-h) +g-\operatorname{div}\hat{z}_{x_j}) (T_b(u)-T_b(\hat{u}_{x_j}))_+\,\mathrm{d}y\leq 0. \end{align}\]

Proof of the Claim 1 We recall that the term \(g-\operatorname{div}\hat{z}_{x_j}\) is \(\gtrsim 1\) by the definition of \(\hat{z}_{x_j}\) and ?? .

Hence, using the monotonicity of the truncation, we see that in every \(D_x^\epsilon\) it holds that \[\begin{align} \MoveEqLeft(\lambda(u-h)+g-\operatorname{div}\hat{z}_{x})(T_b(u)-T_b(\hat{u}_{x}))_+\\ &\geq C(T_b(u)-T_b(\hat{u}_{x}))_++\lambda\big((u-\hat{u}_{x})+(\hat{u}_{x}-h)\big)(T_b(u)-T_b(\hat{u}_{x}))_+\\ &\gtrsim (T_b(u)-T_b(\hat{u}_{x}))_+-Cb\lambda |\hat{u}_{x}-h| \end{align}\] yielding, if we use 55 and Claim 1, that \[\begin{align} \MoveEqLeft\limsup_{\delta,\max_j\epsilon_j\searrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}(T_b(u)-T_b(\hat{u}_{x_j}))_+\,\mathrm{d}y\lesssim_b\limsup_{\delta,\max_j\epsilon_j\searrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\lambda|\hat{u}_{x_j}-h|\,\mathrm{d}y.\label{lim32est} \end{align}\tag{66}\]

By the following Lemma, these weighted integrals converge to the traces of the corresponding functions in a uniform way up to a small cutoff of the radius.

The proof of the lemma is found in Section 5.6.

Now by e.g.the coarea formula, we have \(|\mathrm{D}\left((T_b(u)-T_b(\hat{u}_{x}))_+\right)|\leq |\mathrm{D}u|+|\mathrm{D}\hat{u}_{x}|\) for all \(x\) and \(b\). Using 66 , we hence see that for each \(\rho\) we have \[\begin{align} \MoveEqLeft\limsup_{\delta,\max_j\epsilon_j\searrow 0}\int_{\bigcup_j C_{x_j}^{\rho\epsilon_j}} (T_b(u)-T_b(u_0))_+\,\mathrm{d}\mathcal{H}^{d-1}\\ &\lesssim_{\rho,b} \limsup_{\delta,\max_j\epsilon_j\searrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\lambda|\hat{u}_{x_j}-h|\,\mathrm{d}y+\sum_j |\mathrm{D}u|(D_{x_j}^{\epsilon_j})+|\mathrm{D}\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j}). \end{align}\] Since the \(D_{x_j}^{\epsilon_j}\) are disjoint by Lemma 8 c) and their union vanishes in the limit, we have \[\begin{align} \sum_j |\mathrm{D}u|(D_{x_j}^{\epsilon_j})\rightarrow 0, \end{align}\] and the same holds for \(h\) if \(h\in BV\). Similarly, by ?? and the disjointness it holds that \[\begin{align} \sum_j|\mathrm{D}\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j})\lesssim \big(\max_j\epsilon_j^2\big)\sum_j |\mathrm{D}u_0|(C_{x_j}^{\epsilon_j})\lesssim\max_j\epsilon_j^2|\mathrm{D}u_0|(U')\rightarrow 0. \end{align}\] Now applying ?? with \(w=h-\hat{u}_{x_j}\) if \(h\in BV\) or ?? with \(w=\hat{u}_{x_j}\) if \(h\) is continuous, we see from the assumption that \(h=u_0\) on the boundary that \[\begin{align} \limsup_{\delta,\max_j\epsilon_j\searrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\lambda|\hat{u}_{x_j}-h|\,\mathrm{d}y\lesssim\begin{cases} \sum_j|\mathrm{D}\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j})+|\mathrm{D}h|(D_{x_j}^{\epsilon_j})&\text{ if h\in BV}\\ \sum_j|\mathrm{D}\hat{u}_{x_j}|(D_{x_j}^{\epsilon_j})+\omega(2\max_j \epsilon_j)&\text{ if h\in C^0}. \end{cases} \end{align}\] In either case, this goes to \(0\) as \(\max_j \epsilon_j\rightarrow 0\). This shows that \[\begin{align} \label{fin32lim} \limsup_{\delta,\max_j\epsilon_j\searrow 0}\int_{\bigcup_j C_{x_j}^{\rho\epsilon_j}} (T_b(u)-T_b(u_0))_+\,\mathrm{d}\mathcal{H}^{d-1}=0. \end{align}\tag{67}\] Finally, to deal with the \(\rho\), we note that by ?? and 54 it holds that\[\begin{align} \label{est32def} \sum_j\mathcal{H}^{d-1}(C_{x_j}^{\epsilon_j}\backslash C_{x_j}^{\rho\epsilon_j})\lesssim \sum_j (1-\rho)\epsilon_j^{d-1}\lesssim 1-\rho \end{align}\tag{68}\] and combining 67 with the fact that the \(C_{x_j}^{\epsilon_j}\) cover \(U''\cap \{u>u_0\}\) up to a \(\mathcal{H}^{d-1}\)-zero set by construction yields that for every \(\rho\in (0,1)\) it holds that \[\begin{align} \limsup_{V\subset U',\, \mathcal{H}^{d-1}(V)\leq C(1-\rho)} \int_{U''\backslash V} (T_b(u)-T_b(u_0))_+\,\mathrm{d}\mathcal{H}^{d-1}=0. \end{align}\] Letting first \(\rho\nearrow 1\) and then \(b\rightarrow +\infty\) yields that \(u=u_0\) in \(U''\), showing this case of the theorem.

4.4 Proof of Theorem 1 if \(u_0\in W^{\alpha,p}\)↩︎

We use the same geometric setup and the same field \(\hat{z}_{x}\) as defined in 44 . We will again only show that \(u\leq u_0\) a.e.on the boundary, the other case again follows from symmetry. Instead of \(u-\hat{u}_{x}\) we use \(T_1(u-a)_{+}^{p}\), where \(T_b(y)=\min(b,\max(y,-b))\) is again the trunctation and \(a\) is some real number to be determined later. This function lies in \(BV\) by the chain rule for \(BV\)-functions as explained in Section 2.2. We again use \(z\) fulfilling the conditions in the characterisation of the subdifferential in Lemma 3 and the same \(\hat{z}_x\), defined in 44 . We see by the Gauss-Green formula 23 that \[\begin{align} \MoveEqLeft\int_{C_x^\epsilon} [z-\hat{z}_{x},\nu_y]T_1(u-a)_{+}^{p}\,\mathrm{d}\mathcal{H}^{d-1}(y)+\int_{E_x^\epsilon} [z-\hat{z}_{x},-\nu_x]T_1(u-a)_{+}^{p}\,\mathrm{d}\mathcal{H}^{d-1}\\ &=\int_{D_x^\epsilon}(\lambda(u-h)+g-\operatorname{div}\hat{z}_{x})T_1(u-a)_{+}^{p}\,\mathrm{d}y+\int_{D_x^\epsilon}(z-\hat{z}_x,\mathrm{D}T_1(u-a)_{+}^{p}) \end{align}\] where we have again used that \(-\nu_x\) is the outer normal on \(E_x^\epsilon\). The integral over \(E_x^\epsilon\) is non-positive as in the \(BV\)-case, because \([\hat{z}_x,\nu_x]\geq [z,\nu_x]\) thanks to ?? , 36 and the definition 44 of \(\hat{z}_x\), and hence we have the estimate \[\begin{align}\label{part32int322} \MoveEqLeft\int_{D_x^\epsilon}(\lambda(u-h)+g-\operatorname{div}\hat{z}_{x})T_1(u-a)_{+}^{p}\,\mathrm{d}y\\ &\leq \int_{C_x^\epsilon} [z-\hat{z}_{x},\nu]T_1(u-a)_{+}^{p}\,\mathrm{d}\mathcal{H}^{d-1}(y)-\int_{D_x^\epsilon}(z-\hat{z},\mathrm{D}T_1(u-a)_{+}^{p}). \end{align}\tag{69}\] Observe that the integrand in the integral over \(C_x^\epsilon\) can only be positive when \(a\leq u\). Further observe that \([z-\hat{z}_{x},\nu_y]\leq 0\) if \(u\geq u_0\), as already established in 48 and 49 earlier. Hence, the integral over \(C_x^\epsilon\) is only positive at points with \(a\leq u\leq u_0\), yielding that \[\begin{align} [z-\hat{z}_{x},\nu_y]T_1(u-a)_{+}^{p}\lesssim |u_0-a|^p \quad\text{\mathcal{H}^{d-1}-a.e.\;on C_x^\epsilon}\label{zu32est} \end{align}\tag{70}\] since we also have that \(z\) and \(\hat{z}_{x}\) are uniformly bounded by 46 and 38 . It follows from the Lemma 1 that \[\begin{align} \MoveEqLeft\int_{D_x^\epsilon} |(z-\hat{z},\mathrm{D}T_1(u-a)^p)_-|\leq p \int_{D_x^\epsilon} |(z-\hat{z},\mathrm{D}u)_-|\lesssim \int_{D_x^\epsilon} k(|z-\hat{z}_{x}|)\,\mathrm{d}y\label{ka32part}, \end{align}\tag{71}\] where we also used Lemma 11 b).

We again take \(\delta>0\) fixed. We can again use the Morse covering theorem and Radon-Nikodym [64] to find \(C_{x_j}^{\epsilon_j}\) (depending on \(\delta\)) which are disjoint, cover \(U''\cap \{u>u_0\}\) up to a \(\mathcal{H}^{d-1}\)-zero set and are such that ?? (and therefore also ?? ) holds on each one of them.⁷

Then we again have, as in 54 , that\[\begin{align} \sum_j\epsilon_j^{d-1}\lesssim 1.\label{sum32eps2} \end{align}\tag{72}\] Summing up and combining 70 and 71 , we see that for each fixed \(p\) we have \[\begin{align} \MoveEqLeft\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}(\lambda (u-h)+g-\operatorname{div}\hat{z}_{x_j})T_{1}(u-a_j)_+^p\,\mathrm{d}y\\ &\lesssim \sum_j \epsilon_j^{-2} \int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|^p\,\mathrm{d}\mathcal{H}^{d-1}+\sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}} k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\label{eff32est322} \end{align}\tag{73}\] for numbers \(a_j\) which we define as the average values \[\begin{align} \label{a32avg} a_j:=\strokedint_{C_{x_j}^{\epsilon_j}} u_0\,\mathrm{d}\mathcal{H}^{d-1}. \end{align}\tag{74}\] We use the same basic approach as in the \(BV\)-case, that is, showing that the right-hand side of 73 vanishes in the limit, while the left-hand side controls the trace.

Proof that the right-hand side in 73 vanishes in the limit Jensen’s inequality lets us estimate the first integral on the right-hand side in 73 as \[\begin{align} \MoveEqLeft\epsilon_j^{-2} \int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|^p\,\mathrm{d}\mathcal{H}^{d-1}=\epsilon_j^{-2} \int_{C_{x_j}^{\epsilon_j}}\left|u_0(y)-\strokedint_{C_{x_j}^{\epsilon_j}} u_0(y')\,\mathrm{d}\mathcal{H}^{d-1}(y')\right|^p\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &\leq \epsilon_j^{-2}\int_{C_{x_j}^{\epsilon_j}}\strokedint_{C_{x_j}^{\epsilon_j}} \left|u_0(y)-u_0(y')\right|^p\,\mathrm{d}\mathcal{H}^{d-1}(y)\,\mathrm{d}\mathcal{H}^{d-1}(y'). \end{align}\] Using the property 31 , with a function \(r\) (depending on \(\max_j\epsilon_j\)) fulfilling 31 with \(\epsilon_j\) in place of \(\delta\), we see that, since \(\alpha p\geq 2\) it holds that \[\begin{align} \MoveEqLeft\epsilon_j^{-2} \int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|^p\,\mathrm{d}\mathcal{H}^{d-1}(y)\lesssim \int_{C_{x_j}^{\epsilon_j}}\strokedint_{C_{x_j}^{\epsilon_j}} r(y)^p+r(y')^p\,\mathrm{d}\mathcal{H}^{d-1}(y)\,\mathrm{d}\mathcal{H}^{d-1}(y')\nonumber\\ &=2\int_{C_{x_j}^{\epsilon_j}}r(y)^p\,\mathrm{d}\mathcal{H}^{d-1}(y),\label{est32ua} \end{align}\tag{75}\] since \(|y-y'|\leq 2\epsilon_j\). As we only need 31 for points with distance \(\leq 2\max_j\epsilon_j\), we see from 32 and summing up that in the limit \[\begin{align} \sum_j\epsilon_j^{-2} \int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|^p\,\mathrm{d}\mathcal{H}^{d-1}\lesssim \int_{U'} r(y)^p\,\mathrm{d}\mathcal{H}^{d-1}(y)\xrightarrow{\max_j\epsilon_j\rightarrow 0}0.\label{lim321st32part} \end{align}\tag{76}\] In the other sum in 73 , we split the sum into the small and big parts of \(|z-\hat{z}_{x_j}|\) to see that \[\begin{align} \sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}} k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\leq \sum_j k(\sigma)\epsilon_j^{-2}\mathcal{L}^{d}(D_{x_j}^{\epsilon_j})+\sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\mathbb{1}_{|z-\hat{z}_{x_j}|\geq \sigma}\,\mathrm{d}y \end{align}\] where \(\sigma\) is a real number. By ?? , 72 and the fact that \(\lim_{t\searrow 0} k(t)=0\) the first summand goes to \(0\) if \(\sigma\rightarrow 0\). The second summand is estimated by Chebyshev’s inequality and ?? as \[\begin{align} \sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\mathbb{1}_{|z-\hat{z}_{x_j}|\geq \sigma}\,\mathrm{d}y\lesssim \sum_j\frac{1}{\sigma^2}\left(\epsilon_j^{d-1}(\epsilon_j+\delta)+\int_{D_{x_j}^{\epsilon_j}}|\operatorname{div}z|\,\mathrm{d}y\right), \end{align}\] if we let \(\frac{1}{\sigma^2}\rightarrow \infty\) slow enough (depending on \(\max_j\epsilon_j\) and \(\delta\)), then this goes to \(0\) as already established in 59 , 60 .

