¹

1 Introduction↩︎

Schauder estimates are fundamental tools in the Analysis of PDEs and in the Calculus of Variations, with applications to free boundary problems, evolutionary PDEs, and global regularity. The smoothing effect that (autonomous) elliptic operators have on solutions is the cornerstone of regularity theory. However, plugging in ingredients like forcing or transport terms, or space-depending coefficients, might inhibit the regularizing process. Think e.g.of the Laplace operator. By Weil’s lemma, harmonic maps² are smooth — in particular, the regularity of solutions self-improves. The presence of external ingredients is a clear obstruction to this mechanism, so it is natural to wonder how much of such regularity survives when coefficients are plugged in. Schauder theory answers this basic question. Specifically, given a bounded, elliptic matrix \(\textrm{\texttt{A}}\colon \Omega\to \mathbb{R}^{n\times n}\), \(\textrm{\texttt{A}}\approx \mathbb{I}\), the natural guess for weak energy solutions to \(\,{\rm div}(\textrm{\texttt{A}}(x)Du)=0\) is \[\label i.0 \textrm{\texttt{A}}\in C^0,\alpha_\operatorname{loc}(\Omega,\mathbb{R}^n\times n) \;\Longrightarrow \;u\in C^1,\alpha_\operatorname{loc}(\Omega),\qquadwith \;\;\alpha\in (0,1).\] Linear Schauder theory dates back to the end of the ’20’s of the previous century, with the classical results of Hopf, Caccioppoli, and Schauder, heavily relying on potential theory — techniques later on streamlined by Campanato (via suitable function spaces), L. Simon (by means of blow up methods), and Trudinger (using convolution arguments). The key aspect emerging from these works is the perturbative nature of Schauder estimates for linear, uniformly elliptic equations. Analogous phenomena hold for nonautonomous, quasilinear elliptic PDEs³ of the type \(\,{\rm div}(\textrm{\texttt{c}}(x)\lvert Du\rvert^{p-2}Du)=0\), with \(1\lesssim \textrm{\texttt{c}}(\cdot)\in C^{0,\alpha}_{\operatorname{loc}}(\Omega)\) and \(1<p<\infty\). This is due to Manfredi [2], after DiBenedetto [3], and Giaquinta & Giusti [4], based on the fundamental contributions of Ural’tseva [1] and Uhlenbeck [5]. The unifying feature of the equations and functionals treated in the aforementioned works lays in the uniform boundedness of the related ellipticity ratio. Specifically, for nonlinear elliptic PDEs of the type⁴ \(\,{\rm div}(\textrm{\texttt{B}}(x,Du))=0\), the ellipticity ratio is defined as \[\label er \mathcal{R}(x,z):=\frac{highest eigenvalue of }{\partial} \textrm{\texttt{B}}(x,z)lowest eigenvalue of \partial \textrm{\texttt{B}}(x,z),\] and measures how the gradient variable affects the growth/ellipticity properties of field \(\textrm{\texttt{B}}\). The extension of [er] to functionals as in [fun] goes through the Euler-Lagrange equation, solved by minima. In fact, replacing \(\textrm{\texttt{B}}\) with the derivative in the gradient variable of a (suitably regular, strictly convex) integrand \(\textrm{\texttt{f}}\) in [er], we are led to consider the pointwise ratio between the highest and the lowest eigenvalue of the Hessian matrix \(\partial^{2}\textrm{\texttt{f}}\). Equations or functionals characterized by a uniformly bounded ellipticity ratio are classified as uniformly elliptic, and Schauder theory is well-established, with three key aspects worth highlighting.

• Schauder theory for uniformly elliptic problems always holds, in the sense that to Hölder continuous coefficients always correspond solutions with Hölder continuous gradient.

• The smoother the better: in nondegenerate, uniformly elliptic problems, smooth ingredients grant smooth solutions.

• Uniformly elliptic Schauder estimates are achieved via perturbation arguments.

The above paradigm dramatically fails in the nonuniformly elliptic setting. Nonuniformly elliptic equations or functionals feature unbounded ellipticity ratios, which blow up, in several significant cases, as a positive power of the gradient variable. This class of nonuniformly elliptic PDEs is extremely rich. The most celebrated model is the nonparametric area integral \[w\mapsto \int_{\Omega}\sqrt{1+\lvert Dw\rvert^{2}}\,{\rm d}x,\] featuring quadratic ellipticity ratio, see Bombieri, De Giorgi & Miranda [6], Ladyzhenskaya & Ural’tseva [7], Trudinger [8], and L. Simon [9], for relevant regularity theory. Anisotropic energies [10]–[12], \[w\mapsto \int_{\Omega}\lvert Dw\rvert^{p}+\sum_{i=1}^{n}\lvert \partial_{i}w\rvert^{q_{i}}\,{\rm d}x,\qquad\quad 1<p\le q_{1}\le \cdots\le q_{n}<\infty,\] and slow-growing integrals [13], [14], \[w\mapsto \int_{\Omega}\lvert Dw\rvert\log(1+\lvert Dw\rvert)\,{\rm d}x,\] feature power-type nonuniformity as well. The key idea in this setting is to link the growth/ellipticity features of the nonlinear vector field \(\textrm{\texttt{B}}\) to the rate of blow-up of the ellipticity ratio [er]. Indeed, by imposing that the lowest and the highest eigenvalues of \(\partial \textrm{\texttt{B}}\) behave as two different powers of the gradient variable: \[\label pq \begin arrayc \displaystyle \lvert z\rvert^p-2\mathbb{I}\lesssim \partial\textrm{\texttt{B}}(x,z)\lesssim \lvert z\rvert^q-2\mathbb{I} \;\; in the sense of forms,\\[8pt]\displaystylefor some \;\;1<p<q<\infty \;\; and all \;\;\lvert z\rvert\ge 1,\;x\in \Omega, \end array\] the behavior at infinity of the related ellipticity ratio can be controlled in terms of the difference \(q-p\), \[\label rpq \mathcal{R}(x,z)\approx_\lvert z\rvert\ge 1\lvert z\rvert^q-p.\] The growth rate of \(\mathcal{R}\) can then be reduced by choosing \(q-p\) sufficiently close to zero, so the asymptotic in [rpq] suggests that a moderate blow-up rate of \(\mathcal{R}\) grants hope for regular solutions. Marcellini, Giaquinta and Hong proved that a restriction of the type \(q-p\lesssim 1/n\) is both necessary [11], [15], [16] and sufficient [10], [11] condition for regularity. Since these breakthroughs, autonomous nonuniformly elliptic theory flourished, see [12], [17]–[21] in the scalar setting, [22]–[27] in the vectorial one for an (incomplete) list of advances, and [28]–[30] for general overviews. Most notably, the construction in [16] provides a strongly convex, nonuniformly elliptic, functional with too large nonuniformity rate and unbounded scalar minima. This novel fact breaks the uniformly elliptic orthodoxy, as it points out that regularity should not always be expected for nonuniformly elliptic problems already in the autonomous, strongly convex setting, and suggests that Schauder theory might fail already in the most simple situations. In this respect, the first seminal results date back to the end of the ’60’s, due to Trudinger [31], Ladyzhenskaya & Ural’tseva [7], L. Simon [9], take also [32] as a general reference. In these foundational works, the use of strong solutions, the total differentiability of the equation, and the smoothness of ingredients was unavoidable, thus escaping the classical Schauder framework, which prescribes minimal regularity (Hölder continuity) of coefficients,⁵ and the validity of nonuniformly elliptic Schauder theory remained open. New impulses eventually came from Zhikov [35]–[37], who introduced novel, elementary models like the \(p(x)\)-Laplacian \[\label px.2 w\mapsto \int_\Omega\lvert D\rvert w^p(x)\,{\rm d}x,\qquad\quad 1<p(\cdot)\in L^\infty(\Omega),\] and the Double Phase energy \[\label dp.2 \begin cases \displaystyle \;w\mapsto \int_\Omega\lvert D\rvert w^p+a(x)\lvert D\rvert w^q\,{\rm d}x 1.5mm\\ \displaystyle \;0\le a(\cdot)\in C^0,\alpha(\Omega), \;\;\alpha\in (0,1] 1.5mm\\ \displaystyle \;1<p\le q<\infty, \end cases\] in the setting of homogenization and to study the possible occurrence of Lavrentiev phenomenon. Functionals [px.2]–[dp.2] are uniformly elliptic as their ellipticity ratio [er] stays uniformly bounded. However, a very mild amount of nonuniformity emerges due to the (mis)behavior of coefficients, see [38], which disclose a new, sharp phenomenology for Schauder theory to hold. In fact, building on early 2-d examples of Zhikov [35], Fonseca, Malý & Mingione [39] proved that as soon as \[\label pqn 1<p<n<n+\alpha<q<\infty \;\Longrightarrow \;\frac{q}{p}>1+\frac{\alpha}{n},\] the functional [dp.2] admits minima whose set of essential discontinuity points has almost maximal Hausdorff dimension, while, subject again to [pqn], Esposito, Leonetti & Mingione [40] exihibited a minimizer of [dp.2] with a one point singularity preventing its membership of \(W^{1,q}_{\operatorname{loc}}\), thus implying the failure of Schauder theory. The examples in [35], [39], [40] were eventually perfected by Balci, Diening & Surnachev [41], [42], who constructed minima of [dp.2] that cannot be better than \(W^{1,p}\)-regular, and the Schauder paradigm outlined below [er] starts crumbling. In fact, a closer inspection of [35], [39]–[42] reveals that the same construction works if in [dp.2] the coefficient \(a\) is \(C^{\alpha}\)-regular with any \(\alpha>0\), and the integrand is nondegenerate. This means that, despite the Hölder continuity of the ingredients, Schauder theory may not always hold now. In particular, even for nondegenerate integrands with smooth coefficients, minima need not be smooth, therefore two out of three distinctive features of classical Schauder theory dramatically fail, already in presence of very weak nonuniformity types as those in [px.2]–[dp.2] (keep in mind that both models are pointwise uniformly elliptic). In the case of [px.2]–[dp.2], only the perturbative approach to gradient regularity survives, thanks to the pointiwise uniform ellipticity of the governing integrands, cf. [43], [44]. Further extensions appear in [45]–[55], and a unifying (perturbative) approach to the maximal regularity of a general class of models sharing analogous weak nonuniformity as in [px.2]–[dp.2] can be found in [56]–[58], by the third author and Ok. However, owing to the lack of homogeneous reference estimates, perturbations are no longer feasible in the genuinely nonuniformly elliptic setting, i.e., when the ellipticity ratio [er] blows up. The longstanding⁶ problem of establishing optimal Schauder theory in the nonuniformly elliptic setting was settled by the first author and Mingione [30], [59], [60], by designing novel nonlinear potential theoretic techniques that allow bypassing the structural obstructions generated by polynomial nonuniformity and yield maximal regularity results within the sharp nonuniformity range \[\label qpqp \frac{q}{p}<1+\frac{\alpha}{n},\] cf. [pqn]. The approach of [59], [60] can be further expanded to push Schauder theory as close as possible to linear growth. The regularity of minima of nonautonomous variational integrals at linear growth is notoriously very delicate. In fact, Giaquinta, Modica & Souček [61] constructed (remarkably, one-dimensional) examples showing that nonautonomous, area-type functionals with almost \(C^{2}\)-regular coefficients might admit minimizers with jump discontinuities, which therefore belong to \(BV\setminus W^{1,1}\). On the other hand, \(C^{2}\)-regular coefficients guarantee (at least) \(C^{1}\)-regular minima, cf. [7]. Area-type integrals are specific instances of \(\mu\)-elliptic problems, a degenerate form of nonuniform ellipticity, typical of functionals at linear or nearly linear growth. Being the limiting configurations between linear and power growth, the latter class of models is rather common in materials science: the theories of Prandtl-Eyring fluids and of plastic materials with logarithmic hardening are prominent instances of applications cf. Frehse & Seregin [62]. We further refer to Fuchs & Mingione [13], Bildhauer & Fuchs [14], Schmidt [33], Beck & Schmidt [34], [63], Beck & Gmeineder & Schäffner [64], Gmeineder [65], [66], and Gmeineder & Kristensen [27], [67], [68] on deep regularity results for minima of general \(\mu\)-elliptic functionals, and [69] for an overview. Schauder-type results can be achieved for problems at nearly linear growth. In fact, the first author and Mingione [70] provided the first set of Schauder estimates for functionals at nearly linear growth as \[w\mapsto \int_{\Omega}\textrm{\texttt{c}}(x)\lvert Dw\rvert\log(1+\lvert Dw\rvert)\,{\rm d}x,\qquad \quad 1\lesssim \textrm{\texttt{c}}(\cdot)\in C^{0,\beta}(\Omega),\;\;\beta\in (0,1],\] and Log-Double Phase models \[\label dp \begin cases \displaystyle \;w\mapsto\int_\Omega\lvert D\rvert w\log(1+\lvert D\rvert w)+a(x)\lvert D\rvert w^q\,{\rm d}x 1.5mm\\ \displaystyle \;0\le a(\cdot)\in C^0,\alpha(\Omega),\;\;\alpha\in (0,1],\qquad \quad 1<q<\infty, \end cases\] subject to \[\label aq1 q<1+\frac{\alpha}{n},\] coherently with [qpqp],⁷ see also [71] for the Log-Multi Phase case, [mp] below. Later on, the first two authors and Piccinini [72] obtained sharp regularity for bounded minima of [dp] and highlighted via fractal counterexamples that nearly linear growing functionals may be the limiting configurations in which \(\mu\)-ellipticity, convex anisotropies and Hölder continuous coefficients can coexist, see [72] and [69]. This motivates our analysis. In fact, aim of this paper is to provide a comprehensive Schauder theory for vector-valued solutions to a large class of nonautonomous variational problems at nearly linear growth of the type \[\label fun W^1,1_\operatorname{loc}(\Omega,\mathbb{R}^N)\ni w\mapsto \mathcal{F}(w;\Omega):=\int_\Omega\textrm{\texttt{f}}(x,Dw)\,{\rm d}x.\] Here, \(\Omega\subset \mathbb{R}^{n}\) is an open subset, with \(n\ge 2\), \(N\ge 1\). We assume the radial (Uhlenbeck [5]) structure condition \(\textrm{\texttt{f}}(x,z):=A(x,\lvert z\rvert)\), cf. Section 2.5, which is fundamental to achieve full regularity in the multidimensional setting, [73]. The notion of (local) minimizer we shall use is the standard one.

We indeed develop a nonlinear potential theoretic framework to attack the regularity of a general class of anisotropic variational integrals at nearly linear growth in the vectorial setting. Specifically, being tailored on the Double Phase structure of [dp], the methodology in [70], [71] fails to cover very natural models, for which we deliver Schauder theory. For instance, we are able to treat variable exponent functionals like \[\label px w\mapsto \int_\Omega\left(\lvert D\rvert w\log(1+\lvert D\rvert w)\right)^p(x)\,{\rm d}x,\] where the (Hölder continuous) exponent \(p\) is now allowed to attain one, a scenario previously forbidden in the literature, cf. [74]. Our result in this respect reads as follows.

We furthermore offer the vectorial parallel of the results in [70], which naturally extends to functionals with multiple phases [71].

Another application of our results covers integral \[\label dppx \begin cases \displaystyle \; w\mapsto \int_\Omega\left(\lvert D\rvert w\log(1+\lvert D\rvert w)\right)^p(x)+a(x)\lvert D\rvert w^q(x)\,{\rm d}x 1.5mm\\ \displaystyle \;0\le a(\cdot)\in C^0,\alpha, \;\;\alpha\in (0,1] 1.5mm\\ \displaystyle \;1\le p(\cdot)\le q(\cdot)\in L^\infty(\Omega),\;\;p,q\in C^0,\sigma(\Omega),\;\;\sigma\in (0,1]. \end cases\] We indeed have the following theorem.

Let us point out that the outcome of Theorem 3 aligns with [46], [55], where the uniformly elliptic counterpart of [dppx] is studied. Moreover, notice that Theorems 1 and 3 are new already in the scalar case, see Section 3 for more models we can deal with. Overall, Theorems 1–3 are specific instances of a general result, which goes beyond the examples listed above and grants optimal, intrinsic Schauder estimates.

We refer to Section 2.5 for a detailed description of the (minimal) set of assumptions in force. From a technical standpoint, Theorem 4 relies on the development of a general nonlinear potential-theoretic machinery yielding intrinsic Lipschitz estimates that fully preserve the structural information of the underlying integrand. Unlike previous approaches [70], [71], which crucially relied on the splitting and power-type structure of integrands of the form [dp], our method does not depend on rigid assumptions. Instead, it provides a unified treatment of a broad class of variational integrals characterized by a few common structural properties. These include nearly linear growth below, \(\mu\)-ellipticity, and a quantified control on the oscillation of the coefficients. The latter condition plays a crucial role, as it ensures the absence of the Lavrentiev phenomenon and, consequently, allows for the construction of suitable approximation schemes for the problem. We conclude with an outline of the content of the paper.

Outline of the paper↩︎

In Section 2 we describe our notation and collect some auxiliary results that will be helpful at various stages of the paper. In Section 3 we list some examples to which our results apply, and possible generalizations. Section 4 is devoted to the construction of suitable power-type, uniformly elliptic approximating integrands retaining in a sharp, quantitative fashion all the structural information of the original one in [fun]. In Section 5 we develop the basic Lipschitz regularity for certain auxiliary frozen problems, that will be crucial in the proof of our main result. Section 6 is the core of the paper as it contains the intrinsic Lipschitz bounds, a fundamental step for our main result. Section 7 completes the proof of our vectorial Schauder estimates, of course in an a priori form, and finally Section 8 provides the approximation scheme which ultimately leads to the proof of Theorem 4.

2 Preliminaries↩︎

In this section we display our notation, collect some well-known functional analytic tools that will be useful throughout the paper, and list the main structural assumptions ruling integral [fun].

2.1 Notation↩︎

In this paper, \(\Omega\subset \mathbb{R}^{n}\), \(n\ge 2\), denotes an open, bounded domain with Lipschitz regular boundary. We denote by \(c\) a general constant larger than \(1\). Diverse occurrences from line to line will be still indicated by \(c\). Special occurrences will be denoted by \(c_*, \tilde{c}\) or the like. Relevant dependencies on parameters will be as usual emphasized by putting them in parentheses. Sometimes we shall use symbols "\(\gtrsim\)", "\(\lesssim\)" with subscripts, to indicate that a certain inequality holds up to constants whose dependencies are marked in the subscript. We denote by \(B_r(x_0):= \{x \in \mathbb{R}^n : |x-x_0|< r\}\) the open ball with center \(x_0\) and radius \(r>0\); we omit the center when it is not necessary or irrelevant, i.e., \(B \equiv B_r \equiv B_r(x_0)\); this especially happens when various balls in the same context share the same center. With \(B\) being a given ball with radius \(r\) and \(\theta\) being a positive number, we denote by \(\theta B\) the concentric ball with radius \(\theta r\) and, analogously, \(B/\theta \equiv (1/\theta)B\). We further denote \(\ell_{s}(t):=s+t\) for all \(s,t\in [0,\infty)\). Whenever \(\tilde{\Omega} \subset \mathbb{R}^{n}\) is a measurable subset with bounded positive measure \(0<|\tilde{\Omega}|<\infty\), and \(f \colon \tilde{\Omega} \to \mathbb{R}^{k}\), \(\mathbb{N}\ni k\geq 1\), is a measurable map, we use \[(f)_{\tilde{\Omega}}=\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{\tilde{\Omega}}f(x)\,{\rm d}x:= \lvert \tilde{\Omega}\rvert^{-1}\int_{\tilde{\Omega}} f(x) \,{\rm d}x\] to indicate the integral average. If \(f\in L^{p}(\tilde{\Omega},\mathbb{R}^{k})\), for some \(1\le p<\infty\), we shorten its averaged norm as \[\mathpalette\@thickbar{\lVert} f \rVert_{L^{p}(\tilde{\Omega})}:=\left(\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{\tilde{\Omega}}\lvert f\rvert^{p}\,{\rm d}x\right)^{\frac{1}{p}},\] while if \(f\in W^{s,p}(\tilde{\Omega})\) with \(1\le p<\infty\) and \(s\in (0,1)\), its averaged Sobolev–Slobodeckiǐ seminorm will be denoted by \[\mathpalette\@thickbar{[} f ]_{s,p;\tilde{\Omega}}:=\left(\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{\tilde{\Omega}}\int_{\tilde{\Omega}}\frac{\lvert f(x)-f(y)\rvert^{p}}{\lvert x-y\rvert^{n+sp}}\,{\rm d}x\,{\rm d}y\right)^{\frac{1}{p}}.\] Moreover, given any open set \(\tilde{\Omega}\Subset \Omega\), to simplify the notation we collect the main parameters related to the problems under investigation in the shorthands \[\begin{cases} \displaystyle \;\textrm{\texttt{data}}_{0}:=\left(n,N,A,\textrm{\texttt{g}},\mu,\gamma,\vartheta\right),\qquad \quad \textrm{\texttt{data}}:=(\textrm{\texttt{data}}_{0},\alpha,\beta,\vartheta_{*}),\\ \;\textrm{\texttt{l}}(\tilde{\Omega},\Omega):=(\,{\rm dist}(\tilde{\Omega},\partial\Omega),\,{\rm diam}(\tilde{\Omega}),\,{\rm diam}(\Omega)), \end{cases}\] we refer to Section 2.5 for an outline of the various quantities appearing above. We conclude by introducing some notation that will be helpful when specializing the forthcoming estimates to the \(2\)-d supercritical setting. Specifically, we set \[\begin{align} \label{1111} \begin{array}{c} \displaystyle \mathbb{1}_{1}:=1,\qquad\qquad\quad \mathbb{1}_{2}:=1\\[10pt]\displaystyle \mathbb{1}_{3}:=\begin{cases} \displaystyle \;1\quad &if \;\;n\ge 3 \quad or \quad n=2, \;\;0<\alpha<2/3\\ \displaystyle \;0 \quad &if \;\;n=2 \;\;and \;\;\alpha\ge 2/3, \end{cases}\qquad\qquad \quad \tilde{\mathbb{1}}:=1-\mathbb{1}_{3}. \end{array} \end{align}\tag{1}\]

2.2 Nonlinear potentials↩︎

A key role in this paper is played by a general class of nonlinear potentials, first introduced by Havin & Maz’ya [75]. Recently, nonlinear potentials became crucial tools in the regularity theory of nonuniformly elliptic problems [17], [20], [25], [38], [59], [60], [70]–[72] - specifically, we refer to [59] for the potential theoretic technical toolbox needed here, and to [29], [76] for an overview. For a ball \(B_{r}(x_{0})\subset \mathbb{R}^{n}\), parameters \(\sigma>0\), \(\vartheta\geq 0\), and a function \(f\in L^{1}(B_{r}(x_{0}))\), we introduce the nonlinear Havin-Maz’ya-Wolff type potential \({\boldsymbol{P}}_{\sigma}^{\vartheta}(f;\cdot)\), i.e.: \[\label defi-P \boldsymbol{P}_\sigma^\vartheta(f;x_0,r) := \int_0^r \varrho^\sigma \left( \mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_B_{\varrho}(x_0) \lvert f\rvert \,{\rm d}x\right)^\vartheta \frac{\,{\rm d}}{\varrho}\varrho \,.\] The mapping properties among function spaces of \({\boldsymbol{P}}_{\sigma}^{\vartheta}(f;\cdot)\) needed here are contained in the next lemma, cf. [59].