Together we see that \[\begin{align} \sum_j \epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}} k(|z-\hat{z}_{x_j}|)\,\mathrm{d}y\xrightarrow{\delta,\max_j\epsilon_j\rightarrow 0}0, \end{align}\] showing together with 76 that the right hand side of 73 vanishes in the limit.

Proof that the left-hand side of 73 controlls the trace Again, it holds that \(g-\operatorname{div}\hat{z}_{x_j}\) is \(\gtrsim 1\) by 44 and ?? , and using the pointwise inequality \(cT_1(c)_+^{p}\geq 0\), we see that \[\begin{align} &\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}(\lambda (u-h)+g-\operatorname{div}\hat{z}_{x_j})T_1(u-a_j)_+^p\,\mathrm{d}y\\ &=\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\left(\lambda ((u-a_j)+(a_j-h))+g-\operatorname{div}\hat{z}_{x_j}\right)T_{1}(u-a_j)_{+}^{p}\,\mathrm{d}y\nonumber\\ &\geq C\sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}T_{1}(u-a_j)_{+}^{p}\,\mathrm{d}y-C'\sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda(y)|h-a_j|\,\mathrm{d}y, \end{align}\] where we have also used that \(T_1(u-a)_+\leq 1\).

The first sum of integrals in the last line can be regarded as a single integral on the set \(\cup D_{x_j}^{\epsilon_j}\) with respect to the measure whose density with respect to the Lebesgue measure in \(D_{x_j}^{\epsilon_j}\) is \(\epsilon_j^{-2}\), in particular we can apply Hölder with respect to this measure to see that \[\begin{align} \MoveEqLeft\left(\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}T_{1}(u-a_j)_+\,\mathrm{d}y\right)^p\\ &\lesssim\left(\sum_j \epsilon_j^{-2}\mathcal{L}^d(D_{x_j}^{\epsilon_j})\right)^\frac{p^2}{p-1}\times\Bigg(\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}\big(\lambda(y) (u-h)+g-\operatorname{div}\hat{z}_{x_j}\big)T_{1}(u-a_j)_{+}^{p}\,\mathrm{d}y\\ &\quad+\sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda(y)|h-a_j|\,\mathrm{d}y\Bigg). \end{align}\] Here, the first sum can be estimated as \[\begin{align} \sum_j \epsilon_j^{-2}\mathcal{L}^d(D_{x_j}^{\epsilon_j})\lesssim \sum_j \epsilon_j^{d-1}\lesssim 1, \end{align}\] by ?? and 72 , and hence this factor may be disregarded.

Together, we have obtained \[\begin{align} \limsup_{\delta,\max_j\epsilon_j\rightarrow 0}\sum_j\epsilon_j^{-2}\int_{D_{x_j}^{\epsilon_j}}T_{1}(u-a_j)_+\,\mathrm{d}y\lesssim&\limsup_{\delta,\max_j\epsilon_j\rightarrow 0} \left(\sum_j \epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda(y)|h-a_j|\,\mathrm{d}y\right)^\frac{1}{p}. \end{align}\] We may then apply the trace Lemma 12 for each \(\rho\in (0,1)\), together with the fact that the total variation of \(T_{1}(u-a_j)_+\) is less of equal than the one of \(u\) by e.g.the coarea formula to conclude in the same way as in the \(BV\)-case that for every fixed \(\rho\in (0,1)\) it holds that \[\begin{align} \limsup_{\delta,\,\max_j\epsilon_j\rightarrow 0}\sum_j \int_{C_{x_j}^{\rho\epsilon_j}}T_{1}(u-a_j)_+\,\mathrm{d}\mathcal{H}^{d-1}\lesssim_{\rho} \limsup_{\delta,\,\max_j\epsilon_j\rightarrow 0}\left(\sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda(y)|h-a_j|\,\mathrm{d}y\right)^{\frac{1}{p}}. \end{align}\] Regarding the right-hand side, using Lemma 12 and arguing as in the \(BV\)-case, we have \[\begin{align} \limsup_{\max_j\epsilon_j,\delta\rightarrow 0} \sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda|h-a_j|\,\mathrm{d}y\lesssim \limsup_{\max_j\epsilon_j,\delta\rightarrow 0}\left\lVert \lambda \right\rVert_{L^\infty}\sum_j\int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|\,\mathrm{d}\mathcal{H}^{d-1}, \end{align}\] but we also have, by arguing as for 75 , that\[\begin{align} \sum_j\left\lVert \lambda \right\rVert_{L^\infty}\int_{C_{x_j}^{\epsilon_j}}|u_0-a_j|\,\mathrm{d}\mathcal{H}^{d-1}\lesssim \left\lVert \lambda \right\rVert_{L^\infty}\max_j\epsilon_j^\alpha\int_{U'} r(y)\,\mathrm{d}\mathcal{H}^{d-1}(y)\rightarrow 0, \end{align}\] yielding \[\begin{align} \lim_{\max_j\epsilon_j\rightarrow 0}\sum_j\epsilon_j^{-2} \int_{D_{x_j}^{\epsilon_j}}\lambda(y)|h-a_j|\,\mathrm{d}y=0, \end{align}\] and \[\begin{align} \lim_{\delta,\,\max_j\epsilon_j\rightarrow 0}\sum_j \int_{C_{x_j}^{\rho\epsilon_j}}T_{1}(u-a_j)_+\,\mathrm{d}\mathcal{H}^{d-1}= 0.\label{lim32est2} \end{align}\tag{77}\] It remains to estimate \(\mathcal{H}^{d-1}(\{u> u_0\})\) with this sum. Observe that by Chebyshev and the triangle inequality we have, for any \(a\), \[\begin{align} \MoveEqLeft\mathcal{H}^{d-1}(\{y\in C_x^\epsilon\,\big|\,u(y)\geq a+1\})= \mathcal{H}^{d-1}(\{y\in C_x^\epsilon\,\big|\,T_{1}(u(y)-a)_+\geq 1\})\\ &\leq \int_{C_x^\epsilon}T_{1}(u(y)-a)_+\,\mathrm{d}\mathcal{H}^{d-1}.\label{tru32est} \end{align}\tag{78}\] On the other hand, we also have \[\begin{align} (u(y)-u_0(y))_+\leq T_{1}(u(y)-a)_++|u_0(y)-a| \quad \text{if u(y)\leq a+1} \end{align}\] hence, by Chebyshev \[\begin{align} &\mathcal{H}^{d-1}(\{y\in C_x^\epsilon\,\big|\,u(y)\leq a+1,\, u(y)-u_0(y)\geq c \})\leq\frac{1}{c}\int_{C_x^\epsilon} (T_{1}(u)-a)_++|u_0-a|\,\mathrm{d}\mathcal{H}^{d-1}\label{tru32est2} \end{align}\tag{79}\] for each \(c> 0\).

Similarly as for 75 above, we have that \[\begin{align} \sum_j \int_{C_{x_j}^{\epsilon_j}} \left|u_0(y)-a_j\right|\,\mathrm{d}\mathcal{H}^{d-1}(y)\lesssim \max_j\epsilon_j^\alpha\int_{U'} r\,\mathrm{d}\mathcal{H}^{d-1}\rightarrow 0.\label{u0a} \end{align}\tag{80}\] Hence, using 78 , 79 and 80 , we see that \[\begin{align} \MoveEqLeft\mathcal{H}^{d-1}\left(U''\cap\{u-u_0\geq c\}\cap \bigcup_j C_{x_j}^{\rho\epsilon_j}\right)\\ &\leq \sum_j \mathcal{H}^{d-1}\left(\{y\in C_{x_j}^{\rho\epsilon_j}\,\big|\,u(y)\geq a_j+1\}\right)+\mathcal{H}^{d-1}\left(\{y\in C_{x_j}^{\rho\epsilon_j}\,\big|\,u(y)\leq a_j+1,\, u(y)-u_0(y)\geq c \}\right)\\ & \leq\frac{1}{c} \sum_j \int_{C_{x_j}^{\rho\epsilon_j}}(T_{1}(u(y)-a_j)_+\,\mathrm{d}\mathcal{H}^{d-1}+\frac{1}{c}\max_j\epsilon_j^\alpha\int_{U'} r\,\mathrm{d}\mathcal{H}^{d-1}. \end{align}\] This converges to \(0\) for every fixed \(c>0\) by 77 . Since we have \[\begin{align} \mathcal{H}^{d-1}\left(\bigcup_j(C_{x_j}^{\epsilon_j}\backslash C_{x_j}^{\rho\epsilon_j})\right)\xrightarrow{\rho\nearrow 1} 0, \end{align}\] as in 68 and because the \(C_{x_j}^{\epsilon_j}\) cover \(U''\cap \{u>u_0\}\) up to a \(\mathcal{H}^{d-1}\)-zero set, we conclude the theorem in this case.

4.5 Proof of Thm.1 in the case \(u_0\in C^0\)↩︎

Yet again, it suffices to consider the positive part of \(u-u_0\).

We start with 69 (which did not use the regularity of \(u_0\) at all and therefore still holds here), where we again take \[\begin{align} a=\strokedint_{C_x^\epsilon} u_0\,\mathrm{d}\mathcal{H}^{d-1}. \end{align}\] Then, by the extra assumption in [A6] that in this case either \(\lambda=0\) or \(h\) is a continuous extension of \(u_0\) (near \(U\)) we have \(\left\lVert \lambda(a-h) \right\rVert_{L^\infty(D_x^\epsilon)}\rightarrow 0\) as \(\epsilon\searrow 0\) yielding, together with ?? , that in \(D_x^\epsilon\) it holds that \[\begin{align} \MoveEqLeft(\lambda (u-h)+g-\operatorname{div}\hat{z}_{x})T_1(u-a)_+^{p}\\ &\geq \left(\lambda ((u-a)+(a-h))+g-\operatorname{div}\hat{z}_{x}\right)T_1(u-a)_+^{p}\gtrsim T_1(u-a)_+^{p}, \end{align}\] uniformly in \(p\), where we have again used that \((u-a)T_1(u-a)_+\geq 0\). We now take the limit \(p\rightarrow \infty\) of the \(p\)-th root of 69 , since the limit of the \(L^p\)-norm is the \(L^\infty\)-norm, we see that \[\begin{align} \MoveEqLeft\mathop{\mathrm{esssup}}_{y\in C_x^\epsilon} T_1(u(y)-a)_+\leq\mathop{\mathrm{esssup}}_{y\in D_x^\epsilon} T_1(u(y)-a)_+\\ &\lesssim \limsup_{p\rightarrow \infty}\left( \int_{D_x^\epsilon}(\lambda (u-h)+g-\operatorname{div}\hat{z}_{x})T_1(u-a)_+^p\,\mathrm{d}y\right)^\frac{1}{p}\\ &\lesssim \limsup_{p\rightarrow \infty}\left(\int_{C_x^\epsilon} [z-\hat{z}_{x},\nu]T_1(u-a)_+^p\,\mathrm{d}\mathcal{H}^{d-1}\right)^\frac{1}{p}+\left(\int_{D_x^\epsilon}|(z-\hat{z}_{x},\mathrm{D}(T_1(u-a)_+^p))_-|\right)^\frac{1}{p}\nonumber\\ &\lesssim\sup_{y\in C_x^\epsilon} T_1(u_0-a)_++\limsup_{p\rightarrow \infty}\left(\int_{D_x^\epsilon}|(z-\hat{z}_{x},\mathrm{D}T_1(u-a)_+^p)_-|\right)^\frac{1}{p}. \end{align}\] Using Lemma 1 and that we can take \(k=0\) in Lemma 11 b), due to the extra assumption of positive \(1\)-homogeneity of \(f\) in [A6], we see that the second summand vanishes.

Now if we let \(\epsilon\rightarrow 0\), then the right-hand side goes to \(0\) by continuity, while \(a\) converges to \(u_0(x)\) by the continuity of \(u_0\), showing that \(u(x)\leq u_0(x)\) whenever \(x\) is a Lebesgue point of \(u|_{\partial\Omega}\).

4.6 Proof of Theorem 3↩︎

Throughout the entire proof, we assume that [B1]-[B3] hold, in particular, the lemmata might require them even though we don’t explicitly mention it.

We use a similar scheme as in the \(BV\) case for Theorem 1, but without transforming the domain. As the statement of the theorem is local, we again only need to show that for every point \(x^0\in U\), we can find a neighborhood where \(u=u_0\).

We shall consider \(U'\subset U\) of the form \(U'=\partial\Omega\cap B\) for some (open) ball \(B\). We can also assume that \(\Omega\cap B\) is convex by choosing \(B\) sufficiently small, thanks to the Assumption [B1] on \(U\).

Since \(U\) is uniformly convex and \(C^2\), we can also assume that \(U'\) is uniformly convex in the sense that it can be written as the graph of a uniformly convex \(C^2\) function. It is also not restrictive to assume that such a graph is taken over the plane perpendicular to \(\nu_{x_0}\) for some \(x_0\in U'\) after potentially making \(U'\) smaller.

We use the following adaptions of the sets for \(x\in U'\) and small \(\epsilon\): \[\begin{align} &\tilde{E}_x^\epsilon:=\{y\in \overline{\Omega}\cap B\,\big|\, \langle y-x,\nu_x \rangle=-\epsilon^2\}\\ &\tilde{D}_x^\epsilon:=\{y\in \Omega\cap B\,\big|\, \langle y-x,\nu_x \rangle>-\epsilon^2\}\\ &\tilde{C}_x^\epsilon=\partial\tilde{D}_x^\epsilon\backslash\tilde{E}_x^\epsilon=\{y\in\partial\Omega\cap B\,\big|\, \langle y-x,\nu_x \rangle>-\epsilon^2\}. \end{align}\] Figure 2 shows a sketch. We have the following geometric properties.

The lemma is proven in Section 5.2.

During the rest of the proof, we fix such an \(U''\) and show that \(u=u_0\) holds in \(U''\), which then yields the theorem because \(U\) can be covered with such sets. We will only consider \(\epsilon<<\min(\mathop{\mathrm{dist}}(U'',\partial B), \mathop{\mathrm{dist}}(U'',\partial\Omega\backslash U'))\) so that \(\tilde{C}_x^\epsilon\subset U'\), there is no intersection between \(\tilde{D}_x^\epsilon\) and \(\partial\Omega\backslash U'\), and the statements of the Lemma hold.

We use the functions \(\mu_1,\mu_2\) from the Assumption 13 on \(f\) and define for \(x\in U'\) \[\begin{align} \zeta_x:=\mu_2(\nu_x) \end{align}\] Observe that, by the definition 13 of \(\mu_1\) and \(\mu_2\), the property 29 and the growth condition 26 , it holds that \[\begin{align} \langle \zeta_x,\nu_x \rangle=\frac{1}{\langle \mu_1(a),a \rangle}f^\infty(a\otimes \nu_x)\gtrsim 1,\label{pos32scalar} \end{align}\tag{81}\] where \(a\) is arbitrary. Therefore, by the implicit function theorem, we can write every point in \(\Omega\) in a neighborhood of \(x\) uniquely in the form \[\begin{align} y=y_0^x+t\zeta_x,\label{imp32norm} \end{align}\tag{82}\] where \(t\leq 0\) and \(y_0^x\in \partial\Omega\) is some point near \(x\). Since all possible values of \(\zeta_x\) lie in a compact set, it is possible to pick \(\epsilon\) so small that for every \(x\in U''\), the set \(\tilde{D}_x^\epsilon\) lies in a neighborhood of the boundary where such a unique representation is possible. It follows from 81 and the definition of the sets that for \(y\in \tilde{D}_x^\epsilon\), the point \(y_0^x\) must lie in \(\tilde{C}_x^\epsilon\).

We then set \[\begin{align} \bar{u}_x(y):=u_0(y_0^x). \end{align}\]

The lemma is proven in Section 5.3.

To define the comparison tensor field \(\bar{z}_x\), we first note that every point \(y\) in \(\tilde{D}_x^\epsilon\) can also uniquely be written as \(y_1^x+s'\zeta_x\) for \(y_1^x\in \tilde{E}_x^\epsilon\). Indeed this representation is unique as \(\zeta_x\) is not tangential to \(\tilde{E}_x^\epsilon\) by 81 and every point can be written in this way because the ray \(y+s\zeta_x\) with \(s\leq 0\) must intersect the boundary of \(\tilde{D}_x^\epsilon\), yet can not hit \(\tilde{C}_x^\epsilon\) because of the uniqueness of the representation 82 .

We then set \[\begin{align} \bar{z}_x(y):=-\mu_1\left(u(y_1^x)-\bar{u}_x(y_1^x)\right)\otimes \zeta_x,\label{def32barz} \end{align}\tag{83}\] where we set \(\mu_1(0)=0\). If \(u(y_1^x)-\bar{u}_x(y_1^x)\neq 0\), then by the definition 13 of \(\mu_1\) and \(\mu_2\) it holds that \[\begin{align} \bar{z}_x(y)=-\mathrm{D}_\xi f^\infty\big((u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x\big). \end{align}\] This field is (row-wise) divergence-free, since every row is a multiple of \(\zeta_x\) and the field is constant along lines with direction \(\zeta_x\), and it is furthermore in the class \(X_2(\tilde{D}_x^\epsilon,\mathbb{R}^{n\times d})\).

By the following lemma, the normal trace of \(\bar{z}_x\) is the one that one would expect from the definition.

We now fix a \(z\) fulfilling the conditions in Lemma 4 (with \(v=0\), as it is a minimizer).

We then use the same approach as in the scalar-valued case and see from the Gauss-Green formula 23 that \[\begin{align} \int_{\tilde{C}_x^\epsilon} [z-\bar{z}_x,\nu]\cdot(u-u_0)\,\mathrm{d}\mathcal{H}^{d-1}+\int_{\tilde{E}_x^\epsilon} [z-\bar{z}_x,\nu_{\tilde{E}_x^\epsilon}]\cdot(u-\bar{u}_x)\,\mathrm{d}\mathcal{H}^{d-1}= \int_{\tilde{D}_x^\epsilon} (z-\bar{z}_x,\mathrm{D}u-\mathrm{D}\bar{u}_x).\label{par32int} \end{align}\tag{84}\] By ?? , the definition of \(\bar{z}_x\), and 29 we know that on \(\tilde{E}_x^\epsilon\) it holds that \[\begin{align} [\bar{z}_x,\nu_{\tilde{E}_x^\epsilon}]\cdot(u-\bar{u}_x)=\big(\mu_1\big(u(y_1^x)-\bar{u}_x(y_1^x)\big)\otimes \mu_2(\nu_x)\big):\big((u-\bar{u}_x)\otimes \nu_x\big)=f^\infty((u-\bar{u}_x)\otimes \nu_x) \end{align}\] since \(y=y_1^x\) on \(\tilde{E}_x^\epsilon.\) Together with 36 , ?? and 29 this shows that \[\begin{align} [z-\bar{z}_x,\nu_{\tilde{E}_x^\epsilon}]\cdot(u-\bar{u}_x)\leq 0 \quad \text{ on \tilde{E}_x^\epsilon}. \end{align}\] Hence, using 84 , we see that \[\begin{align} -\int_{\tilde{C}_x^\epsilon} [z-\bar{z}_x,\nu]\cdot(u-u_0)\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq \int_{\tilde{D}_x^\epsilon} (z-\bar{z}_x,\mathrm{D}\bar{u}_x)-(z-\bar{z}_x,\mathrm{D}u).\label{dadada} \end{align}\tag{85}\] It follows from the definition of \(\bar{z}_x\), the equation ?? and the formulas for the density of the Anzelotti pairing in Section [sec:S23] that \[\begin{align} (\bar{z}_x,\mathrm{D}\bar{u}_x)=0. \end{align}\]

The analogue of Lemma 11 holds:

The proof can be found in Sections 5.4 and 5.5.

We note that if \(u(y)\neq u_0(y)\) at some \(y\in \partial\Omega\), then, if we plug 13 into 37 , it holds that \([z,\nu](y)= \langle \zeta_y,\nu_y \rangle\mu_1(v)\) with \(v=\frac{u_0(y)- u(y)}{|u_0(y)- u(y)|}\). Since we also have \(\langle \zeta_y,\nu_y \rangle\gtrsim 1\) by 81 , we see from Radon-Nikodym and the Lebesgue point theorem that for \(\mathcal{H}^{d-1}\)-a.e.\(x\) in the set \(\{u\neq u_0\}\), the condition ?? holds for every sufficiently small \(\epsilon\) (depending on \(\delta\) and \(x\)).

Now we fix \(\delta>0\) and use a covering argument as in the scalar case. We can again find \(\epsilon_{0,x}>0\) such that for \(\mathcal{H}^{d-1}\)-a.e.\(x\in U''\cap \{u\neq u_0\}\) the property ?? holds for \(\epsilon<\epsilon_{0,x}\) and a \(v\) depending on \(\epsilon\), as explained in the previous paragraph, and furthermore such that \(|\mathrm{D}^su_0|(\tilde{C}_x^\epsilon)\leq \delta \mathcal{H}^{d-1}(\tilde{C}_x^\epsilon)\). Indeed, this follows from Radon-Nikodym.

We can apply the Morse covering theorem [64] to the projections of the \(\tilde{C}_x^\epsilon\) onto the plane perpendicular to \(\nu_{x^0}\) since the assumptions of the theorem are true by Lemma 13 d). Since \(U'\) can be written as a graph over the plane, we have that two different \(\tilde{C}_x^\epsilon\) are disjoint if their projections are disjoint. This yields that we can find again find a (countable) cover of \(U''\cap \{u\neq u_0\}\), up to a \(\mathcal{H}^{d-1}\)-zero set with disjoint \(\tilde{C}_{x_j}^{\epsilon_j}\) (depending on \(\delta\)) such that ?? holds for each of these and such that \[\begin{align} |\mathrm{D}^su_0|(\tilde{C}_{x_j}^{\epsilon_j})\leq \delta \mathcal{H}^{d-1}(\tilde{C}_{x_j}^{\epsilon_j}). \end{align}\] The analogue of 54 does still hold here for the same reason, i.e.\[\begin{align} \sum_j \epsilon_j^{d-1}\approx \sum_j \mathcal{H}^{d-1}(\tilde{C}_{x_j}^{\epsilon_j})\leq \mathcal{H}^{d-1}(U')\lesssim 1.\label{sum32eps3} \end{align}\tag{86}\]

We hence see that as in 55 , 56 it follows from 85 that \[\begin{align} \MoveEqLeft\sum_j-\epsilon_j^{-2}\int_{\tilde{C}_x^\epsilon} [z-\bar{z}_{x_j},\nu]\cdot(u-u_0)\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq\sum_j\epsilon_j^{-2}\left(\int_{\tilde{D}_{x_j}^{\epsilon_j}} (z,\mathrm{D}\bar{u}_{x_j})+|(z-\bar{z}_{x_j},\mathrm{D}u)_-|\right)\\ &\lesssim \sum_j \epsilon_j^{-2}\int_{\tilde{D}_{x_j}^{\epsilon_j}}|z-\mu_1(v_j)\otimes\zeta_{x_j}||\mathrm{D}^a \bar{u}_{x_j}|\,\mathrm{d}y\\ &\quad+\epsilon_j^{-2}|\mathrm{D}^s\bar{u}_{x_j}|(\tilde{D}_{x_j}^{\epsilon_j})+\epsilon_j^{-2} \int_{\tilde{D}_{x_j}^{\epsilon_j}}k(|z-\mu_1(v_j)\otimes\zeta_{x_j}|)\,\mathrm{d}y, \end{align}\] where we have used ?? to bring in the \(\mu_1(v_j)\otimes\zeta_{x_j}\).

The same arguments as in the proof of Claim 1 in the scalar case earlier show that this goes to \(0\) as \(\delta,\max_j\epsilon_j\searrow 0\), in fact, the argument even simplifies quite a bit due to the vanishing divergence of \(z\).

It remains to control the trace with the left-hand side.

*Proof that the left-hand side controls the trace Let us analyze the integrand on the left-hand side. We distinguish the cases \(u(y_1^x)-\bar{u}_x(y_1^x)=0\) and \(u(y_1^x)-\bar{u}_x(y_1^x)\neq 0\). At points \(y\in \tilde{C}_x^\epsilon\) with \(u(y_1^x)-\bar{u}_x(y_1^x)=0\) it holds that \([\bar{z}_x,\nu](y)=0\) by ?? . Then it follows from ?? and 26 that \[\begin{align} -\left[z-\bar{z}_x,\nu\right]\cdot(u-u_0)=f^\infty((u-u_0)\otimes \nu_y)\gtrsim |u-u_0|.\label{str32bd} \end{align}\tag{87}\] In the other case, it either holds that \(u=u_0\) or, as one sees from 37 and the definition 13 of \(\mu_1,\mu_2\), it holds that \[[z,\nu]\cdot (u-u_0)(y)=-\left(\mu_1(u-u_0)\otimes \mu_2(\nu_y)\right):((u-u_0)\otimes \nu_y)\] as \(\mu_1\) is odd by definition.