Finally, we record a nonlinear potential theoretic iteration à la De Giorgi, whose basic prototype can be found in [77] and [78] — we shall record it in the form of a quantified reverse Hölder inequality, first appeared in [59], see also [17].

2.3 Embeddings of fractional Sobolev spaces↩︎

For \(w \colon \Omega \to \mathbb{R}^{k}\), \(k\ge 1\), \(\texttt{t}>0\) and \(h \in \mathbb{R}^n\), we set \(\Omega_{\texttt{t}\lvert h\rvert}:=\left\{x\in \Omega\colon \,{\rm dist}(x,\partial \Omega)>\texttt{t}\lvert h\rvert\right\}\), and introduce the finite difference operator \(\tau_{h}\colon L^{1}(\Omega;\mathbb{R}^{k})\to L^{1}(\Omega_{|h|};\mathbb{R}^{k})\), defined as \(\tau_{h}w(x):=w(x+h)-w(x)\). Given another map \(v\colon \Omega\to \mathbb{R}^{k}\), the discrete Leibniz rule reads as \[\label prod \tau_h(vw)(x)=w(x+h)\tau_hv(x)+v(x)\tau_hw(x)\,.\] Moreover, if \(B_{\varrho}\Subset B_{r}\) are concentric balls and \(w\in W^{1,p}(B_r;\mathbb{R}^{k})\), \(p\ge 1\) and \(\lvert h\rvert\leq r-\varrho\), then \[\label gh \lVert \tau \rVert_{h}w_L^{p}(B_{\varrho})\le \lvert h\rvert\lVert D \rVert w_L^{p}(B_{r})\,.\] Two function spaces that will play a key role in this paper are Sobolev–Slobodeckij and Nikol’skii spaces.

Next, a combination of the embedding of Nikol’skii spaces into fractional Sobolev spaces, and the Sobolev-Morrey embedding of fractional Sobolev spaces, [79].

2.4 Tools for degenerate/singular problems↩︎

Here we collect some basic tools of common use when dealing with singular or degenerate problems. More precisely, for \(s\in [0,1]\) and \(0<p<\infty\), we introduce the field \(V_{s,p}\colon \mathbb{R}^{N\times n}\to \mathbb{R}^{N\times n}\) defined as \(V_{s,p}(z):=(s^{2}+\lvert z\rvert^{2})^{(p-2)/4}z\). It is well-known that \(V_{s,p}\) satisfies \[\label Vm \lvert V\rvert_{s,p}(z_{1})-V_{s,p}(z_{2})\approx_n,p(s^2+\lvert z\rvert_{1}^2+\lvert z\rvert_{2}^2)^\frac{p-2}{4}\lvert z\rvert_{1}-z_{2}\qquadfor all \;\;z_1,z_2\in \mathbb{R}^N\times n,\] cf. [80]. Next, the technical equivalence \[\label l60 (\lvert z\rvert_{1}^2+\lvert z\rvert_{2}^2+\omega^2)^-t/2 \approx_n,t \int_0^1(\lvert z\rvert_{2}+\tau (z_{1}-z_{2})^2+\omega^2)^-t/2 \,{\rm d}\tau\,,\] holds for all \(z_{1},z_{2}\in \mathbb{R}^{N\times n}\), and any \(t<1\). The following is the multidimensional counterpart of the integration-by-parts trick from [60], after [81].

We close this section with the "simple, but fundamental" iteration lemma from [4].

2.5 Structural assumptions↩︎

The high degree of generality we aim to achieve in this paper requires a careful description of the abstract integrand in [fun] that incorporates all the relevant features of the main models listed in Section 3 below. For this reason, we split the remainder of this section in several paragraph each devoted to the description of a key aspect of functional \(\mathcal{F}\).

The main integrand: regularity and basic properties↩︎

We assume that \(\textrm{\texttt{f}}\colon \Omega\times \mathbb{R}^{N\times n}\to \mathbb{R}\) has radial (Uhlenbeck [5]) structure, i.e., there exists a function \(A\colon \Omega\times [0,\infty)\to [0,\infty)\) such that \[\label a.1 \begin cases \displaystyle \;\textrm{\texttt{f}}(x,z)=A(x,\lvert z\rvert)\;\; & for all \;\;(x,z)\in \Omega\times \mathbb{R}^N\times n 0.5mm\\ \displaystyle \;t\mapsto A(x,t)\in C^2_\operatorname{loc}(0,\infty)\cap C^1_\operatorname{loc}[0,\infty) \;\;& for all \;\;x\in \Omega0.5mm\\ \displaystyle \;x\mapsto A'(x,t) \;\; &continuous for all\;\;t\ge 0. \end cases\] In general,⁸ \(A\), \(A'\) are continuous on \(\Omega\times [0,\infty)\), while \(A''\) is Caratéodory-regular on the same set. We shall further suppose that the basic monotonicity properties \[\label a.2 0<\inf_x\in \Omega A(x,1)\le \sup_x\in \Omega A(x,1)<\infty,\] and \[\label a.2.x \begin cases \displaystyle \;t\mapsto \frac{A}{(}\cdot,t)t \;\; almost increasing for all \;\;t\in (0,\infty) 1.5mm\\ \displaystyle \;t\mapsto \frac{A}{(}\cdot,t)t^{\gamma}, \;\; almost decreasing for all \;\;t\in [1,\infty), \end cases\] hold⁹ uniformly in \(x\in \Omega\) for some \(1<\gamma<\infty\). Notice that [a.2.x]\(_{1}\) implies that \(A(x,0)=0\) for all \(x\in \Omega\). For \((x,t)\in \Omega\times (0,\infty)\), we set for simplicity \(\mathcal{a}(x,t):=A'(x,t)t^{-1}\).

Closeness to linear growth↩︎

To measure how the growth/ellipticity of \(\textrm{\texttt{f}}\) is close to linear, we introduce a function \(\textrm{\texttt{g}}\colon [0,\infty)\to [0,\infty)\) \[\label a.4 \begin cases \;0\le\textrm{\texttt{g}}(\cdot)\in C[0,\infty),\qquad t\textrm{\texttt{g}}(t) \;\; is convex,1mm\\ \;\textrm{\texttt{g}} \;\; is nondecreasing, unbounded, and concave, 1mm\\ \;\textrm{\texttt{g}}(t)\lesssim_\textrm{\texttt{g}},\omega\ell_1(t)^\omega\qquadfor all \;\;\omega>0, \;\;t\ge 0 1.5mm\\ \displaystyle \;t\textrm{\texttt{g}}(t)\le \textrm{\texttt{c}}A(x,t)+\textrm{\texttt{c}}, \end cases\] for all \((x,t)\in \Omega\times [0,\infty)\). Notice that, letting \(b(t):=t\textrm{\texttt{g}}(t)\), \(\eqref{a.4}_{1,2}\) implies that \[\label binf \lim_t\to \infty\frac{b}{(}t)t=\infty,\] which will be useful to gain compactness in the \(W^{1,1}\)-setting.

Measuring growth/ellipticity and oscillation↩︎

The growth/ellipticity features of \(\textrm{\texttt{f}}\) will be described by two positive continuous functions \(\lambda,\Lambda\colon \Omega\times (0,\infty)\to [0,\infty)\) such that \[\label a.5.1 \textrm{\texttt{c}}^-1\lambda(x,\lvert z\rvert)\lvert \xi\rvert^2\le \langle\partial^2\textrm{\texttt{f}}(x,z)\xi,\xi\rangle\qquadand\qquad \lvert \partial\rvert^{2}\textrm{\texttt{f}}(x,z)\le \textrm{\texttt{c}}\Lambda(x,\lvert z\rvert),\] for all \(z\in \mathbb{R}^{N\times n}\setminus \{0\}\), \(\xi\in \mathbb{R}^{N\times n}\), and \(x\in \Omega\). The functions \(\lambda\) and \(\Lambda\) are the lowest and the highest eigenvalues of \(\partial^{2}\textrm{\texttt{f}}\) respectively, and satisfy the minimal set of assumptions \[\label a.5 \begin cases \displaystyle \;t\mapsto t^\mu \lambda(\cdot,t) \;\;& is almost increasing for all \;\;t\in (0,\infty) 1.5mm\\ \displaystyle \;t\mapsto \max\{t^-\vartheta,1\}\lambda(\cdot,t)\;\;& is almost decreasing for all \;\;t\in (0,\infty), \end cases\] for some \(\vartheta\in [0,1)\), and \[\label a.5.x \inf_x\in \Omega\lambda(x,1)\in (0,\infty)\qquadand\qquad \frac{\Lambda}{(}x,t)\lambda (x,t)\le \textrm{\texttt{c}}+\textrm{\texttt{c}}\textrm{\texttt{g}}(t)\ell_1(t)^\mu-1=:\textrm{\texttt{c}}\textrm{\texttt{r}}_*(t),\] for all \(x\in \Omega\), \(t\in (0,\infty)\), where \(1\le \mu<2\) is a number, and \(\textrm{\texttt{g}}\) is as in [a.4]. Finally, let \(B\subset \Omega\) be a ball and \[\label atat \alpha\in (0,1],\qquad \quad \vartheta_*\in \left(0,\alpha/n\right),\] be numbers. We assume that \[\label a.3 \lvert \lambda\rvert(x_{1},t)-\lambda(x_{2},t)t+\lvert \mathcal{\rvert{a}}(x_{1},t)-\mathcal{a}(x_{2},t)t\le \textrm{\texttt{c}}\lvert B\rvert^\frac{\alpha}{n}+\textrm{\texttt{c}}\lvert B\rvert^\frac{\alpha}{n}\left(\inf_x\in BA(x,t)\right)^\vartheta_{*},\] for all \(x_{1},x_{2}\in B\), \(t\in (0,\infty)\).

Continuity of second derivatives near the origin↩︎

To gain full regularity in the vectorial setting, we need to control the oscillations of second derivatives. In fact, we prescribe that there exists \(\beta\in (0,1]\), such that for every \(M>0\) there is a constant \(\textrm{\texttt{c}}_{M}\equiv \textrm{\texttt{c}}_{M}(M,A)\) satisfying \[\label a.5.2 \lvert A\rvert''(x,t+\tau)-A''(x,t)\le \textrm{\texttt{c}}_MA''(x,t)\left(\frac{\lvert}{\tau}\rvert t\right)^\beta,\] for all \(x\in \Omega\), \(t\in (0,M)\), \(\tau\in \mathbb{R}\) such that \(0<\lvert \tau\rvert< t/2\).

A slight enforcement in \(2\)-d supercritical cases↩︎

To beat the criticality of the embedding of nonlinear potentials in low dimension, if \(n=2\) and \(\alpha\ge 2/3\) we reinforce [a.3] by requiring \(\vartheta_{*}\in (0,\gamma-1]\) in [a.3] and also \[\label a.5.2s \lvert \mathcal{\rvert{a}}(x_{1},t)-\mathcal{a}(x_{2},t)t\le \textrm{\texttt{c}}\lvert B\rvert^\frac{\alpha}{n}\left(1+t^\gamma-1\right),\qquad \quad \gamma<1+\frac{\alpha}{n}.\] Notice that such a limitation is coherent with the usual ones imposed in the unbalanced polynomial growth setting [70], [71], [83].

3 Models↩︎

In this section we briefly discuss the models displayed in Section 1, and show that they all satisfy the structural conditions in Section 2.5.

Variable exponent↩︎

We focus on \(p(x)\)-type functionals, specifically on the case in which \(\inf_{x\in \Omega}p(x)=1\). In fact, if \(\inf_{x\in \Omega}p(x)>1\), the Hölder continuity of the gradient of minima is well-known, [43]. Since our results are local in nature, we can work on balls \(B(\equiv B_{r}(x_{0})\Subset \Omega)\) with radius \(r\in (0,r_{*}]\) where \(r_{*}\in (0,1)\) is a threshold parameter to be determined in a few lines, such that \(\min_{x\in \bar{B}}p(x)=1\). Let us consider integral [px] with [px.11], and verify that the assumptions listed in Section 2.5 are satisfied.

3.1 Verification of [a.1]–[a.4]↩︎

For \((x,t)\in \Omega\times [0,\infty)\), set \(\textrm{\texttt{P}}(x,t):=(t\log(1+t))^{p(x)}\). The very definition and the strict convexity of integrand \(\textrm{\texttt{P}}\), yield that [a.1]–[a.2] are satisfied. Moreover, \(\eqref{a.2.x}_{1}\) holds as \(p(x)\ge 1\) for all \(x\in \Omega\) and \(\eqref{a.2.x}_{2}\) is verified for any \(\gamma>\lVert p \rVert_{L^{\infty}(\Omega)}\). Condition [a.4]\(_{4}\) holds with \(\textrm{\texttt{g}}(t):=\log(1+t)\) up to constants depending on \(\lVert p \rVert_{L^{\infty}(\Omega)}\), while [a.4]\(_{1,2,3}\) follow from the basic properties of logarithms.

3.2 Verification of [a.5.1]–[a.5.x] and [a.5.2]↩︎

Fix any \(\vartheta_{*}\in (0,\alpha/n)\), restrict \(r_{*}\in (0,1)\) so much that \[\label r* [p]_0,\alpha;\Omega r_*^\alpha\le \frac{\vartheta}{_}{*}4 \;\Longrightarrow \;\lVert p \rVert_L^{\infty}(\Omega)\le 1+\frac{\vartheta}{_}{*}4,\] and, for all \((x,t)\in \Omega\times (0,\infty)\), record \[\begin{cases} \displaystyle \;\frac{\textrm{\texttt{P}}'(x,t)}{t}=p(x)\left(t\log(1+t)\right)^{p(x)-1}\left(\frac{\log(1+t)}{t}+\frac{1}{1+t}\right)\\ \displaystyle \;\textrm{\texttt{P}}''(x,t)=\frac{(p(x)-1)}{\log(1+t)}\left(\log(1+t)+\frac{t}{1+t}\right)\left(\frac{\textrm{\texttt{P}}'(x,t)}{t}\right)+p(x)\left(t\log(1+t)\right)^{p(x)-1}\left(\frac{2+t}{(1+t)^{2}}\right). \end{cases}\] Next, set \[\begin{cases} \displaystyle \;\lambda(x,t):=\frac{(t\log(1+t))^{p(x)-1}}{1+t}\\ \displaystyle \;\Lambda(x,t):=t^{p(x)-2}(\log(1+t))^{p(x)-2}\max\left\{1,(\log(1+t))^{2}\right\}. \end{cases}\] A direct computation yields estimates for the eigenvalues \[\min\left\{\frac{\textrm{\texttt{P}}'(x,t)}{t},\textrm{\texttt{P}}''(x,t)\right\}\ge \lambda(x,t),\qquad \quad \max\left\{\frac{\textrm{\texttt{P}}'(x,t)}{t},\textrm{\texttt{P}}''(x,t)\right\}\lesssim_{\lVert p \rVert_{L^{\infty}(\Omega)}} \Lambda(x,t),\] and [a.5.1] follows, up to constants depending on \((n,N,\lVert p \rVert_{L^{\infty}(\Omega)})\). Moreover, \(\eqref{a.5.x}_{1}\) is a consequence of the definition of \(\lambda\), and \[\frac{\Lambda(x,t)}{\lambda(x,t)}\lesssim_{\lVert p \rVert_{L^{\infty}(\Omega)}}1+\log(1+t),\] which implies \(\eqref{a.5.x}_{2}\) for all \(\mu\ge 1\). Estimate [a.5.2] is obtained by the mean value theorem after observing that \(\lvert \textrm{\texttt{P}}'''(t)\rvert\lesssim_{\lVert p \rVert_{L^{\infty}(\Omega)}} t^{-1}\textrm{\texttt{P}}''(t)\) and exploiting the very definition of \(\textrm{\texttt{P}}''\). Finally, let us take care of [a.5]. A direct computation (take first derivatives), shows that \(t\mapsto t^{\mu}\lambda(\cdot,t)\) is nondecreasing for all \(t\in (0,\infty)\) and any \(\mu\ge 1\). Concerning \(\eqref{a.5}_{2}\), set \(\vartheta:=2\left(\lVert p \rVert_{L^{\infty}(\Omega)}-1\right)\in [0,1)\), cf.[r*], and notice that if \(t\in (0,1)\), then \((1+t)^{-1},\log(1+t)t^{-1}\approx 1\), thus \(\max\{t^{-\vartheta},1\}\lambda(x,t)=t^{-\vartheta}\lambda(x,t)\approx_{\lVert p \rVert_{L^{\infty}(\Omega)}} t^{-\vartheta+2(p(x)-1)}\), which is decreasing for all \((x,t)\in \Omega\times (0,1)\). On the other hand, if \(t\ge 1\), then \(\max\{t^{-\vartheta},1\}\lambda(x,t)=\lambda (x,t)\), which is decreasing for all \(x\in \Omega\), therefore \(\eqref{a.5}_{2}\) comes by combining the two previous observations.

3.3 Verification of [a.3] and [a.5.2s]↩︎

Let \(B_{\varrho}\subseteq B\) be any ball, set \(\mathcal{p}(x,t):=(t\log(1+t))^{p(x)-1}\) and notice that, given the explicit expressions of \(\textrm{\texttt{P}}'(x,t)t^{-1}\) and \(\lambda(x,t)\), it is enough to estimate \(\mathop{\mathrm{osc}}_{x\in B_{\varrho}}\mathcal{p}(x,t)\). Note that [r*] implies that \[\label c.3.1 \mathcal{p}(x,t)\lesssim_\lVert p \rVert_{L^{\infty}(\Omega)} \max\left\{1,\left(t\log(1+t)\right)\right\}^\frac{\vartheta_{*}}{4},\] and \[\begin{align} \label{c463462} \lvert \mathcal{p}(x_{1},t)-\mathcal{p}(x_{2},t)\rvert&\le& [p]_{0,\alpha;\Omega}\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\log\left(\max\left\{1,t\log(1+t)\right\}\right)\max\left\{1,\left(t\log(1+t)\right)\right\}^{\lVert p \rVert_{L^{\infty}(\bar{B})}-1}\nonumber \\ &\le&c([p]_{0,\alpha;\Omega},\vartheta_{*})\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\max\left\{1,\left(t\log(1+t)\right)\right\}^{\frac{\vartheta_{*}}{2}}, \end{align}\tag{4}\] for all \(x_{1},x_{2}\in B_{\varrho}\), where we also used well-known properties of logarithms. We then obtain \[\begin{align} \label{c463463} &\lvert \lambda(x_{1},t)-\lambda(x_{2},t)\rvert t+\lvert \textrm{\texttt{P}}'(x_{1},t)-\textrm{\texttt{P}}'(x_{2},t)\rvert\nonumber \\ &\qquad \qquad \quad \le\lvert p(x_{1})-p(x_{2})\rvert\lVert \mathcal{p}(\cdot,t) \rVert_{L^{\infty}(\Omega)}\left(\log(1+t)+\frac{t}{1+t}\right)\nonumber \\ &\qquad\qquad \quad \quad +\lVert p \rVert_{L^{\infty}(\Omega)}\left(\log(1+t)+\frac{t}{1+t}\right)\lvert \mathcal{p}(x_{1},t)-\mathcal{p}(x_{2},t)\rvert\nonumber \\ &\qquad \qquad \quad \quad +\left(\frac{t}{1+t}\right)\lvert \mathcal{p}(x_{1},t)-\mathcal{p}(x_{2},t)\rvert\nonumber \\ &\qquad \qquad \quad \le c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\left(1+\log(1+t)+\frac{t}{1+t}\right)\max\left\{1,\left(t\log(1+t)\right)\right\}^{\frac{\vartheta_{*}}{2}}\nonumber \\ &\qquad \qquad \quad \le c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}+c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}(t\log(1+t))^{\frac{3\vartheta_{*}}{4}}\le c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}+c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\inf_{x\in B_{\varrho}}\textrm{\texttt{P}}(x,t)^{\vartheta_{*}}, \end{align}\tag{5}\] and [a.3] holds with \(c\equiv c(\lVert p \rVert_{C^{0,\alpha}(\Omega)},\alpha,\vartheta_{*})\). To validate [a.5.2s], we look back at the last-but-one line of the previous display, and use the slow growth of logarithms to bound \[\lvert \textrm{\texttt{P}}'(x_{1},t)-\textrm{\texttt{P}}'(x_{2},t)\rvert\le c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}+c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}(t\log(1+t))^{\frac{3\vartheta_{*}}{4}}\le c\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\left(1+t^{\vartheta_{*}}\right),\] for \(c\equiv c(\lVert p \rVert_{C^{0,\alpha}(\Omega)},\alpha,\vartheta_{*})\). Letting \(\vartheta_{*}=\gamma-1\), any \(1\le\lVert p \rVert_{L^{\infty}(\Omega)}<\gamma<1+\alpha/2\), together with [r*] works.

The assumptions of Theorem 4 are then verified by merging the content of Sections 3.1–3.3, and Theorem 1 follows.

Log-Double Phase↩︎

Our setting enc also the Log-Double Phase energy, that is functional [dp], subject to [aq1]. Maximal regularity for minima of functional \(\eqref{dp}_{1}\) has been obtained in [70], [72] in the scalar setting \(N=1\), here we show that our approach grants the vectorial counterpart of [70].¹¹

3.4 Verification of [a.1]–[a.4]↩︎

For \((x,t)\in \Omega \times [0,\infty)\), set \(\textrm{\texttt{H}}(x,t):=t\log(1+t)+a(x)t^{q}\), with coefficient \(a\) and exponent \(q\) as in [dp]–[aq1]. Verifying [a.1]–[a.2] is straightforward, given the definition of integrand \(\textrm{\texttt{H}}\). The monotonicity conditions in [a.2.x] hold for any \(\gamma\ge q\). Moreover, the conditions in [a.4] are satisfied with \(\textrm{\texttt{g}}(t):=\log(1+t)\).