Combining this with ?? and the Definition 83 of \(\bar{z}_x\), we see, using the oddness of \(\mu_1\) again, that\[\begin{align} \MoveEqLeft-\left[z-\bar{z}_x,\nu\right]\cdot(u-u_0)(y)\\ &=\langle \mu_1(u-u_0)\otimes \mu_2(\nu_y)-\mu_1\left(u(y_1^x)-\bar{u}_x(y_1^x)\right)\otimes \mu_2(\nu_x),(u-u_0)\otimes \nu_y \rangle\\ &\gtrsim|(u-u_0)\otimes \nu_y|\left|\mu_1(u-u_0)\otimes \mu_2(\nu_y)-\mu_1\left(u(y_1^x)-\bar{u}_x(y_1^x)\right)\otimes \mu_2(\nu_x)\right|^2, \end{align}\] where the last step used Lemma 2 and the positive \(0\)-homogeneity of \(\mu_1\) and \(\mu_2\). Again, using the Definition 13 of \(\mu_1\) and \(\mu_2\), we see that \[\begin{align} \MoveEqLeft\left|\mu_1(u-u_0)\otimes \mu_2(\nu_y)-\mu_1\left(u(y_1^x)-\bar{u}_x(y_1^x)\right)\otimes \mu_2(\nu_x)\right|\nonumber\\ &=\left|\mu_1\left(\frac{u-u_0}{f^\infty((u-u_0)\otimes \nu_y)}\right)\otimes \mu_2(\nu_y)-\mu_1\left(\frac{u(y_1^x)-\bar{u}_x(y_1^x)}{f^\infty((u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x)}\right)\otimes \mu_2(\nu_x)\right|\nonumber\\ &=\frac{1}{2}\left|\mathrm{D}_\xi( (f^\infty)^2)\left(\frac{(u-u_0)\otimes \nu_y}{f^\infty((u-u_0)\otimes \nu_y)}\right)-\mathrm{D}_\xi ( (f^\infty)^2)\left(\frac{(u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x}{f^\infty((u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x)}\right)\right|.\label{rewrite32grad} \end{align}\tag{88}\] Since \((f^\infty)^2\) is uniformly convex by assumption, we know that for all \(\xi_1,\xi_2\in \mathbb{R}^{n\times d}\backslash \{0\}\) it holds that \[\begin{align} \MoveEqLeft\left|\mathrm{D}_\xi( (f^\infty)^2)(\xi_1)-\mathrm{D}_\xi( (f^\infty)^2)(\xi_2)\right|\\ &\geq \frac{1}{|\xi_1-\xi_2|}\langle \mathrm{D}_\xi( (f^\infty)^2)(\xi_1)-\mathrm{D}_\xi( (f^\infty)^2)(\xi_2),\xi_1-\xi_2 \rangle\\ &\gtrsim |\xi_1-\xi_2|. \end{align}\] Combining this estimate with 88 , we see that \[\begin{align} \MoveEqLeft\big|\mu_1(u-u_0)\otimes \mu_2(\nu_y)-\mu_1\left(u(y_1^x)-\bar{u}_x(y_1^x)\right)\otimes \mu_2(\nu_x)\big|\\ &\gtrsim \left|\frac{(u-u_0)\otimes \nu_y}{f^\infty((u-u_0)\otimes \nu_y)}-\frac{(u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x}{f^\infty((u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x)}\right|. \end{align}\] Using the growth bound 26 for \(f^\infty\) and that it holds \(|a\otimes v_1-b\otimes v_2|\gtrsim \min(|a|,|b|)\min(|v_1-v_2|,|v_1+v_2|)\) for all unit vectors \(v_1\) and \(v_2\), we see that \[\begin{align} \left|\frac{(u-u_0)\otimes \nu_y}{f^\infty((u-u_0)\otimes \nu_y)}-\frac{(u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x}{f^\infty((u(y_1^x)-\bar{u}_x(y_1^x))\otimes \nu_x)}\right|\gtrsim \min(|\nu_x-\nu_y|,\,|\nu_x+\nu_y|). \end{align}\] In sum, we have obtained that \[\begin{align} -\left[z-\bar{z}_x,\nu\right]\cdot(u-u_0)\gtrsim |u-u_0| \min(|\nu_x-\nu_y|,\,|\nu_x+\nu_y|)^2. \end{align}\] Using that \(|\nu_x-\nu_y|<<1\) as \(|x-y|<<1\) and that the normal is continuous by the assumed regularity of the boundary, we see that \(|\nu_x+\nu_y|\geq |\nu_x-\nu_y|\) for small \(\epsilon\) and hence, thanks to Lemma 13 e), we see that this is \[\begin{align} \gtrsim_\rho\epsilon^2 |u-u_0|\quad \text{ in \tilde{C}_x^\epsilon\backslash \tilde{C}_x^{\rho\epsilon}}\label{norm32est32} \end{align}\tag{89}\] uniformly in \(x\) and \(\epsilon\) for \(\rho\in (0,1)\) for \(u(y_1^x)-\bar{u}_x(y_1^x)\neq 0\). This is also true in the case \(u(y_1^x)-\bar{u}_x(y_1^x)= 0\) from earlier, as 87 is clearly much stronger and hence 89 holds on all of \(\tilde{C}_x^\epsilon\backslash \tilde{C}_x^{\rho\epsilon}\). This yields that \[\begin{align} \limsup_{\delta,\max_j\epsilon_j\searrow 0} \int_{\bigcup_j\tilde{C}_x^\epsilon\backslash \tilde{C}_x^{\rho\epsilon}}|u-u_0|\,\mathrm{d}\mathcal{H}^{d-1}=0.\label{lim32est323} \end{align}\tag{90}\] We note that, thanks to the disjointness of the sets, it holds that \[\begin{align} \mathcal{H}^{d-1}\left(\bigcup_j \tilde{C}_{x_j}^{\rho\epsilon_j}\right)\leq \sum_j\mathcal{H}^{d-1}(\tilde{C}_{x_j}^{\rho\epsilon_j})\approx \rho^{d-1}\sum_j\epsilon_j^{d-1}\lesssim \rho^{d-1}, \end{align}\] where we have used ?? and 86 .

We therefore obtain that \[\begin{align} \sup_{V\subset U', \mathcal{H}^{d-1}(V)\leq C\rho^{d-1}}\int_{U''\backslash V} |u-u_0|\,\mathrm{d}\mathcal{H}^{d-1}=0. \end{align}\] Letting \(\rho\searrow 0\) yields the theorem.

5 Proofs of the technical Lemmata↩︎

5.1 Proof of the diffeomorphism Lemma 7↩︎

a) We first check that the assumptions hold for the transformed data. [A1] is trivial. [A6] is trivial if \(u_0\in C^0\), if \(u_0\in BV\) it follows from the chain rule for \(BV\)-functions, see e.g.[53]. For \(u_0\in W^{\alpha,p}\) it follows e.g.from applying the transformation rule to the integral in the definition and using that \(\Phi\) is Lipschitz.

Regarding [A2], we can check the different points as follows:

• Convexity of \(f_\Phi\) is trivial from the definition.

• The growth bound 6 holds because \(\Phi\) is \(C^1\) and, since it is a diffeomorphism, the \(C^1\) norm of the inverse is also bounded, yielding the bound directly from the definition.

• \(f_\Phi\) being \(C^2\) in the \(\xi\)-variable outside a large ball follows from \(\Phi\) being \(C^2\). We also see directly from the definition and the chain rule that \[\begin{align} \langle \mathrm{D}_\xi^2 f_\Phi(y,\xi)\frac{\xi}{|\xi|},\frac{\xi}{|\xi|} \rangle=|\det \mathrm{D}\Phi^{-1}|\langle \mathrm{D}_\xi^2 f\big(\Phi^{-1}(y),(\mathrm{D}\Phi^{-1})^{-T}\xi\big)\frac{(\mathrm{D}\Phi^{-1})^{-T}\xi}{|\xi|},\frac{(\mathrm{D}\Phi^{-1})^{-T}\xi}{|\xi|} \rangle \end{align}\] which yields 7 for \(f_\Phi\) after applying 7 with \((\mathrm{D}\Phi^{-1})^T\xi\) in place of \(\xi\) and using the boundedness of the derivatives of \(\Phi^{-1}\).

• Continuity of \(f_\Phi\) follows from the regularity of \(\Phi\).

• The recession function of \(f_\Phi\) is \[\begin{align} f_\Phi^\infty(y,\xi)=|(\det \mathrm{D}\Phi^{-1})y|f^\infty\big(\Phi^{-1}(y),(\mathrm{D}\Phi^{-1}(y))^{-T}\xi\big) \end{align}\] and the convergence of \(\frac{1}{t}f_\Phi(y,t\cdot)\) to it follows from the continuity of \(\Phi\) and the boundedness of the involved derivatives.

• The regularity of the recession function follows from the chain rule, the homogeneity of the recession function, and \(\Phi\) being \(C^2\).

It follows directly from the fact that \(\Phi\) is \(C^2\) that \(\lambda_\Phi\) and \(h_\Phi\) have the regularity/integrability required by [A4]-[A5].

It follows from the transformation rule that \(g_\Phi\in L^{\max(2,d)}(\Omega')\). In order to establish the curvature condition in [A3], we write \(H_{\partial\Omega', f_\Phi}\) to denote the generalized curvature with respect to \(\Omega'\) and \(f_\Phi\), defined analogously to 8 . We claim that for \(y\in U\) it holds that \[\begin{align} H_{\partial\Omega', f_\Phi}(\Phi(y))=|\det \mathrm{D}\Phi(y)|^{-1}H_{\partial\Omega,f}(y),\label{cur32ch} \end{align}\tag{91}\] which then implies [A3].

To show this, we first claim that for any \(C^1\) vector field \(z_0\), defined in a neighborhood of \(U\), for which it holds that \(z_0(y)=c(y)\nu_y\) for some \(c(y)\in \mathbb{R}_{<0}\) on \(U\), it holds that \[\begin{align} \operatorname{div}\mathrm{D}_\xi f^\infty(y,z_0(y))=\operatorname{div}\mathrm{D}_{\xi} f^\infty(y,\mathrm{D}\mathfrak{d}(y))\quad\text{ for y\in U.}\label{n32field32claim} \end{align}\tag{92}\] Indeed, to see this, we first note that we can assume that \(z_0(y)=-\nu_y=\mathrm{D}\mathfrak{d}(y)\) on the boundary because \(\mathrm{D}_\xi f^\infty\) is positively \(0\)-homogeneous. Therefore, we only need to show that the part of the divergence that is not determined by the values on the boundary is the same for both fields.

We can rewrite the divergence at each point \(x\in U\), thanks to its frame invariance, as \(\operatorname{div}=\nu\cdot\partial_\nu+ \mathfrak{pr}_{T_x}(\nabla-\partial_\nu)\), where \(\mathfrak{pr}_{T_x}\) is the orthogonal projection onto the tangent space of \(\partial\Omega\) at \(x\) and \(\partial_\nu\) the normal derivative. The second summand is determined by the boundary values only.

By the chain rule, it holds that \[\begin{align} \nu_y\cdot \partial_\nu \mathrm{D}_\xi f^\infty(y,z_0(y))=\nu_y\otimes \nu_y:\partial_y\mathrm{D}_\xi f^\infty(y,z_0(y))+\nu_y\cdot \mathrm{D}_\xi^2 f^\infty(y,-\nu_y)\cdot \partial_\nu z_0(y). \end{align}\] The first term is determined by the values of \(z_0(y)\) at the boundary. To see that the second term vanishes, we note that the function \[\begin{align} -\nu_y\cdot \mathrm{D}_\xi f^\infty(y,v) \end{align}\] has a local maximum in the variable \(v\) at \(v=-\nu_y\) by e.g.Lemma 2 and therefore its derivative must vanish, yielding \(\nu_y\cdot \mathrm{D}_\xi^2 f^\infty(y,-\nu_y)\cdot \partial_\nu z_0(y)=0\). This implies 92 .

To proceed with the proof of the equivalence, we let \(\mathfrak{d}_{\Omega'}\) denote the distance to the boundary in \(\Omega'\). We note that, thanks to the identity \(|\det \mathrm{D}\Phi|^{-1}\operatorname{div}(\psi\circ\Phi)=\operatorname{div}(|\det \mathrm{D}\Phi^{-1}|(\mathrm{D}\Phi^{-1}(\cdot))^{-1}\psi)\circ \Phi\) which holds for every vector field \(\psi\), we have \[\begin{align} \MoveEqLeft\big(\operatorname{div}\mathrm{D}_\xi f_\Phi^\infty(\cdot,\mathrm{D}\mathfrak{d}_{\Omega'}(\cdot))\big)\circ \Phi\\ &=\operatorname{div}\Big(|\det \mathrm{D}\Phi^{-1}| (\mathrm{D}\Phi^{-1}(\cdot))^{-1}\mathrm{D}_\xi f^\infty\left(\Phi^{-1}(\cdot),(\mathrm{D}\Phi^{-1}(\cdot))^{-T}\mathrm{D}\mathfrak{d}_{\Omega'}(\cdot)\right)\Big)\circ \Phi\\ &=|\det \mathrm{D}\Phi|^{-1}|\operatorname{div}\left(f^\infty(\cdot,(\mathrm{D}\Phi^{-1})^{-T}\circ \Phi\mathrm{D}\mathfrak{d}_{\Omega'}\circ \Phi )\right). \end{align}\] The claim 92 is applicable to the field \((\mathrm{D}\Phi^{-1})^{-T}\circ \Phi\mathrm{D}\mathfrak{d}_{\Omega'}\circ \Phi=\mathrm{D}(\mathfrak{d}_{\Omega'}\circ \Phi)\). Indeed, this field is trivially normal to the boundary as \(\Phi\) maps the boundary to the boundary, the interior to the interior, and is bilipschitz. This yields \[\begin{align} \MoveEqLeft(\operatorname{div}\mathrm{D}_\xi f_\Phi^\infty(\cdot,\mathrm{D}\mathfrak{d}_{\Omega'}(x)))\circ \Phi=|\det \mathrm{D}\Phi|^{-1}\operatorname{div}\mathrm{D}_\xi f^\infty(\cdot, \mathrm{D}\mathfrak{d}(\cdot)). \end{align}\] 91 follows by applying the same argument to \(-\mathfrak{d}_{\Omega'}\) and noting that this is the term in the definition 8 of the generalized curvature.