3.5 Verification of [a.5.1]–[a.5.x] and [a.5.2]↩︎

For \((x,t)\in \Omega\times (0,\infty)\), we have \[\begin{cases} \displaystyle \;\frac{\textrm{\texttt{H}}'(x,t)}{t}=\frac{\log(1+t)}{t}+\frac{1}{1+t}+qa(x)t^{q-2}\\ \displaystyle \;\textrm{\texttt{H}}''(x,t)=\frac{2+t}{(1+t)^{2}}+q(q-1)a(x)t^{q-2}. \end{cases}\] We then define \[\begin{cases} \displaystyle \;\lambda(x,t):=\frac{1}{1+t}+a(x)t^{q-2}\\ \displaystyle \;\Lambda(x,t):=\frac{1+\log(1+t)}{1+t}+a(x)t^{q-2}, \end{cases}\] so that \[\min\left\{\frac{\textrm{\texttt{H}}'(x,t)}{t},\textrm{\texttt{H}}''(x,t)\right\}\ge \lambda(x,t),\qquad \quad \max\left\{\frac{\textrm{\texttt{H}}'(x,t)}{t},\textrm{\texttt{H}}''(x,t)\right\}\lesssim_{q}\Lambda(x,t),\] thus [a.5.1] and [a.5.x] are satisfied up to constants depending on \((n,N,q)\), cf. [70]. Furthermore, [a.5] is true for \(\mu=1\) and any \(\vartheta\ge0\) - recall that, by [aq1], \(1<q<2\). Finally, the splitting structure of \(\textrm{\texttt{H}}\) guarantees the validity of the Hölder condition [a.5.2] of \(\textrm{\texttt{H}}''\). In fact, [a.5.2] can be verified by treating separately the (weighted) \(q\)-power term, for which [a.5.2] is standard [84], and the nearly linear growing one, that is subject to considerations analogous to those in Section 3.2.

3.6 Verification of [a.3] and [a.5.2s]↩︎

Fix any ball \(B_{\varrho}\subset \Omega\). By the splitting structure of \(\textrm{\texttt{H}}\), it is enough to control \[t^{q-1}\lvert a(x_{1})-a(x_{2})\rvert\le [a]_{0,\alpha;\Omega}\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}t^{q-1}\stackrel{\eqref{aq1}}{\le}[a]_{0,\alpha;\Omega}\lvert B_{\varrho}\rvert^{\frac{\alpha}{n}}\left(\inf_{x\in B_{\varrho}}\textrm{\texttt{H}}(x,t)\right)^{\vartheta_{*}},\] with \(\vartheta_{*}:=q-1\), admissible by [aq1]. Finally, [a.5.2s] holds by direct computation via [aq1].

Log-Multi Phase↩︎

The prototypical model we have in mind is \[\label mp \begin cases \displaystyle \;w\mapsto \int_\Omega\lvert D\rvert w\log(1+\lvert D\rvert w)+\sum_i=1^ka_i(x)\lvert D\rvert w^q_{i}\,{\rm d}x 1.5mm\\ \displaystyle \;0\le a_i(\cdot)\in C^0,\alpha_{i}(\Omega), \;\;i\in \{1,\cdots,k\}, \end cases\] for exponents \(q_{i}>1\), \(\alpha_{i}\in (0,1]\) for all \(i\in \{1,\cdots, k\}\), verifying \[q_{i}<1+\frac{\alpha_{i}}{n}\qquad for all \;\;i\in \{1,\cdots,k\}.\] Recalling the third bullet of Remark 5, the verification of the assumptions listed in Section 2.5 is analogous to the Log-Double Phase case (with obvious structural variations accounting for the multiple phases), thus Theorem 4 grants the validity of Theorem 2 (and of its Log-Multi Phase counterpart). In particular, Schauder theory for vector-valued minimizers of the Log-Multi Phase energy [mp] holds true, see [71] for the scalar counterpart.

Log-Double Phase with variable exponents↩︎

This is integral [dppx], under assumptions [pxqx]. We focus on the case \(1=\inf_{x\in \Omega}p(x)\), otherwise the gradient Hölder continuity of minima follows by more standard means, cf. [46], [55] and references therein. We can therefore proceed as in Section 3.2, set \(\textrm{\texttt{J}}(x,t):=(t\log(1+t))^{p(x)}+a(x)t^{q(x)}\), \(q_{0}:=\inf_{x\in \Omega}q(x)\), fix any \(\vartheta_{*}\in (\lVert q \rVert_{L^{\infty}(\Omega)}-1,\min\{\alpha,\sigma\}/n)\) and work on balls \(B(\equiv B_{r}(x_{0}))\Subset \Omega\) such that \(\min_{x\in \bar{B}}p(x)=1\), \(q_{0;B}:=\min_{x\in \bar{B}}q(x)>1\) and with radius \(r\in (0,r_{*}]\) for some threshold \(r_{*}\in (0,1)\) satisfying \[\label px.1 \max\{[p]_0,\beta;\Omega,[q]_0,\beta;\Omega\}r_*^\sigma\le \frac{\vartheta}{_}{*}8 \;\Longrightarrow \; \lVert p \rVert_L^{\infty}(\bar{B})\le 1+\frac{\vartheta}{_}{*}8.\] Let us verify the validity of the conditions in Section 2.5.

3.7 Verification of [a.1]–[a.4]↩︎

Assumptions [a.1]–[a.2.x]\(_{1}\) immediately follow from the definition of integrand \(\textrm{\texttt{J}}\), \(\eqref{a.2.x}_{2}\) is satisfied for any \(\gamma>\lVert q \rVert_{L^{\infty}(\bar{B})}\), and [a.4] holds with \(\textrm{\texttt{g}}(t):=\log(1+t)\).

3.8 Verification of [a.5.1]–[a.5.x] and [a.5.2]↩︎

For \((x,t)\in \Omega\times (0,\infty)\), \[\begin{cases} \displaystyle \;\frac{\textrm{\texttt{J}}'(x,t)}{t}=p(x)\left(t\log(1+t)\right)^{p(x)-1}\left(\frac{\log(1+t)}{t}+\frac{1}{1+t}\right)+q(x)a(x)t^{q(x)-1}\\ \displaystyle \;\textrm{\texttt{J}}''(x,t)=\left(\frac{\textrm{\texttt{J}}'(x,t)}{t}\right)\left(\frac{(p(x)-1)}{\log(1+t)}\left(\log(1+t)+\frac{t}{1+t}\right)\right) \\ \displaystyle \qquad\qquad\quad+p(x)\left(t\log(1+t)\right)^{p(x)-1}\left(\frac{2+t}{(1+t)^{2}}\right)+q(x)(q(x)-1)a(x)t^{q(x)-2}. \end{cases}\] Letting \[\begin{cases} \displaystyle \;\lambda(x,t):=\frac{(t\log(1+t))^{p(x)-1}}{1+t}+a(x)t^{q(x)-2}\\ \displaystyle \;\Lambda(x,t):=t^{p(x)-2}(\log(1+t))^{p(x)-2}\max\left\{1,(\log(1+t))^{2}\right\}+a(x)t^{q(x)-2}, \end{cases}\] a direct computation gives \[\min\left\{\frac{\textrm{\texttt{J}}'(x,t)}{t},\textrm{\texttt{J}}''(x,t)\right\}\ge \lambda(x,t),\qquad \quad \max\left\{\frac{\textrm{\texttt{J}}'(x,t)}{t},\textrm{\texttt{J}}''(x,t)\right\}\lesssim_{q}\Lambda(x,t),\] and [a.5.1] follows, with constants depending on \((n,N,\lVert p \rVert_{L^{\infty}(\Omega)},\lVert q \rVert_{L^{\infty}(\Omega)},q_{0})\). The monotonicity properties in [a.5] hold with \(\vartheta:=2(\lVert p \rVert_{L^{\infty}(\Omega)}-1)\), by combining Sections 3.2 and 3.5. Moreover, [a.5.x]\(_{1}\) trivially follows from the definition of \(\lambda\), and \[\frac{\Lambda(x,t)}{\lambda(x,t)}\lesssim_{\lVert p \rVert_{L^{\infty}(\Omega)},\lVert q \rVert_{L^{\infty}(\Omega)}}1+\log(1+t),\] which is \(\eqref{a.5.x}_{2}\), valid for all \(\mu\ge 1\). Finally, thanks to the splitting structure of integrand \(\textrm{\texttt{J}}\), [a.5.2] follows by merging again Sections 3.2 and 3.5.

3.9 Verification of [a.3] and [a.5.2s]↩︎

Let \(\mathcal{p}\) be the auxiliary function introduced in Section 3.3, set \(\mathcal{q}(x,t):=a(x)t^{q(x)-1}\) and let \(B_{r}\subseteq B\). We then estimate, via [c.3.1] and [px.1], \[\begin{align} \mathcal{p}(x,t)+\mathcal{q}(x,t)&\le&c\max\left\{1,\left(t\log(1+t)\right)\right\}^{\frac{\vartheta_{*}}{4}}+ca(x)\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1}\nonumber \\ &\le&c\ell_{1}(t)^{\frac{\vartheta_{*}}{2}}+c\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1}, \end{align}\] for \(c\equiv c(n,\lVert p \rVert_{L^{\infty}(\Omega)},\lVert q \rVert_{L^{\infty}(\Omega)},\alpha)\), and \[\begin{align} \lvert \mathcal{q}(x_{1},t)-\mathcal{q}(x_{2},t)\rvert&\le &c\lvert q(x_{1})-q(x_{2})\rvert\lvert \log(t)\rvert t^{\lVert q \rVert_{L^{\infty}(\Omega)}-1}\nonumber \\ &&+c\lvert a(x_{1})-a(x_{2})\rvert\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1}\le c\lvert B_{\varrho}\rvert^{\frac{\min\{\alpha,\sigma\}}{n}}\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1+\epsilon}, \end{align}\] where \(\epsilon\in (\lVert q \rVert_{L^{\infty}(\Omega)}-1,\vartheta_{*})\) is any number and \(c\equiv c(n,\lVert a \rVert_{C^{0,\alpha}(\Omega)}, \lVert q \rVert_{C^{0,\sigma}(\Omega)},\alpha)\). By the splitting structure of \(\textrm{\texttt{J}}\) and 5 we get \[\begin{align} \lvert \lambda(x_{1},t)-\lambda(x_{2},t)\rvert t+\lvert \textrm{\texttt{J}}'(x_{1},t)-\textrm{\texttt{J}}'(x_{2},t)\rvert&\le&c\lvert B_{\varrho}\rvert^{\frac{\sigma}{n}}\left(1+(t\log(1+t))^{\frac{3\vartheta_{*}}{4}}\right)\nonumber \\ &&+c\lvert \mathcal{q}(x_{1},t)-\mathcal{q}(x_{1},t)\rvert+c\lvert q(x_{1})-q(x_{2})\rvert\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1}\nonumber \\ &\le&c\lvert B_{\varrho}\rvert^{\frac{\sigma}{n}}+c\lvert B_{\varrho}\rvert^{\frac{\sigma}{n}}\left(\inf_{x\in B_{\varrho}}\textrm{\texttt{J}}(x,t)\right)^{\vartheta_{*}}\nonumber \\ &&+c\lvert B_{\varrho}\rvert^{\frac{\min\{\alpha,\sigma\}}{n}}\ell_{1}(t)^{\lVert q \rVert_{L^{\infty}(\Omega)}-1+\epsilon}\nonumber \\ &\le&c\lvert B_{\varrho}\rvert^{\frac{\min\{\sigma,\alpha\}}{n}}+c\lvert B_{\varrho}\rvert^{\frac{\min\{\sigma,\alpha\}}{n}}\left(\inf_{x\in \bar{B}}\textrm{\texttt{J}}(x,t)\right)^{\vartheta_{*}}, \end{align}\] with \(c\equiv c(n,\lVert a \rVert_{C^{0,\alpha}(\Omega)}, \lVert q \rVert_{C^{0,\sigma}(\Omega)},\alpha)\), and [a.3] is verified. Concerning [a.5.2s], we just need to restrict the size of \(\gamma\) in Section 3.7: any \(\gamma\in (\lVert q \rVert_{L^{\infty}(\Omega)},1+\vartheta_{*})\) works.