Regarding ?? , we note that \[\begin{align} (\Phi^*\mathcal{F}_{u_0\circ \Phi^{-1}})(w\circ \Phi^{-1})=\mathcal{F}_{u_0}(w) \end{align}\] by the chain rule whenever \(w\in W_{u_0}^{1,1}\). By using an approximating sequence and Proposition 6 both for \(\Phi^*\mathcal{F}_{u_0\circ \Phi^{-1}}\) and \(\mathcal{F}_{u_0}\), we see that this also holds for \(w\in BV(\Omega)\). This implies by the transformation rule and the definition of the subdifferential that \(v\in \partial\mathcal{F}_{u_0}(w)\) if and only if \(|\det \mathrm{D}\Phi^{-1}|v\circ\Phi^{-1}\in \partial(\Phi^*\mathcal{F}_{u_0\circ \Phi^{-1}})(w\circ \Phi^{-1})\). This in turn implies ?? by unraveling the definitions.

b) We first show the first two points by constructing \(\Phi=\Phi_\kappa\) depending on the (small) parameter \(\kappa>0\) and then show that the third point holds automatically if we pick \(\kappa\) small enough.

By translation, it is not restrictive to assume that \(x_0=e_d\). We let \(U'\) be a neighborhood of \(x_0\) in \(U\) such that in this neighborhood \(\partial\Omega\) can be written as a graph of a strictly positive \(C^2\)-function \(\phi\) over the unit ball in \(\mathbb{R}^{d-1}\) (after a rotation and a rescaling) and that \[\begin{align} \left\{y\,\big|\,(y_1,\dots, y_{d-1})\in B_{1}^{\mathbb{R}^{d-1}}(0)\text{ and } y_d\in (0,\,\phi(y_1,\dots, y_{d-1})))\right\}\subset \Omega, \end{align}\] as well as \[\begin{align} \left\{y\,\big|\,(y_1,\dots, y_{d-1})\in B_{1}^{\mathbb{R}^{d-1}}(0)\text{ and } y_d\in (\phi(y_1,\dots, y_{d-1}),3)\right\}\cap\Omega=\emptyset. \end{align}\] By rescaling and potentially making \(U'\) smaller, it also not restrictive to assume that \(\phi\) takes values in \((\frac{9}{10},\frac{11}{10})\). Pick \(\delta>0\) so that \(U'\) is contained in the graph over \(B_{1-\delta}^{\mathbb{R}^{d-1}}(0)\) and let \(\psi_1:\mathbb{R}^{d-1}\rightarrow \mathbb{R}_{\geq 0}\) denote a smooth, nonnegative cutoff function which equals \(1\) on \(B_{1-\delta}^{\mathbb{R}^{d-1}}(0)\) and is supported on \(B_{1-\frac{1}{2}\delta}^{\mathbb{R}^{d-1}}(0)\). Let \(\psi_2\) denote a smooth, non-negative cutoff function which equals \(1\) on \((\frac{1}{2},\frac{3}{2})\) and is supported in \((\frac{1}{4},2)\).

We can then use the following diffeomorphism \[\begin{align} &(\Phi_\kappa(y)_1,\cdots, \Phi_\kappa(y)_{d-1}):=(y_1,\cdots, y_{d-1})=:y'\\ &\Phi_\kappa(y)_d:=\psi_1(y')\psi_2(y_d)\frac{y_d}{\phi(y')}\left(\sqrt{\frac{1}{\kappa^2}-|y'|^2}-\left(\frac{1}{\kappa}-1\right)\right)+\big(1-\psi_1(y')\psi_2(y_d)\big)y_d. \end{align}\] We can extend this to the whole \(\Omega\) as the identity, which does not affect the regularity due to the cut-off. We define \(\Omega'\) (depending on \(\kappa\)) as the image of \(\Omega\) under \(\Phi_\kappa\).

It holds that \[\begin{align} \left\lVert \sqrt{\frac{1}{\kappa^2}-|y'|^2}-\frac{1}{\kappa} \right\rVert_{C^k(B_1(0))}=\left\lVert \frac{\kappa|y'|^2}{\sqrt{1-\kappa^2|y'|^2}+1} \right\rVert_{C^k(B_1(0))}\leq C(k,d) \kappa \end{align}\] for sufficiently small \(\kappa\) and \(k\in \mathbb{N}_{\geq 0}\), where the constant \(C(k,d)\) does not depend on \(\kappa\).

Therefore, we see that this is indeed a \(C^2\)-diffeomorphism for sufficiently small \(\kappa\) and that \(U'\) is mapped to the graph of \(\sqrt{\frac{1}{\kappa^2}-|y'|^2}-\frac{1}{\kappa}-1\), which a subset of the boundary of a ball of radius \(\frac{1}{\kappa}\) and this yields ?? after translating.

Furthermore, it is straightforward to verify that \(\left\lVert \Phi_\kappa \right\rVert_{C^2}\) and \(\left\lVert \Phi_\kappa^{-1} \right\rVert_{C^2}\) are bounded uniformly in small \(\kappa\). This shows the first two points.

We move on to the proof of the third property. We set \[\begin{align} c:=\inf_{x\in U}\mathop{\mathrm{essinf}}_{y\rightarrow x} H_{\partial\Omega,f}(x)-|g(y)|, \end{align}\] which is \(>0\) by assumption. It follows from part a) and the uniform estimates on \(\Phi_\kappa\) that we also have \[\begin{align} \inf_{x\in \Phi_{\kappa}(U')}\mathop{\mathrm{essinf}}_{y\rightarrow x} H_{\partial\Omega',f_{\Phi_\kappa}}(x)-|g_{\Phi_\kappa}(y)|\geq c|\det\mathrm{D} \Phi_\kappa^{-1}|\geq c' \label{essinf32re} \end{align}\tag{93}\] for some \(c'>0\), uniformly in small \(\kappa\). This also implies that for every open subset \(V\subset \Phi_{\kappa}(U')\) with a positive distance to the boundary of \(\Phi_{\kappa}(U')\), there is an \(\epsilon_1=\epsilon_1(\kappa)>0\) such that \[\begin{align} \inf_{x\in V}\mathop{\mathrm{essinf}}_{y:\, |y-x|\leq \epsilon_1}H_{\partial\Omega',f_{\Phi_\kappa}}(x)-|g_{\Phi_\kappa}(y)|\geq \frac{c'}{2},\label{act32inf} \end{align}\tag{94}\] if not, we could find a sequence of \(x_n\) for which this difference is \(\leq \frac{c'}{2}\) on a set of positive measure with distance \(\leq \frac{1}{n}\) to \(x_n\), which by compactness and continuity of \(H_{\partial\Omega',f_{\Phi_\kappa}}\) would show that 93 is wrong at any accumulation point of the \(x_n\).

We now claim that \[\begin{align} \left|\operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\pm\mathrm{D}\mathfrak{d}_{\Omega'}(y))\big|_{y=x}- \operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\mp\nu_x^{\Omega'})\big|_{y=x}\right|\xrightarrow{\kappa\searrow 0} 0,\label{curv32claim} \end{align}\tag{95}\] uniformly in \(x\in V\), where \(\mathfrak{d}_{\Omega'}\) denotes the signed distance to the boundary in \(\Phi_\kappa(\Omega)\) and \(\nu_x^{\Omega'}\) is the outer normal of \(\Omega'\) at \(x\). To show the claim, we first note that it holds that \[\begin{align} |\mathrm{D}^2\mathfrak{d}_{\Omega'}(x)|\leq C(d) \kappa,\label{kappa32bd} \end{align}\tag{96}\] where the constant \(C(d)\) does not depend on \(\kappa\), for \(x\in \Phi(U')\), since \(\Phi(U')\) is a subset of a sphere of curvature \(\kappa\) by construction.

We can then estimate, using the chain rule and that \(\nu_x^{\Omega'}=-\mathrm{D}\mathfrak{d}_{\Omega'}(x)\) does not depend on \(y\) \[\begin{align} \MoveEqLeft\left|\operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\pm\mathrm{D}\mathfrak{d}_{\Omega'}(y))\big|_{y=x}- \operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\mp\nu_x^{\Omega'})\big|_{y=x}\right|\\ &\leq C(d)|\mathrm{D}_{\xi}^2f_{\Phi_\kappa}^\infty(x,\mathrm{D}\mathfrak{d}_{\Omega'}(x))||\mathrm{D}_y^2\mathfrak{d}_{\Omega'}(x))|\\ &\leq C(d)\left\lVert f_{\Phi_\kappa}^\infty(x,\cdot) \right\rVert_{C_\xi^2(B_2(0)\backslash B_{\frac{1}{2}}(0))}|\mathrm{D}^2\mathfrak{d}_{\Omega'}(x)|\leq C(d)\kappa\xrightarrow{\kappa\rightarrow 0} 0, \end{align}\] where the last step follows from 96 and the fact that, thanks to step a) and the uniform estimates on \(\Phi_\kappa\), the norm \(\left\lVert f_{\Phi_\kappa}^\infty(y,\cdot) \right\rVert_{C_\xi^2(B_2(0)\backslash B_{\frac{1}{2}}(0))}\) is bounded uniformly in \(\kappa\).

Now, we see from 95 and the the definition 8 of the generalized curvature that, if we choose \(\kappa\) sufficiently small, it holds that \[\begin{align} \pm\operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\pm\nu_x^{\Omega'})\big|_{y=x}-H_{\partial\Omega',f_{\Phi_\kappa}}(x)\geq-\frac{c'}{8} \end{align}\] for all \(x\in V\). It further follows from the fact that \(f_{\Phi_\kappa}^\infty\) is \(C_y^1C_\xi^1\), that, after potentially lowering \(\epsilon_1\) (which can be done uniformly in \(x\in V\)), we also have \[\begin{align} \mathop{\mathrm{essinf}}_{y:\, |y-x|\leq \epsilon_1}\pm\operatorname{div}_y \mathrm{D}_\xi f_{\Phi_\kappa}^\infty(y,\pm\nu_x^{\Omega'})-H_{\partial\Omega',f_{\Phi_\kappa}}(x)\geq-\frac{c'}{4} \end{align}\] which, together with 94 , yields ?? .

5.2 Proof of the geometric Lemmata 8 and 13↩︎

We only prove Lemma 13 in dimension \(d\geq 2\). The Lemma 8 is merely a special case of this if \(d\geq 2\) and trivial if \(d=1\), while the Lemma 13 is a void statement if \(d=1\) as the boundary of a \(1\)-dimensional set can not be uniformly convex.

5.3 Proof of the Lemmata 9, 10 and 14↩︎

We prove all three Lemmata together: We first construct \(\hat{u}_{x}\) and then show the bounds for \(\hat{u}_{x}\) and \(\bar{u}_x\) in parallel.

Construction of \(\hat{u}_{x}\): First observe that \(\hat{z}_x\) is \(C^1\) by the assumption [A2] and its definition 44 , hence the characteristics of \(\hat{z}_{x}\) exist in a neighborhood of the boundary (cf.[65]). We furthermore have \[\begin{align} - \langle \hat{z}_x(y),\nu_x \rangle=\langle \mathrm{D}_\xi f^\infty(y,-\nu_x),-\nu_x \rangle=f^\infty(y,-\nu_x)\gtrsim 1,\label{prod32est} \end{align}\tag{103}\] thanks to 29 and 26 .

This implies that no characteristic can intersect \(E_x^\epsilon\) in more than one point and, since by continuity, for small \(\epsilon>0\) it also holds \(\langle \hat{z}_x,\nu_y \rangle<0\) in a neighborhood of size \(O(1)\) of \(x\), that no characteristic can intersect \(C_x^\epsilon\) twice.

Therefore, there is a well-defined map \(\Psi:\overline{D_x^\epsilon}\rightarrow \overline{C_x^\epsilon}\times [0,1]\), depending on \(x\) and \(\epsilon\), which is a \(C^1\)-diffeomorphism onto its image, defined through its inverse fulfilling the ODE \[\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\Psi^{-1}(y,t)=\hat{z}_{x}(\Psi^{-1}(y,t))\quad\text{and} \quad \Psi^{-1}(y,0)=y.\label{charode} \end{align}\tag{104}\] We extend \(u_0\) to a function on \(\overline{C_x^\epsilon}\times [0,1]\) which does not depend on the last variable. We denote the extension by \(u_0^e\).

We then set \[\begin{align} \hat{u}_x=u_0^e\circ \Psi.\label{def32pb} \end{align}\tag{105}\] This fulfills ?? and ?? trivially if \(u_0\) is \(C^1\), the general case follows from the fact that the Anzelotti pairing with \(\hat{z}_{x}\) is just the pointwise product since \(\hat{z}_{x}\) is \(C^1\), once we have established that \(\hat{u}_{x}\in BV\).

Estimates on \(\Psi\): We next have to show the estimate ?? , and in the case of Lemma 14, the estimates ?? and ?? . In order to do so, we will first show that \(\Psi\) is bi-Lipschitz uniformly in \(x\) and \(\epsilon\), which is quite standard, but we provide the proof here anyway for the sake of completeness.

We start with \(\left\lVert \Psi \right\rVert_{C^1}\), in order to bound it, it suffices to estimate the Lipschitz constants of the two components of \(\Psi\) uniformly.

The (backwards) characteristic starting at \(y\in D_x^\epsilon\) can be written as \(\Psi^{-1}(\Psi_1(y),\Psi_2(y)-t)\), where the indices denote the components. As \(\hat{z}_x\) is \(C^1\), uniformly in \(x\) and \(\epsilon\), we therefore have by Gronwall, applied to 104 \[\begin{align} \left|\Psi^{-1}(\Psi_1(y),\Psi_2(y)-t)-\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t)\right|\leq e^{\left\lVert \hat{z}_x \right\rVert_{W^{1,\infty}}t}|y-y'|\label{gw1} \end{align}\tag{106}\] for \(t\leq \min(\Psi_2(y),\Psi_2(y'))\), where \(y,y'\in D_x^\epsilon\) are arbitrary. Now, if we take \(t_y=\Psi_2(y)\leq \Psi_2(y')\), which is not restrictive by symmetry, then it holds that \(\Psi^{-1}(\Psi_1(y),\Psi_2(y)-t_y)=\Psi_1(y)\in C_x^\epsilon\) and hence 106 gives \[\begin{align} \mathop{\mathrm{dist}}(\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t_y),C_x^\epsilon)\lesssim |y-y'|, \end{align}\] where we could absorb the exponential factor into the implicit constant because \(t_y\lesssim 1\). By the estimate 103 and the fact that the curve \(\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t)\) is the characteristic through \(y'\), we have \[\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\mathop{\mathrm{dist}}\big(\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t),C_x^\epsilon\big)\leq -\min_{x'\in C_x^\epsilon}\langle \hat{z}_x,\nu_{x'} \rangle\lesssim -1, \end{align}\] this implies that the time \(\Psi_2(y')\) at which the curve \(\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t)\) hits \(C_x^\epsilon\) is \(\lesssim |y-y'|\) apart from \(t_y\), yielding \[\begin{align} |t_y-\Psi_2(y')|=|\Psi_2(y)-\Psi_2(y')|\lesssim |y-y'|. \end{align}\] By the boundedness of \(\hat{z}_x\), combined with 106 \[\begin{align} \MoveEqLeft|\Psi_1(y)-\Psi_1(y')|=|\Psi^{-1}(\Psi_1(y),0)-\Psi^{-1}(\Psi_1(y'),0)|\\ &\leq \left|\Psi^{-1}(\Psi_1(y),\Psi_2(y)-t_y)-\Psi^{-1}(\Psi_1(y'),\Psi_2(y')-t_y)\right|+\left\lVert \hat{z}_x \right\rVert_{L^\infty}\left|\Psi_2(y')-t_y\right|\\ &\lesssim |y-y'|, \end{align}\] proving that \(\left\lVert \Psi \right\rVert_{C^1}\lesssim 1\), uniformly in \(x\).

Similarly, it holds that \[\begin{align} &|\Psi^{-1}(y,t)-\Psi^{-1}(y,t')|\leq \left\lVert \hat{z}_x \right\rVert_{L^\infty}|t-t'|\lesssim |t-t'|\\ &|\Psi^{-1}(y,t)-\Psi^{-1}(y',t)|\leq e^{\left\lVert \hat{z}_x \right\rVert_{W^{1,\infty}}t}|y-y'|, \end{align}\] showing that \(\left\lVert \Psi^{-1} \right\rVert_{C^1}\lesssim 1\) uniformly in \(x\) and \(\epsilon\).

The same argument can also be made for the projection in direction \(\zeta_x\), which is used in Lemma 14, where one uses 81 in place of 103 . We omit the details.

Proof of the bounds: We only show the \(BV\)-bounds for \(\hat{u}_x\), the \(BV\)-bounds for \(\bar{u}_x\) proceed in the same way, since one has the same estimates for the characteristics in both cases. The fact that the \(\bar{u}_x\in L^2\) is a trivial consequence of the construction and the assumption that \(u_0\in L^2\) in the vectorial setting.

We first note that the image of \(D_x^\epsilon\) under \(\Psi\) actually lies in \(C_x^\epsilon\times (0,C\epsilon^2)\) for some \(C\) bounded uniformly in \(x\) and \(\epsilon\), since \(D_x^\epsilon\) is contained in a strip of width \(\lesssim \epsilon^2\) in direction \(\nu_x\) by definition and since the characteristics have a speed \(\gtrsim 1\) in the \(\nu_x\)-direction by the estimate 103 .

By e.g.the chain rule and smooth approximation (see e.g.[53] for more details), it holds \(\hat{u}_x\in BV\) and that \[\begin{align} \mathrm{D}^a\hat{u}_x(y)=\mathrm{D}^au_0^e(\Psi(y)) \mathrm{D}\Psi(y) \end{align}\] and \[\begin{align} (\mathrm{D}^s\hat{u}_x)(A)=\int_{\Psi(A)}|\det \mathrm{D}\Psi|^{-1} \mathrm{D}\Psi^T\,\mathrm{d}\mathrm{D}^su_0^e, \end{align}\] for any Borel set \(A\). Therefore, using that both the \(C^1\)-norms of \(\Psi\) and \(\Psi^{-1}\) are uniformly bounded by the previous step, we see that for any nondecreasing function \(i_0:\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}\) with \(i_0(2t)\lesssim i_0(t)\) it holds that \[\begin{align} \MoveEqLeft\int_{D_x^\epsilon} i_0(|\mathrm{D}^a\hat{u}_x(y)|)\,\mathrm{d}y\lesssim \int_{C_x^\epsilon\times (0,C\epsilon^2) }i_0(|\mathrm{D}^au_0^e(y)|)\,\mathrm{d}(\mathcal{H}^{d-1}\times \mathcal{L}^1)(y)\nonumber\\ & \approx \epsilon^2\int_{C_x^\epsilon}i_0(|\mathrm{D}^au_0(y)|)\,\mathrm{d}\mathcal{H}^{d-1}(y)\label{i32est} \end{align}\tag{107}\] and \[\begin{align} |\mathrm{D}^s(\hat{u}_x)|(D_x^\epsilon)\lesssim \epsilon^2|\mathrm{D}^su_0|(C_x^\epsilon) \end{align}\] since \(u_0^e\) is a constant extension of \(u_0\) onto a strip of \(\lesssim \epsilon^2\).

In particular, this shows the estimate ?? by setting \(i_0=|\cdot|\). By the de La Vallée Poussin Lemma [66]⁸, we can pick \(i:\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}\) non-decreasing, such that \(i(2t)\lesssim i(t)\) and \[\begin{align} \lim_{t\rightarrow+\infty} \frac{i(t)}{t}=+\infty \end{align}\] and \[\begin{align} \int_{U'} i(|\mathrm{D}^au_0|(y))\,\mathrm{d}\mathcal{H}^{d-1}(y)<\infty. \end{align}\] Setting \(i=i_0\) in 107 then shows the Lemma.

5.4 Proof of the Lemmata 11 and 16↩︎

Part a) We first focus on the statement for \(z\) in Lemma 11 a), and show Lemma 16 a) afterwards. We first partially integrate \(z-\hat{z}_{x}\), using 23 , against the function \(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\), which vanishes on \(E_x^\epsilon\) by its definition.

This yields that \[\begin{align} \MoveEqLeft\int_{D_x^\epsilon} \langle z-\hat{z}_x,\nu_x \rangle\,\mathrm{d}y=-\int_{D_x^\epsilon} \operatorname{div}(z-\hat{z}_x)\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\,\mathrm{d}y\\ &+\int_{C_x^\epsilon}[z-\hat{z}_x,\nu_y]\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\,\mathrm{d}\mathcal{H}^{d-1}(y).\label{pint32lem} \end{align}\tag{108}\] We next realize that, thanks to \(f^\infty\) being \(C^1\) by [A2], and because \(|\nu_x-\nu_y|\lesssim\epsilon\) in \(C_x^\epsilon\), as it is a spherical cap of base radius \(\leq\epsilon\), it holds that \[\begin{align} \left|\mathrm{D}_\xi f^\infty(y,-\nu_y)-\hat{z}_x\right|= \left|\mathrm{D}_\xi f^\infty(y,-\nu_y)-\mathrm{D}_\xi f^\infty(y,-\nu_x)\right|\lesssim \epsilon. \end{align}\] It holds that \[\begin{align} |\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2|\leq \frac{1}{2}\kappa\epsilon^2\label{kappaeps} \end{align}\tag{109}\] on \(D_x^\epsilon\) and \(C_x^\epsilon\), by definition (see 42 ) and since the scalar product is maximized at \(y=x\) by convexity. Now, using the assumed inequality ?? , we see that \[\begin{align} \MoveEqLeft\left|\int_{C_x^\epsilon}[z-\hat{z}_x,\nu_y]\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\,\mathrm{d}\mathcal{H}^{d-1}(y)\right|\\ &\leq \int_{C_x^\epsilon}\left|[\mathrm{D}_\xi f^\infty(y,-\nu_y)-\hat{z}_x,\nu_y]\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\right|\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &\quad+\int_{C_x^\epsilon}\mathbb{1}_{[z,\nu_y](y)\neq \langle \mathrm{D}_\xi f^\infty(y,- \nu_y),\nu_y \rangle}\left|[z-\hat{z}_x,\nu_y]\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\right|\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &\lesssim \epsilon^3\mathcal{H}^{d-1}(C_x^\epsilon)+(\left\lVert z \right\rVert_{L^\infty(D_x^\epsilon)}+\left\lVert \hat{z}_{x} \right\rVert_{L^\infty(D_x^\epsilon)})\epsilon^2\delta\mathcal{H}^{d-1}(C_x^\epsilon)\\ &\lesssim (\epsilon+\delta)\epsilon^{d+1}, \end{align}\] where we also used the bound ?? on the measure of \(C_x^\epsilon\) and that \(\left\lVert z \right\rVert_{L^\infty(D_x^\epsilon)}+\left\lVert \hat{z}_{x} \right\rVert_{L^\infty(D_x^\epsilon)}\lesssim 1\) by 27 , 38 and 46 .

Combining 108 and 109 to estimate the other integral, this shows that \[\begin{align} \MoveEqLeft\int_{D_x^\epsilon} \langle z-\hat{z}_x,\nu_x \rangle\,\mathrm{d}y\lesssim\epsilon^2\left(\int_{D_x^\epsilon}|\operatorname{div}z|\,\mathrm{d}y+(\epsilon+\delta)\epsilon^{d-1}\right)+\left\lVert \operatorname{div}\hat{z} \right\rVert_{L^\infty}\mathcal{L}^d(D_x^\epsilon)\epsilon^2\\ & \lesssim\epsilon^2\left(\int_{D_x^\epsilon}|\operatorname{div}z|\,\mathrm{d}y+(\epsilon+\delta)\epsilon^{d-1}\right). \end{align}\] where we also used that \(\left\lVert \operatorname{div}\hat{z}_x \right\rVert_{L^\infty}\lesssim \left\lVert f^\infty \right\rVert_{C^1_yC^1_\xi}\lesssim 1\) by its Definition 44 and the Assumption [A2] and that \(\mathcal{L}^d(D_x^\epsilon)\lesssim \epsilon^{d+1}\) by ?? .

We note that by the definition 44 of \(\hat{z}_x\), the pointwise constraint ?? on \(z\) and the inequality ?? , it holds that \[\begin{align} \langle z(y)-\hat{z}_x(y),\nu_x \rangle=\langle \mathrm{D}f^\infty(y,-\nu_x)-z(y),-\nu_x \rangle\gtrsim |z-\hat{z}_x|^2, \end{align}\] yielding ?? .