4 Degeneracy/growth corrections↩︎

In this section we construct suitable regularized counterparts of the integrands introduced in Section 2.5. Let \(s\mapsto \sigma_{s}\) be a decreasing function with \(\sigma_{0}=0\), \(\delta,\varepsilon\in (0,1/4)\) be positive numbers, and, for \((x,t)\in \Omega\times [0,\infty)\), define \[\label ade.1 \begin cases \displaystyle \;\mathcal{a}_\delta(x,t):=\mathcal{a}(x,\ell_\delta(t)),\qquad \quad &\mathcal{a}_\delta^\varepsilon(x,t):=\mathcal{a}_\delta(x,t)+4\gamma\sigma_\varepsilon\ell_1(t^2)^2\gamma-1, 1.5mm\\ \displaystyle \;\bar\mathcal{a}_\delta(x,t):=\mathcal{a}_\delta(x,t)+t\mathcal{a}_\delta'(x,t),\qquad \quad &\bar\mathcal{a}_\delta^\varepsilon(x,t):=\mathcal{a}_\delta^\varepsilon(x,t)+t(\mathcal{a}_\delta^\varepsilon)'(x,t), \end cases\] set corresponding integrands \[\begin{align} \label{adade} A_{\delta}(x,t):=\int_{0}^{t}\mathcal{a}_{\delta}(x,s)s\,{\rm d}s,\qquad and\qquad A_{\delta}^{\varepsilon}(x,t):=A_{\delta}(x,t)+\sigma_{\varepsilon} \ell_{1}(t^{2})^{2\gamma}, \end{align}\tag{6}\] for all \((x,z)\in \Omega\times \mathbb{R}^{N\times n}\), and name \[\begin{align} &\textrm{\texttt{f}}_{\delta}(x,z):=A_{\delta}(x,\lvert z\rvert),\qquad \qquad \quad \textrm{\texttt{f}}_{\delta}^{\varepsilon}(x,z):=A_{\delta}^{\varepsilon}(x,\lvert z\rvert), \end{align}\] see [84], [85]. Define also the corrected “eigenfunctions” \[\begin{align} &\lambda_{\delta}(x,t):=\lambda(x,\ell_{\delta}(t)),\qquad \quad \;\lambda_{\delta}^{\varepsilon}(x,t):=\lambda_{\delta}(x,t)+\sigma_{\varepsilon}\ell_{1}(t^{2})^{2\gamma-1}\\ &\Lambda_{\delta}(x,t):=\Lambda(x,\ell_{\delta}(t)),\qquad \quad \Lambda_{\delta}^{\varepsilon}(x,t):=\Lambda_{\delta}(x,t)+\sigma_{\varepsilon}\ell_{1}(t^{2})^{2\gamma-1}, \end{align}\] that now make sense for all \((x,t)\in \Omega\times [0,\infty)\), and the primitive \[\textrm{\texttt{a}}_{\delta}(x,t):=\int_{0}^{t}\lambda_{\delta}(x,s)s\,{\rm d}s\qquad and\qquad \textrm{\texttt{a}}_{\delta}^{\varepsilon}(x,t):=\int_{0}^{t}\lambda_{\delta}^{\varepsilon}(x,s)s\,{\rm d}s.\] By construction, [a.1] and [a.3], we have \[\label regreg z\mapsto \textrm{\texttt{f}}_\delta^\varepsilon(\cdot,z)\in C^2_\operatorname{loc}(\mathbb{R}^N\times n)\qquadand\qquad x\mapsto \textrm{\texttt{f}}_\delta^\varepsilon(x,\cdot)\in C^0,\alpha(\Omega).\] In fact, \(\eqref{regreg}_{1}\) follows directly by construction, while \(\eqref{regreg}_{2}\) is detailed in [a12.x]\(_{1}\) below. Let us show that the newly defined maps preserve the original structural features of \(A\) and \(b\).

In the next lemma we describe the main features of the regularized eigenvalues

We then collect the main growth/ellipticity properties of our approximating integrands.

We highlight some mutual bounds for the auxiliary function \(\textrm{\texttt{a}}_{\delta}^{\varepsilon}\), the main integrand \(A_{\delta}^{\varepsilon}\) and eigenvalue \(\lambda_{\delta}^{\varepsilon}\).

The main oscillation properties of integrand \(A_{\delta}^{\varepsilon}\) are described in the following lemma.

Let us discuss the limiting behavior of \(A_{\delta}^{\varepsilon}\) as \(\delta\to 0\), \(\varepsilon\to 0\).

Let us show that integrand \(\textrm{\texttt{f}}_{\delta}^{\varepsilon}\) can be traced back to standard, controlled polynomial growth condition, that of course will hold in a nonuniform fashion with respect to \(\delta\) and \(\varepsilon\).

The standard uniformly elliptic setting [84], [86] can be recovered on bounded subsets.

We conclude this section with a direct consequence of Lemma 13.

4.1 Regularized Dirichlet problems↩︎

Let \(u\in W^{1,1}_{\operatorname{loc}}(\Omega,\mathbb{R}^{N})\) be a local minimizer of functional \(\mathcal{F}\), and \(B\Subset \Omega\) be a ball. Recalling [a.3] and the first bullet of Remark 5, [48] applies¹³ and yields a sequence \(\tilde{u}_{\varepsilon}\in C^{\infty}(B,\mathbb{R}^{N})\) such that \[\label conv \tilde{u}_\varepsilon\to u \;\; strongly in \;\;W^1,1(B,\mathbb{R}^N)\qquadand\qquad \mathcal{F}(\tilde{u}_\varepsilon;B)\to \mathcal{F}(u;B).\] Let us denote \(\sigma_{\varepsilon}:=\left(10+\varepsilon^{-1}+\varepsilon^{-1}\lVert D\tilde{u}_{\varepsilon} \rVert_{L^{4\gamma}(B)}^{8\gamma}\right)^{-1}\), observe that \[\label oe \texttt o(\varepsilon)\equiv \sigma_\varepsilon\lVert D \rVert\tilde{u}_{\varepsilon}_L^{4\gamma}(B)^4\gamma\to 0,\] and introduce functional \[\label funed W^1,4\gamma(B,\mathbb{R}^N)\ni w\mapsto \mathcal{F}_\delta^\varepsilon(w;B):=\int_B\textrm{\texttt{f}}_\delta^\varepsilon(x,Dw)\,{\rm d}x,\] with integrand \(\textrm{\texttt{f}}_{\delta}^{\varepsilon}\) constructed as in Section 4, corresponding to the choice of \(\sigma_{\varepsilon}\) made above. By strict convexity, cf. \(\eqref{lala}_{1}\), and direct methods, the Dirichlet problem \[\label pded \tilde{u}_\varepsilon+W^1,4\gamma_0(B,\mathbb{R}^N)\ni w\mapsto \min_\tilde{u}_{\varepsilon}+W^{1,4\gamma}_{0}(B,\mathbb{R}^{N})\mathcal{F}_\delta^\varepsilon(w;B)\] admits a unique solution \(u_{\delta}^{\varepsilon}\in \tilde{u}_{\varepsilon}+W^{1,4\gamma}_{0}(B,\mathbb{R}^{N})\), that is a weak solution to system \[\label elsed \int_B\langle\partial\textrm{\texttt{f}}_\delta^\varepsilon(x,Du_\delta^\varepsilon),Dw\rangle\,{\rm d}x=0\qquadfor all \;\;w\in W^1,4\gamma_0(B,\mathbb{R}^N).\] Moreover, by Lemma 12 we see that the integrand \(\textrm{\texttt{f}}_{\delta}^{\varepsilon}\) satisfies standard, controlled growth conditions [corfl.1r], thus by now classical regularity theory applies [87] and \[\label areg u_\delta^\varepsilon\in W^1,\infty_\operatorname{loc}(B,\mathbb{R}^N).\] This will be crucial for deriving uniform Lipschitz estimates, cf. Section 6.

4.2 Quantitative higher integrability↩︎

Let us prove a quantitative higher integrability result for the solution to problem [pded] under (unbalanced) polynomial growth conditions. More precisely, we have the following theorem.

4.3 Blow up↩︎

Let \(B_{\varrho}(x_{0})\Subset B\) be a ball with radius \(\varrho\in (0,1]\). Keeping in mind Remark 12, we scale on \(B_{1}(\equiv B_{1}(0))\) the function \(u\) by letting \(u_{\varrho}(x):=u(x_{0}+\varrho x)\varrho^{-1}\), and integrands \(\textrm{\texttt{f}},\tilde{\textrm{\texttt{f}}}\) and all related control functions by setting¹⁵ \(\mathcal{w}_{\varrho}(x,z):=\mathcal{w}(x_{0}+\varrho x,z)\) if both indexes \(\varepsilon\), \(\delta\) appear in their original expression, and \(\tilde{\mathcal{w}}_{\varrho}(x,t):=\tilde{\mathcal{w}}(x_{0}+\varrho x,z)\) if they depend only on \(\delta\). Such maps are now defined for all \(x\in\left\{x\in \mathbb{R}^{n}\colon x_{0}+\varrho x\in B\right\}=:\mathcal{B}\), and any \(z\in \mathbb{R}^{N\times n}\). Since \(B_{\varrho}(x_{0})\Subset B\), after scaling, \(B_{1}\Subset \mathcal{B}\), and thanks also to [areg], \(u_{\varrho}\in W^{1,\infty}(B_{1},\mathbb{R}^{N})\) minimizes the integral \[\label funrr W^1,4\gamma(B_1,\mathbb{R}^N)\ni w\mapsto \mathcal{F}_\varrho(w;B_1):=\int_B_{1}\textrm{\texttt{f}}_\varrho(x,Dw)\,{\rm d}x,\] and solves the related Euler–Lagrange system \[\label elrr \int_B_{1}\langle\partial\textrm{\texttt{f}}_\varrho(x,Du_\varrho),Dw\rangle\,{\rm d}x=0\qquadfor all \;\;w\in W^1,4\gamma_0(B_1,\mathbb{R}^N).\] Notice that the integrand \(\textrm{\texttt{f}}_{\varrho}\) is of the same type as those described in Section 4, with obvious, minimal variations impacting only the oscillation properties [a12.x], in which now the bounding constant \(c\) is replaced by \(c(A,\vartheta_{*})\varrho^{\alpha}\) and \(B_{r}\) is any ball contained in \(\mathcal{B}\).