The vector-valued case in Lemma 16 is very similar. We partially integrate against the weight \(v\otimes(\langle y-x,\nu_x \rangle+\epsilon^2)\), which vanishes on \(\tilde{E}_x^\epsilon\) by definition, and use the fact that the divergences vanish to see that \[\begin{align} \MoveEqLeft\int_{\tilde{D}_x^\epsilon} \langle z-\mu_1(v)\otimes \zeta_x,v\otimes\nu_x \rangle\,\mathrm{d}y= \int_{\tilde{C}_x^\epsilon}\langle [z-\mu_1(v)\otimes \zeta_x,\nu_y],v \rangle\left(\langle y-x,\nu_x \rangle+\epsilon^2\right)\,\mathrm{d}\mathcal{H}^{d-1}(y). \end{align}\] Since \(\mu_2\) inherits the \(C^1\)-regularity of \(f^\infty\), we also have \[\begin{align} |\mu_1(v)\otimes \zeta_y-\mu_1(v)\otimes \zeta_x|=\left|\mu_1(v)\otimes \mu_2(\nu_y)-\mu_1(v)\otimes\mu_2(\nu_x)\right|\lesssim \epsilon \end{align}\] on \(\tilde{C}_x^\epsilon\). Similarly as above, the assumed inequality ?? then shows that \[\begin{align} \MoveEqLeft\left|\int_{\tilde{C}_x^\epsilon}\langle [z-\mu_1(v)\otimes\mu_2(\nu_x),\nu_y],v \rangle\left(\langle y-x,\nu_x \rangle+\epsilon^2\right)\,\mathrm{d}\mathcal{H}^{d-1}(y)\right|\\ &\leq \int_{\tilde{C}_x^\epsilon}\big|\langle [\mu_1(v)\otimes \zeta_y-\mu_1(v)\otimes \zeta_x,\nu_y],v \rangle\big|\left|\langle y-x,\nu_x \rangle+\epsilon^2\right|\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &\quad+\int_{\tilde{C}_x^\epsilon}\mathbb{1}_{[z,\nu](y)\in \langle \zeta_y,\nu_y \rangle B_\delta(\mu_1(v))}|[z,\nu_y]-\mu_1(v)\langle \zeta_y,\nu_y \rangle||v||\langle y-x,\nu_x \rangle+\epsilon^2|\,\mathrm{d}\mathcal{H}^{d-1}(y) \\ &\quad+\int_{\tilde{C}_x^\epsilon}\mathbb{1}_{[z,\nu](y)\notin \langle \zeta_y,\nu_y \rangle B_\delta(\mu_1(v))}\left|\langle [z-\mu_1(v)\otimes \zeta_x,\nu_y],v \rangle\right|\left|\langle y-x,\nu_x \rangle+\epsilon^2\right|\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &\lesssim \epsilon^3\mathcal{H}^{d-1}(\tilde{C}_x^\epsilon)+\delta\epsilon^2\mathcal{H}^{d-1}(\tilde{C}_x^\epsilon)+(\left\lVert z \right\rVert_{L^\infty(\tilde{D}_x^\epsilon)}+1)\epsilon^2\delta\mathcal{H}^{d-1}(\tilde{C}_x^\epsilon)\\ &\lesssim (\epsilon+\delta)\epsilon^{d+1}, \end{align}\] where we have used the bound ?? on the measure of \(\tilde{C}_x^\epsilon\), as well as the fact that \(v\) is a unit vector and that \(z\) and the \(\mu\)’s are \(\lesssim 1\) by 38 and the assumption on \(f\). Therefore \[\begin{align} \int_{\tilde{D}_x^\epsilon}\langle z-\mu_1(v)\otimes \zeta_x,v\otimes \nu_x \rangle\,\mathrm{d}y\lesssim (\epsilon+\delta)\epsilon^{d+1} \end{align}\] from which one concludes ?? from Lemma 2 as above.

5.5 Proof of Lemma 11 b) and Lemma 16 b)↩︎

Both proofs are quite similar, we start with the one of Lemma 11 b).

We make use of the pointwise characterisation of \(z\) in Lemma 3. It holds that \[\begin{align} (z-\hat{z}_{x},\mathrm{D}u)&=\langle z-\hat{z}_{x},\mathrm{D}^au \rangle\mathcal{L}^d+ \left(f^\infty(\cdot,\frac{\mathrm{d}\mathrm{D}^su}{\mathrm{d}|\mathrm{D}^su|}(\cdot))-\langle \hat{z}_{x},\frac{\mathrm{d}\mathrm{D}^su}{\mathrm{d}|\mathrm{D}^su|}(\cdot) \rangle\right)|\mathrm{D}^su|,\label{exp32diff} \end{align}\tag{110}\] here we have used that the pairing is linear and that for \(\hat{z}_{x}\) it is just the classical product because \(\hat{z}_{x}\) is \(C^1\) by 45 .

The prefactor of the singular part is non-negative and therefore the singular part may be discarded. Indeed this follows from the fact that \(\hat{z}_{x}\in \mathop{\mathrm{Ran}}(\partial_\xi f^\infty(y,\cdot))\) by definition (see 44 ) and the property 29 .

Furthermore, if \(f(x,\cdot)\) is differentiable, which is e.g.the case for \(|\mathrm{D}^au|>R\) for the \(R\) from Assumption [A2] or if \(f=f^\infty\), then it holds that \(z(y)=\mathrm{D}_\xi f(y,\mathrm{D}^a u(y))\) by ?? . We note that in the special case in which \(f\) is positively \(1\)-homogeneous, the absolutely continuous part of 110 is also non-negative as a consequence of 29 and ?? and because then \(f=f^\infty\).

Regarding the absolutely continuous part, we note that among all values in \(\mathop{\mathrm{Ran}}(\partial_\xi f^\infty(y,\cdot))\) the one maximizing the scalar product with \(\mathrm{D}^au(y)\) is \(\mathrm{D}f^\infty(y,\mathrm{D}^au(y))\) by e.g.@eq:doal and hence, if \(|\mathrm{D}^au|\geq R\) we can estimate, \[\begin{align} \MoveEqLeft\langle \mathrm{D}_\xi f(y,\mathrm{D}^au(y))-\hat{z}_{x},\mathrm{D}^au(y) \rangle\geq \langle \mathrm{D}_\xi f(y,\mathrm{D}^au(y))-\mathrm{D}_\xi f^\infty(y,\mathrm{D}^au(y)),\mathrm{D}^au(y) \rangle\tag{111}\\ &=-\left\langle\int_{|\mathrm{D}^au(y)|}^{\infty}\langle \mathrm{D}_\xi^2f(y,s\frac{\mathrm{D}^au(y)}{|\mathrm{D}^au(y)|})\frac{\mathrm{D}^au(y)}{|\mathrm{D}^au(y)|},\frac{\mathrm{D}^au(y)}{|\mathrm{D}^au(y)|} \rangle\,\mathrm{d}s,\,\mathrm{D}^au(y)\right\rangle,\tag{112} \end{align}\] where we used 30 and the fundamental theorem of calculus in the second step. If we set \[\begin{align} \mathfrak{f}(t):=\sup_{y\in \Omega,\, |\xi|\geq t}|\xi|\int_{|\xi|}^\infty|\langle \mathrm{D}^2f(y, s\frac{\xi}{|\xi|})\frac{\xi}{|\xi|},\frac{\xi}{|\xi|} \rangle|\,\mathrm{d}s, \end{align}\] which is exactly the expression from 7 in the Assumption [A2], then 112 implies that \[\begin{align} \langle \mathrm{D}_\xi f(y,\mathrm{D}^au(y))-\hat{z}_{x},\mathrm{D}^au(y) \rangle\geq -\mathfrak{f}(|\mathrm{D}^au(y)|) \end{align}\] for \(|\mathrm{D}^au(y)|\geq R\). To conclude, we split the integral of the absolutely continous part in 110 into the parts where \(|\mathrm{D}^au|\geq \max(R,|z-\hat{z}_{x}|^{-\frac{1}{2}})\) and where the opposite inequality holds, this then gives \[\begin{align} \MoveEqLeft\int_{D_x^\epsilon}|(z-\hat{z}_{x},\mathrm{D}u)_-|\leq \int_{D_x^\epsilon}\mathbb{1}_{|\mathrm{D}^au|\leq \max(R,|z-\hat{z}_{x}|^{-\frac{1}{2}})} |z-\hat{z}_{x}||\mathrm{D}^au|\,\mathrm{d}y+\int_{D_x^\epsilon} \mathfrak{f}(\max(R,|z-\hat{z}_{x}|^{-\frac{1}{2}}))\,\mathrm{d}y\\ &\leq\int_{D_x^\epsilon} R|z-\hat{z}_{x}|+|z-\hat{z}_{x}|^\frac{1}{2}+\mathfrak{f}(\max(R,|z-\hat{z}_{x}|^{-\frac{1}{2}}))\,\mathrm{d}y. \end{align}\] We then take \[\begin{align} k(t)=\min\big(C,Rt+t^\frac{1}{2}+\mathfrak{f}(\max(R,t^{-\frac{1}{2}}))\big), \end{align}\] where \(C\) is some large fixed constant such that \(C>\left\lVert z \right\rVert_{L^\infty}+\left\lVert \hat{z}_{x} \right\rVert_{L^\infty}\) (which exists by 38 and 46 ). Since \(|z-\hat{z}_{x}|\leq C\), this cutoff does not affect the previous inequalities. The function \(k\) goes to \(0\) at \(0\) because \(\lim_{t\rightarrow \infty} \mathfrak{f}(t)=0\) by the Assumption [A2].

Proof of Lemma 16 b) As in 111 above, we see from the pointwise characterisation in Lemma 4 of \(z\) that \[\begin{align} \langle z(y)-\bar{z}_x,\mathrm{D}^au(y) \rangle\geq \langle \mathrm{D}_\xi f(\mathrm{D}^au(y))-\mathrm{D}_\xi f^\infty(\mathrm{D}^au(y)),\mathrm{D}^au(y) \rangle.\label{easy32est} \end{align}\tag{113}\] By the same argument as earlier\[\begin{align} \bar{\mathfrak{f}}(t):=\max_{ |\xi|\geq t}|\xi|\int_{|\xi|}^\infty|\langle \mathrm{D}^2f( s\frac{\xi}{|\xi|})\frac{\xi}{|\xi|},\frac{\xi}{|\xi|} \rangle|\,\mathrm{d}s \end{align}\] fulfills \(\lim_{t\rightarrow +\infty}\bar{\mathfrak{f}}(t)=0\) by the Assumption [B2] and \[\begin{align} \label{lim32df} \langle \mathrm{D}_\xi f(\xi)-\mathrm{D}_\xi f^\infty(\xi),\xi \rangle\geq -\bar{\mathfrak{f}}(\xi). \end{align}\tag{114}\] We now claim that for every \(t>0\) there is an \(A(t)\) with \(\lim_{t\searrow 0} A(t)=0\) such that whenever there are \(\xi\in \mathbb{R}^{n\times d}\) and \(v_1\in S^{n-1}\) and \(v_2\in S^{d-1}\) fulfilling \[\begin{align} \left|\mu_1(v_1)\otimes \mu_2(v_2)-\mathrm{D}_\xi f(\xi)\right|\leq t, \quad \text{then}\quad \langle \mathrm{D}_\xi f(\xi)-\mathrm{D}_\xi f^\infty(\xi),\xi \rangle\geq -A(t).\label{A32stat} \end{align}\tag{115}\] Indeed, if this were not the case, there would be sequences \(\xi_m\in \mathbb{R}^{n\times d}\) and \(v_{1,m}\in S^{n-1}\) and \(v_{2,m}\in S^{d-1}\), depending on \(t\), with \[\begin{align} \mu_1(v_{1,m})\otimes \mu_2(v_{2,m})-\mathrm{D}_\xi f(\xi_m)\rightarrow 0. \end{align}\] and \[\begin{align} \limsup_{m\rightarrow +\infty} \langle \mathrm{D}_\xi f(\xi_m)-\mathrm{D}_\xi f^\infty(\xi_m),\xi_m \rangle<0.\label{bu1} \end{align}\tag{116}\] However, by 114 , this is only possible if \(\xi_m\) is a bounded sequence and hence these sequences have a limit, at least along a subsequence. Furthermore, \(|\xi_m|\) must have a positive lower bound by 116 and because the derivatives of \(f\) and \(f^\infty\) are uniformly bounded (see e.g.@eq:ball32subdiff ).

Since \(\mu_1,\mu_2,\mathrm{D}_\xi f\) and \(\mathrm{D}_\xi f^\infty\) are all continuous (away from \(0\)) by assumption, this yields that there are \(v_{1,\infty},v_{2,\infty}\) and \(\xi_{\infty}\neq 0\) with \[\begin{align} \mu_1(v_{1,\infty})\otimes \mu_2(v_{2,\infty })=\mathrm{D}_\xi f^\infty(v_{1,\infty}\otimes v_{2,\infty})=\mathrm{D}_\xi f(\xi_\infty) \end{align}\] and \[\begin{align} \mathrm{D}_\xi f(\xi_\infty)\neq \mathrm{D}_\xi f^\infty(\xi_\infty), \end{align}\] which is a contradiction to the last point in Assumption [B2].

The proof of the Lemma now proceeds by first noting that the singular part of the measure is non-negative for the same reason as above in the previous proof with the slight change that, because \(\bar{z}_x\) is not \(C^1\), one has to use that by 21 it holds that \[\begin{align} \frac{\mathrm{d}(\bar{z}_x,\mathrm{D}u)}{\mathrm{d}|\mathrm{D}^s u|}(y)\in\left\{z_0:\frac{\mathrm{d}\mathrm{D}^s u}{\mathrm{d}|\mathrm{D}^s u|}(y)\,\big|\, z_0\in \overline{\mathop{\mathrm{Ran}}(\partial_\xi f)}\right\} \end{align}\] (up to a zero set) which still yields the non-negativity of the singular part by 29 .

We now estimate the integral over the absolutely continuous part, using ?? , as \[\begin{align} \MoveEqLeft\int_{\tilde{D}_x^\epsilon}\left|\langle z(y)-\bar{z}_x(y),\mathrm{D}^a u(y) \rangle_-\right|\,\mathrm{d}y\geq \int_{\tilde{D}_x^\epsilon}\langle \mathrm{D}_\xi f(\mathrm{D}^au(y))-\mathrm{D}_\xi f^\infty(\mathrm{D}^au(y)),\mathrm{D}^au(y) \rangle\,\mathrm{d}y\\ &\geq-\int_{\tilde{D}_x^\epsilon}A(|z-\mu_1(v)\otimes \zeta_x|)\,\mathrm{d}y. \end{align}\] Here we have used 113 in the first step, as well as 115 in the second step.

This shows the Lemma with \(k(t)=A(t)\).

5.6 Proof of the trace Lemma 12↩︎

We note first that for any \(w\in BV(D_x^\epsilon)\) we can partially integrate \(|w|\) against the function \(\epsilon^{-2}(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2)\), which vanishes on \(E_x^\epsilon\) by definition (see 43 ), hence \[\begin{align} \MoveEqLeft\epsilon^{-2}\int_{D_x^\epsilon}|w|\,\mathrm{d}y=\int_{C_x^\epsilon}\epsilon^{-2}\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\langle \nu_x,\nu_y \rangle|w|(y)\,\mathrm{d}\mathcal{H}^{d-1}(y)\\ &-\int_{D_x^\epsilon}\epsilon^{-2}\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\nu_x\cdot\mathrm{D}|w|(y). \end{align}\] Since \(D_x^\epsilon\) is contained in a strip of width \(\frac{1}{2}\kappa\epsilon^2\) in direction \(\nu_x\) by definition, the second integral on the right-hand side is \(\lesssim |\mathrm{D}|w||(D_x^\epsilon)\leq|\mathrm{D}w|(D_x^\epsilon)\). In the first integral, on the other hand, it holds that \(\langle \nu_x,\nu_y \rangle\gtrsim 1\) by continuity of the normal, and in each \(C_x^{\rho\epsilon}\) it holds that \[\begin{align} \epsilon^{-2}\left(\langle y-x,\nu_x \rangle+\frac{1}{2}\kappa\epsilon^2\right)\geq \frac{1}{2}\kappa(1-\rho^2)\gtrsim_{\rho} 1 \end{align}\] by some straightforward geometric computations. Hence \[\begin{align} \int_{C_x^{\rho\epsilon}}|w|\,\mathrm{d}\mathcal{H}^{d-1}\lesssim_\rho\epsilon^{-2}\int_{D_x^\epsilon}|w|\,\mathrm{d}y+|\mathrm{D}w|(D_x^\epsilon), \end{align}\] which shows ?? . ?? follows in the same way.

For ?? on the other hand, we estimate \[\begin{align} \int_{D_x^\epsilon}|w -h|\,\mathrm{d}y\leq\int_{D_x^\epsilon}|w-h(x)|+|h-h(x)|\,\mathrm{d}y,\label{h32split} \end{align}\tag{117}\] the first integrand is again dealt with using ?? since adding a constant does not change the total variation, yielding \[\begin{align} \epsilon^{-2}\int_{D_x^\epsilon}|w-h(x)|\,\mathrm{d}y\lesssim \int_{C_x^\epsilon}|w-h(y)|\,\mathrm{d}\mathcal{H}^{d-1}(y)+\int_{C_x^\epsilon}|h(x)-h(y)|\,\mathrm{d}\mathcal{H}^{d-1}(y)+|\mathrm{D}w|(D_x^\epsilon), \end{align}\] where the second term is estimated as \[\begin{align} \int_{C_x^\epsilon}|h(x)-h(y)|\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq \omega(\epsilon)\mathcal{H}^{d-1}(C_x^\epsilon)\lesssim \omega(2\epsilon)\epsilon^{d-1}, \end{align}\] since \(C_x^\epsilon\) is contained in a ball of radius \(\epsilon\) by definition and where we used the bound ?? . The second summand in 117 is estimated, using ?? , as \[\begin{align} \int_{D_x^\epsilon}|h-h(x)|\,\mathrm{d}y\leq \omega(2\epsilon)\mathcal{L}^{d}(D_x^\epsilon)\approx\omega(2\epsilon)\epsilon^{d+1}, \end{align}\] since \(\mathop{\mathrm{Diam}}(D_x^\epsilon)\leq 2\epsilon\) by definition.

6 Proof of Corollary 5↩︎

We use the following Lemma.

The corollary follows directly from this Lemma, since we know from Proposition 8 that for a given \(u_0\in L^1(\Omega)\) a minimizer of the relaxed problem exists and we know from Theorem 1 and the Lemma that \(T_b(u)=T_b(u_0)\) on \(\partial\Omega\) for every \(b\), yielding that in fact \(u=u_0\) on \(\partial\Omega\).

7 Proof of Theorem 4↩︎

The theorem is a consequence of the following Proposition.

8 Proof of Lemma 2↩︎

We define the dual of \(f^\infty(y,\cdot)\) as \[\begin{align} f_\star(y,\xi^*)=\sup_{\xi:\, f^\infty(y,\xi)\leq 1} \langle \xi^*,\xi \rangle.\label{def32pol} \end{align}\tag{125}\] By classical convex analysis (see e.g.[69]), this can equivalently be written as \[\begin{align} f_\star(y,\xi^*)=\inf \left\{t\,\big|\, \xi^*=\frac{1}{t}\mathrm{D}_\xi f^\infty(y,\xi) \text{ for some \xi}\right\}.\label{desc32d32ball} \end{align}\tag{126}\] Furthermore, by applying the bipolar theorem [56] to the set \(\{f^\infty(x,\cdot)\leq 1\}\) we see that, \[\begin{align} f^\infty(y,\xi)=\sup_{\xi^*: f_\star(y,\xi^*)\leq 1}\langle \xi^*,\xi \rangle.\label{bipo} \end{align}\tag{127}\] Finally, by the growth bound on \(f\) in [A2] and the induced bound 25 for \(f^\infty\), we have \[\begin{align} f^\infty(y,\xi)\approx |\xi|\label{l32bd32fin} \end{align}\tag{128}\] and \[\begin{align} f_\star(y,\xi^*)\approx |\xi^*|\label{bd32pol} \end{align}\tag{129}\] uniformly in \(y\). Then, even though these quantities are not norms as they are not necessarily even, the analogue of the statement “uniform smoothness implies uniform convexity of the dual norm” does still hold here, more precisely, for any given \(t\in (0,1)\) we can estimate \[\begin{align} \MoveEqLeft\sup\left\{ f_\star(y,\xi_1^*+\xi_2^*)+tf_\star(y,\xi_1^*-\xi_2^*)\,\big|\, f_\star(y,\xi_1^*),f_\star(y,\xi_2^*)\leq 1\right\}\\ &=\sup\left\{ \langle \xi_1,\xi_1^*+\xi_2^* \rangle+t\langle \xi_2,\xi_1^*-\xi_2^* \rangle\,\big|\, f_\star(y,\xi_1^*),f_\star(y,\xi_2^*),f^\infty(y,\xi_1),f^\infty(y,\xi_2)\leq 1\right\}\\ &=\sup\left\{ \langle \xi_1+t\xi_2,\xi_1^* \rangle+\langle \xi_1-t\xi_2,\xi_2^* \rangle\,\big|\, f_\star(y,\xi_1^*),f_\star(y,\xi_2^*),f^\infty(y,\xi_1),f^\infty(y,\xi_2)\leq 1\right\}\\ &=\sup\left\{ f^\infty(y,\xi_1+t\xi_2)+f^\infty(y,\xi_1-t\xi_2)\,\big|\, f^\infty(y,\xi_1),f^\infty(y,\xi_2)\leq 1\right\}, \end{align}\] here we have used 127 in the final step. We claim that this supremum is \(\leq 2+Ct^2\) for some \(C\) not depending on \(y\) or \(t\).

To estimate the supremum we distinguish the three cases \(f^\infty(y,\xi_1)\geq \frac{1}{2}\) and \(f^\infty(y,\xi_1)\leq \frac{1}{2}\) and \(t\geq \frac{1}{3}\). In the latter case, we have a trivial upper bound of \(2+Ct^2\), simply by picking \(C\) large enough. In the second case, it trivially holds that \[\begin{align} f^\infty(y,\xi_1+t\xi_2)+f^\infty(y,\xi_1-t\xi_2)\leq \frac{5}{3} \end{align}\] thanks to the convexity and homogeneity if \(t<\frac{1}{3}\). In the case \(f^\infty(y,\xi_1)\geq \frac{1}{2}\) and \(t\leq \frac{1}{3}\) we may use the lower bound 128 for \(f^\infty\) to conclude that the line between \(\xi_1+t\xi_2\) and \(\xi_1-t\xi_2\) lies outside of some \(B_{\frac{1}{C}}(0)\), with a \(C\) not depending on \(y\). This enables us to estimate the sum from above with \(2+Ct^2\), thanks to \(f^\infty\) being \(C^2\) uniformly in \(y\) away from \(0\) by assumption and because \(\xi_2\lesssim 1\) by assumption and the bound 128 .

Either way, we have obtained that \[\begin{align} \sup\left\{ f_\star(y,\xi_1^*+\xi_2^*)+tf_\star(y,\xi_1^*-\xi_2^*)\,\big|\, f_\star(y,\xi_1^*),f_\star(y,\xi_2^*)\leq 1\right\}\leq 2+Ct^2.\label{uni32conv1} \end{align}\tag{130}\] If we pick \(t=\frac{1}{2C}f_\star(y,\xi_1^*-\xi_2^*)\), then we see that \[\begin{align} f_\star(y,\frac{1}{2}\left(\xi_1^*+\xi_2^*\right))\leq 1-\frac{1}{8C}f_\star(y,\xi_1^*-\xi_2^*)^2. \end{align}\] for all \(\xi_1,\xi_2\) with \(f_\star(y,\xi_1^*),f_\star(y,\xi_2^*)\leq 1\). Coming back to the estimate ?? in the lemma, we see that \(f_*(y,\mathrm{D}_\xi f^\infty(y,v))=1\), because of 126 and 127 . Therefore, using the definition 125 of \(f_*\), we can estimate \[\begin{align} \MoveEqLeft\langle \mathrm{D}_\xi f^\infty(y,v)-v^*,v \rangle=2\langle \mathrm{D}_\xi f^\infty(y,v)-\frac{v^*+\mathrm{D}_\xi f^\infty(y,v)}{2},v \rangle\\ &\geq 2f^\infty(y,v)\left(1-f_\star(y,\frac{v^*+\mathrm{D}_\xi f^\infty(y,v)}{2})\right)\\ &\gtrsim f_\star(y,\mathrm{D}_\xi f^\infty(y,v)-v^*)^2\\ &\gtrsim |\mathrm{D}_\xi f^\infty(y,v)-v^*|^2. \end{align}\] Here we have used 130 and that \(f^\infty\) and \(f_\star\) are equivalent to the modulus of their argument up to a constant by 128 129 in the last two steps.

The vectorial case proceeds by exactly the same argument.

Acknowledgment↩︎

The author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme through the grant agreement 862342.

References↩︎

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1. It is for instance quite easy to see by symmetry considerations that for \(\Omega=B_2(0)\backslash B_1(0)\) and \(f(y,\xi)=|\xi|\) with boundary values \(1\) on \(\partial B_1(0)\) and \(0\) on \(\partial B_2(0)\), the limit of any minimizing sequence is \(0\).↩︎

2. Here \(C_x^1C_\xi^1\) denotes the space of continuous functions \(v\) for which the derivatives \(\mathrm{D}_x v,\mathrm{D}_\xi v,\mathrm{D}_x\mathrm{D}_\xi v\) exist as classical derivatives and are continuous in the joint variable \((x,\xi)\)↩︎

3. Here we define the essential supremum near a point as \(\mathop{\mathrm{essinf}}_{\Omega\ni y\rightarrow x}a(y):=\sup_{m\rightarrow \infty}\mathop{\mathrm{essinf}}_{B_{\frac{1}{m}}(x)\cap \Omega} a\) for any measurable function \(a\)↩︎

4. this can, for instance, be seen from the equivalence with the barrier condition, as stated in [23], which holds for all strictly convex sets (see e.g.[30])↩︎

5. The reference only contains the case \(n=1\), the extension to \(n\geq1\) is however trivial by applying the theory for \(n=1\) to each row of \(z\)↩︎

6. The reference only states the theorems for subsets of the Euclidean space, for subsets of \(\partial\Omega\) it easily follows by parametrizing the boundary.↩︎

7. This is done with the sole intent of dealing with the \(k\)-term. In the special case in which \(f\) is positively \(1\)-homogeneous and \(k=0\), one can simplify the proof by dropping \(\delta\) and taking an arbitrary covering.↩︎

8. The property that \(i(2t)\lesssim i(t)\) is not stated in the reference, one can however easily achieve it by replacing \(i(t)\) with \(\min_{\rho\in (0,1]}\rho^{-2}(i+1)(\rho t)\)↩︎