5 Frozen Lipschitz bounds↩︎

In the following, an important role will be played by the constant-coefficient counterpart of functional \(\mathcal{F}_{\varrho}\). Specifically, let \(B_{\sigma}(\equiv B_{\sigma}(x_{\textrm{c}}))\Subset B_{1}\) be a ball, and, for \(z\in \mathbb{R}^{N\times n}\), define \[\begin{align} \label{fi460} \begin{array}{c} \displaystyle \textrm{\texttt{f}}_{\textrm{c}}(z):=\textrm{\texttt{f}}_{\varrho}(x_{\textrm{c}},z),\qquad A_{\textrm{c}}(\lvert z\rvert):=A_{\varrho}(x_{\textrm{c}},\lvert z\rvert),\qquad \textrm{\texttt{a}}_{\textrm{c}}(\lvert z\rvert):=\textrm{\texttt{a}}_{\varrho}(x_{\textrm{c}},\lvert z\rvert),\\ [8pt]\displaystyle \lambda_{\textrm{c}}(\lvert z\rvert):=\lambda_{\varrho}(x_{\textrm{c}},\lvert z\rvert),\qquad \Lambda_{\textrm{c}}(\lvert z\rvert):=\Lambda_{\varrho}(x_{\textrm{c}},\lvert z\rvert). \end{array} \end{align}\tag{14}\] This is nothing but the integrand \(\textrm{\texttt{f}}_{\varrho}\) and related auxiliary functions made autonomous by "freezing" the space-depending component in the center of ball \(B_{\sigma}\). Needles to say, all properties collected in Section 4 continue to hold for \(\textrm{\texttt{f}}_{\textrm{c}}\) (just replace \(x\) with \(x_{\textrm{c}}\) there), while the oscillation conditions [a12.x] will be used to compare \(\textrm{\texttt{f}}_{\textrm{c}}\) to \(\textrm{\texttt{f}}_{\varrho}\) later on. In particular, the convention established in Remark 12 holds also for the frozen rescaled maps above, see also at the beginning of Section 4.3. Next, introduce the variational integral \[W^{1,4\gamma}(B_{\sigma},\mathbb{R}^{N})\ni w\mapsto \mathcal{F}_{\textrm{c}}(w;B_{\sigma}):=\int_{B_{\sigma}}\textrm{\texttt{f}}_{\textrm{c}}(Dw)\,{\rm d}x,\] and, with \(v_{0}\in W^{1,\infty}(\bar{B}_{\sigma},\mathbb{R}^{N})\), let us look at the Dirichlet problem \[\label pd0 v_0+W^1,4\gamma_0(B_\sigma,\mathbb{R}^N)\ni w\mapsto \min_v_{0}+W^{1,4\gamma}(B_{\sigma},\mathbb{R}^{N})\mathcal{F}_\textrm{c}(w;B_\sigma).\] Existence and uniqueness of the solution \(v\in v_{0}+W^{1,4\gamma}(B_{\sigma},\mathbb{R}^{N})\) follows via standard direct methods and strict convexity arguments. We aim at providing some Lipschitz and higher differentiability estimates for problem [pd0] that will play a crucial role in the rest of the paper. This is the content of the following theorem.

6 Intrinsic Lipschitz bounds↩︎

Let \(u_{\varrho}\in W^{1,\infty}(B_{1},\mathbb{R}^{N})\) be a local minimizer of functional \(\mathcal{F}_{\varrho}\) in [funrr]. Recall that \(u_{\varrho}\) solves [elrr] and its Lipschitz continuity is granted by [areg], so, in particular, numbers \[\label mm \textrm{\texttt{M}}\ge \max\left\{\textrm{\texttt{t}},\lVert D \rVert u_L^{\infty}(B_{\varrho}(x_{0}))\right\}\qquadand\qquad \mathcal{M}:=\max\left\{\textrm{\texttt{t}},\lVert \textrm{\texttt{\rVert}}{a}(\cdot,\lvert Du\rvert)_L^{\infty}(B_{\varrho}(x_{0}))\right\}\] are both finite. Let us introduce a series of parameters that will play a crucial role in the reminder of this section.

Step 1: numerology↩︎

Name \[\label ab0 \beta_0:=\frac{2}{2}+\alpha,\qquad \quad \alpha_0:=\frac{\alpha}{2}+\alpha,\] and, with \(i\in \{1,2,3\}\), define numbers \(\omega_{i}, \tilde{\omega}\in (0,1)\) as \[\label omega \begin arrayc \displaystyle \;\omega_i:=1-\frac{i}{2}\left(\frac{\alpha}{n}+\vartheta_*\right),\qquad\quad \tilde{\omega}:=\frac{3}{+}\alpha-\gamma(1+2\alpha)2. \end array\] Next, for \(i\in \{1,2,3\}\), shorten \[\label bbb \begin arrayc \displaystyle \mathcal{b}_0:=36\delta_0(\mu-1+\omega_\mu),\qquad \quad \mathcal{b}_i:=80\gamma\delta_0(\mu-1+\omega_\mu)+\frac{i}{\vartheta}_{*}2+\frac{1}{+}\omega_{i}2\\[10pt]\displaystyle \tilde{\mathcal{}}{b}:=80\gamma\delta_0(\mu-1+\omega_\mu)+\vartheta_*+\frac{1}{+}\tilde{\omega}2. \end array}\] and finally reduce the size of \(\mu\) by requiring that \(1\le \mu<\mu_{\textrm{max}}:=\min\left\{\mu_{0},\mu_{*}\right\},\) where \(\mu_{0}\equiv \mu_{0}(n)\in (1,3/2)\) is the limiting threshold from Theorem 15, cf. [4.10], and \(\mu_{*}\equiv \mu_{*}(\alpha,n,\vartheta_{*})\) is such that \[\label mu* \mu_*:=1+\min\left\{\frac{\chi}{-}1144\delta_{0}\chi,\frac{\chi}{-}11280\gamma\delta_{0}\chi\left(\frac{\alpha}{n}-\vartheta_*\right)\right\}.\] Here we will also use that \(1\le \mu<3/2\), implied by [mu*], to control \(2-\mu>1/2\). Similarly, we pick \(0<\omega_{\mu}<\bar{\omega}_{\mu}\), where \(\omega_{\mu}\) is so small that \[\label tio 0<\bar\omega_\mu<\min\left\{\mu_*-1,\tilde{\omega}_\mu\right\},\] with \(\tilde{\omega}_{\mu}\equiv \tilde{\omega}_{\mu}(n)\) coming from Theorem 15, see [4.10.1]. In the supercritical case \(n=2\) and \(\alpha\ge 2/3\), we restrict further the size of \(\mu_{\textrm{max}}\), \(\bar{\omega}_{\mu}\) by requiring that \[\label muma \begin arrayc \displaystyle \mu_\textrm{max}:=\min\left\{\mu_0,\mu_*,\frac{(}{3}-\alpha+\gamma(2\alpha-3))(\chi-1)1280\delta_{0}\gamma\chi,2+\frac{\alpha}{2}-\gamma\right\}>1,\\ [12pt]\displaystyle 0<\bar\omega_\mu<\min\{\mu_\textrm{max}-1,\tilde{\omega}_\mu\}, \end array\] which makes sense thanks to [a.5.2s].

Step 2: auxiliary estimates↩︎

Pick a vector \(h\in \mathbb{R}^{n}\) with \(\lvert h\rvert\in \left(0,2^{-8/\beta_{0}}\right)\), and for \(x_{\textrm{c}}\in B_{\frac{1}{2}+2\lvert h\rvert^{\beta_{0}}}\) set \(B_{h}:=B_{\lvert h\rvert^{\beta_{0}}}(x_{\textrm{c}})\), which, by construction, satisfies the inclusion \(8B_{h}\Subset B_{3/4}\subset B_{1}\). Next, let \(\textrm{\texttt{f}}_{\textrm{c}}\) be the integrand in Section 5, frozen at point \(x_{\textrm{c}}\), the center of ball \(B_{h}\), and let \(v\in u_{\varrho}+W^{1,4\gamma}_{0}(8B_{h},\mathbb{R}^{N})\) be the solution of Dirichlet problem \[\label 5.0 u_\varrho+W^1,4\gamma_0(8B_h,\mathbb{R}^N)\ni w\mapsto \min_w\in u_{\varrho}+W^{1,4\gamma}_{0}(8B_{h},\mathbb{R}^{N})\int_8B_{h}\textrm{\texttt{f}}_\textrm{c}(Dw)\,{\rm d}x.\] Existence and uniqueness of \(v\) follow by standard direct methods. By minimality, \(v\) solves the Euler-Lagrange system \[\label 5.3 \int_8B_{h}\langle\partial\textrm{\texttt{f}}_\textrm{c}(Dv),Dw\rangle\,{\rm d}x=0\qquadfor all \;\;w\in W^1,4\gamma_0(8B_h,\mathbb{R}^N),\] and satisfies the energy estimate \[\label 5.4 \int_8B_{h}\textrm{\texttt{f}}_\textrm{c}(Dv)\,{\rm d}x\le \int_8B_{h}\textrm{\texttt{f}}_\textrm{c}(Du_\varrho)\,{\rm d}x.\] Moreover, keeping in mind also [areg], Theorem 15 applies, so \(v\in W^{1,\infty}_{\operatorname{loc}}(8B_{h},\mathbb{R}^{N})\cap W^{2,2}_{\operatorname{loc}}(8B_{h},\mathbb{R}^{N})\) with \(\partial\textrm{\texttt{f}}_{\textrm{c}}(Dv)\in W^{1,2}_{\operatorname{loc}}(8B_{h},\mathbb{R}^{N\times n})\) and \(\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)\in W^{1,2}_{\operatorname{loc}}(8B_{h})\), cf. [4.0]. Before entering into the key bounds of this section, let us record a few preliminary estimates. Since \(\textrm{\texttt{a}}_{\textrm{c}}\) is increasing and continuous, we bound \[\begin{align} \label{10460} \textrm{\texttt{a}}_{\textrm{c}}(\textrm{\texttt{M}})&\le&\lVert \textrm{\texttt{a}}_{\textrm{c}}(\lvert Du_{\varrho}\rvert) \rVert_{L^{\infty}(B_{1})}\nonumber \\ &\stackrel{\eqref{a12.x}_{1}}{\le}&\mathcal{M}+c\varrho^{\alpha}\textrm{\texttt{M}}\left(1+\left\|\inf_{x\in B_{1}}\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)\right\|_{L^{\infty}(B_{1})}^{\vartheta_{*}}\right)\nonumber \\ &\stackrel{\eqref{a.7.1.x}_{1}}{\le}&\mathcal{M}+c\varrho^{\alpha}\textrm{\texttt{r}}_{*}(\textrm{\texttt{M}})^{\vartheta_{*}}\textrm{\texttt{M}}\mathcal{M}^{\vartheta_{*}}\nonumber \\ &\stackrel{\eqref{rrr}}{\le}&\mathcal{M}+c\varrho^{\alpha}\textrm{\texttt{M}}^{2\vartheta_{*}(\mu-1+\omega_{\mu})+1}\mathcal{M}^{\vartheta_{*}}\stackrel{\eqref{a.7.1.x}_{4}}{\le}\mathcal{M}+c\varrho^{\alpha}\mathcal{M}^{6(\mu-1+\omega_{\mu})+\vartheta_{*}+1}, \end{align}\tag{21}\] with \(c\equiv c(A,\textrm{\texttt{g}},\mu,\gamma,\alpha)\). Recalling [mm], by 21 , estimates [4.4]–[4.8] can be rewritten as \[\begin{align} \label{5461} \lVert Dv \rVert_{L^{\infty}(6B_{h})}\le c\textrm{\texttt{a}}_{\textrm{c}}\left(\textrm{\texttt{M}}\right)^{(\mu-1+\omega_{\mu})\delta_{0}}\textrm{\texttt{M}}\le c\mathcal{M}^{\delta_{0}(6(\mu-1+\omega_{\mu})+\vartheta_{*}+1)(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}\le \mathcal{M}^{16\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}, \end{align}\tag{22}\] for \(c\equiv c(\textrm{\texttt{data}}_{0},\alpha)\), and, given any ball \(B_{\tau}\subseteq 6B_{h}\) and number \(\,\kappa\ge 0\), \[\begin{align} \label{5462} \tau\lVert D(\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)-\,\kappa)_{+} \rVert_{L^{2}(B_{\tau/2})}&\stackrel{\eqref{5461}}{\le}&c\textrm{\texttt{r}}_{*}(\mathcal{M}^{16\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}})^{\frac{1}{2}}\lVert (\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)-\,\kappa)_{+} \rVert_{L^{2}(B_{\tau})}\nonumber \\ &\stackrel{\eqref{rrr},\eqref{a.7.1.x}_{4}}{\le}&c\mathcal{M}^{36\delta_{0}(\mu-1+\omega_{\mu})}\lVert (\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)-\,\kappa)_{+} \rVert_{L^{2}(B_{\tau})}, \end{align}\tag{23}\] with \(c\equiv c(\textrm{\texttt{data}}_{0},\alpha)\). Moreover, on \(6B_{h}\), \[\begin{align} \label{5466} \textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)&\stackrel{\eqref{5461}}{\le}&\int_{0}^{c\mathcal{M}^{16\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}}\tilde{\lambda}_{\textrm{c}}(s)s\,{\rm d}x+c\sigma_{\varepsilon}\ell_{1}\left(\mathcal{M}^{32\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}^{2}\right)^{2\gamma}\nonumber \\ &\stackrel{\eqref{a.55}_{1}}{\le}& c\tilde{\lambda}_{\textrm{c}}(c\mathcal{M}^{16\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}})\mathcal{M}^{16\mu\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}^{\mu}\int_{0}^{c\mathcal{M}^{16\delta_{0}(\mu-1+\omega_{\mu})}\textrm{\texttt{M}}}s^{1-\mu}\,{\rm d}x\nonumber \\ &&+c\mathcal{M}^{64\gamma\delta_{0}(\mu-1+\omega_{\mu})}\sigma_{\varepsilon}\ell_{1}(\textrm{\texttt{M}}^{2})^{2\gamma}\nonumber \\ &\stackrel{\eqref{a.55}_{2}}{\le}&c\mathcal{M}^{32\delta_{0}(\mu-1+\omega_{\mu})}\tilde{\lambda}_{\textrm{c}}(\textrm{\texttt{M}})\textrm{\texttt{M}}^{2}+c\mathcal{M}^{64\gamma\delta_{0}(\mu-1+\omega_{\mu})}\sigma_{\varepsilon}\ell_{1}(\textrm{\texttt{M}}^{2})^{2\gamma-1}\textrm{\texttt{M}}^{2}\nonumber \\ &\le&c\mathcal{M}^{64\gamma\delta_{0}(\mu-1+\omega_{\mu})}\lambda_{\textrm{c}}(\textrm{\texttt{M}})\textrm{\texttt{M}}^{2} \stackrel{\eqref{a.7.1.x}_{2}}{\le} c\mathcal{M}^{64\gamma\delta_{0}(\mu-1+\omega_{\mu})}\left(\lVert \textrm{\texttt{a}}_{\textrm{c}}(\lvert Du_{\varrho}\rvert) \rVert_{L^{\infty}(B_{1})}+1\right), \end{align}\tag{24}\] where \(c\equiv c(\textrm{\texttt{data}}_{0},\alpha)\). Combining 21 and 24 we obtain \[\label 10.1 \lVert \textrm{\texttt{\rVert}}{a}_{\textrm{c}}(\lvert Dv\rvert)_L^{\infty}(6B_{h})\le c\mathcal{M}^1+64\gamma\delta_{0}(\mu-1+\omega_{\mu})+c\varrho^\alpha\mathcal{M}^70\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1,\] for \(c\equiv c(\textrm{\texttt{data}}_{0},\alpha)\). We shall also need the following estimates. For \(x\in 6B_{h}\), we control oscillation \[\begin{align} \label{5468} \mathcal{A}_{1}&:=&\lvert \textrm{\texttt{a}}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)-\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)\rvert\nonumber \\ &\stackrel{\eqref{a12.x}_{1}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\ell_{\delta}(\lvert Du_{\varrho}\rvert)+c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\left(\inf_{x\in 6B_{h}}\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)\right)^{\vartheta_{*}}\ell_{\delta}(\lvert Du_{\varrho}\rvert)\nonumber \\ &\stackrel{\eqref{a.7.1.x}_{1,4}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}+c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\textrm{\texttt{r}}_{*}(\textrm{\texttt{M}})^{\vartheta_{*}}\tilde{\textrm{\texttt{a}}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{\vartheta_{*}+\frac{1}{2-\mu}}\nonumber \\ &\stackrel{\eqref{rrr}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}+c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{6(\mu-1+\omega_{\mu})}\tilde{\textrm{\texttt{a}}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{\vartheta_{*}+1}, \end{align}\tag{25}\] for \(c\equiv c(A,\textrm{\texttt{g}},\mu,\gamma,\alpha)\). Furthermore, after letting \[\mathcal{D}_{\textrm{c}}:=\sqrt{\lambda_{\textrm{c}}(\lvert Du_{\varrho}\rvert+\lvert Dv\rvert)}\lvert Du_{\varrho}-Dv\rvert,\] by the mean value theorem, \(\eqref{a.7.1.x}_{2}\), [a.55], [l60], 21 , and [10.1] we also have \[\begin{align} \label{5469} \mathcal{A}_{2}&:=&\lvert \textrm{\texttt{a}}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)-\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)\rvert\nonumber\\ &\le&c\left(\int_{0}^{1}\tilde{\lambda}_{\textrm{c}}(\lvert Dv+s(Du_{\varrho}-Dv)\rvert)\lvert Dv+s(Du_{\varrho}-Dv)\rvert\,{\rm d}s\right)\lvert Du_{\varrho}-Dv\rvert\nonumber \\ &&+c\sigma_{\varepsilon}\left(\int_{0}^{1}\ell_{1}(\lvert Dv+s(Du_{\varrho}-Dv)\rvert^{2})^{2\gamma-1}\lvert Dv+s(Du_{\varrho}-Dv)\rvert\,{\rm d}s\right)\lvert Du_{\varrho}-Dv\rvert\nonumber \\ &\le&c\tilde{\lambda}_{\textrm{c}}(\lvert Du_{\varrho}\rvert+\lvert Dv\rvert)\left(\lvert Du_{\varrho}\rvert+\lvert Dv\rvert\right)^{\mu}\left(\int_{0}^{1}\lvert Dv+s(Du_{\varrho}-Dv)\rvert^{1-\mu}\,{\rm d}s\right)\lvert Du_{\varrho}-Dv\rvert\nonumber \\ &&+c\sqrt{\sigma_{\varepsilon}}\ell_{1}(\lvert Du_{\varrho}\rvert^{2}+\lvert Dv\rvert^{2})^{\gamma}\mathcal{D}_{\textrm{c}}\nonumber \\ &\le&c\left(\left(\tilde{\lambda}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)\lvert Du_{\varrho}\rvert^{2}\right)^{\frac{1}{2}}+\left(\tilde{\lambda}_{\textrm{c}}(\lvert Dv\rvert)\lvert Dv\rvert^{2}\right)^{\frac{1}{2}}+\sqrt{\sigma_{\varepsilon}}\ell_{1}(\lvert Du_{\varrho}\rvert^{2}+\lvert Dv\rvert^{2})^{\gamma}\right)\mathcal{D}_{\textrm{c}}\nonumber \\ &\le&c\left(1+\textrm{\texttt{a}}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)^{\frac{1}{2}}+\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)^{\frac{1}{2}}\right)\mathcal{D}_{\textrm{c}}\nonumber \\ &\le& c\left(\mathcal{M}^{\frac{1}{2}+32\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\frac{\alpha}{2}}\mathcal{M}^{35\gamma\delta_{0}(\mu-1+\omega_{\mu})+\frac{\vartheta_{*}+1}{2}}\right)\mathcal{D}_{\textrm{c}}, \end{align}\tag{26}\] with \(c\equiv c(\textrm{\texttt{data}}_{0},\alpha)\). Now we are ready to enter the core of the proof.

Step 3: comparison↩︎

We jump back to [5.3], and see that, upon extension as \(v=u_{\varrho}\) in \(B_{1}\setminus 8B_{h}\), by [areg], \(v-u_{\varrho}\in W^{1,4\gamma}_{0}(8B_{h},\mathbb{R}^{N})\) is an admissible test function in both [elrr] and [5.3], so we bound via the mean value theorem, \[\begin{align} \label{com} \int_{8B_{h}}\mathcal{D}_{\textrm{c}}^{2}\,{\rm d}x&\stackrel{\eqref{lala}_{2}}{\le}&c\int_{8B_{h}}\langle \partial\textrm{\texttt{f}}_{\textrm{c}}(Dv)-\partial\textrm{\texttt{f}}_{\textrm{c}}(Du_{\varrho}),Dv-Du_{\varrho}\rangle\,{\rm d}x\nonumber \\ &\stackrel{\eqref{5.3},\eqref{elrr}}{\le}&c\int_{8B_{h}}\langle\partial\textrm{\texttt{f}}_{\varrho}(x,Du_{\varrho})-\partial\textrm{\texttt{f}}_{\textrm{c}}(Du_{\varrho}),Dv-Du_{\varrho}\rangle\,{\rm d}x\nonumber \\ &\stackrel{\eqref{a12.x}_{2}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\beta_{0}\alpha}\int_{8B_{h}}\left(1+\inf_{x\in 8B_{h}}\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)\right)^{\vartheta_{*}}\lvert Du_{\varrho}-Dv\rvert\,{\rm d}x\nonumber \\ &\stackrel{\eqref{a.7.1.x}_{1,4}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\textrm{\texttt{r}}_{*}(\mathcal{M}^{1/(2-\mu)})^{\vartheta_{*}}\mathcal{M}^{\vartheta_{*}}\int_{8B_{h}}\lvert Du_{\varrho}\rvert+\lvert Dv\rvert\,{\rm d}x\nonumber \\ &\stackrel{\eqref{rrr}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{\vartheta_{*}+4(\mu-1+\omega_{\mu})}\int_{8B_{h}}1+\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)+\tilde{A}_{\textrm{c}}(\lvert Dv\rvert)\,{\rm d}x\nonumber \\ &\stackrel{\eqref{5.4}}{\le}&c\varrho^{\alpha}\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{\vartheta_{*}+4(\mu-1+\omega_{\mu})}\int_{8B_{h}}1+\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)+A_{\textrm{c}}(\lvert Du_{\varrho}\rvert)\,{\rm d}x=:\mathcal{C}, \end{align}\tag{27}\] where \(c\equiv c(\textrm{\texttt{data}})\).

Step 4: uniform bounds in Nikol’skii spaces↩︎

Let \(\omega_{1},\omega_{2},\omega_{3}, \tilde{\omega}\in (0,1)\) be the numbers in [omega]. Basic properties of translations, the \(1\)-Lipschitz character of truncations, the Poincaré inequality, \(\eqref{a.7.1.x}_{1,4}\), [4.0], 23 and 25 –27 yield \[\begin{align} \label{6462146x} \int_{B_{h}}\lvert \tau_{h}(\textrm{\texttt{a}}_{\varrho}(\cdot,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x&\le&c\int_{B_{h}}\lvert \tau_{h}(\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x+c\int_{2B_{h}}\mathcal{A}_{1}^{2}+\mathcal{A}_{2}^{2}\,{\rm d}x\nonumber \\ &\le&c\lvert h\rvert^{2}\int_{B_{h}}\lvert D(\textrm{\texttt{a}}_{\textrm{c}}(\lvert Dv\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x\nonumber \\ &&+c\varrho^{2\alpha}\lvert h\rvert^{2\alpha\beta_{0}}\mathcal{M}^{12(\mu-1+\omega_{\mu})}\int_{2B_{h}}\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{2+2\vartheta_{*}}+1\,{\rm d}x\nonumber \\ &&+c\left(\mathcal{M}^{1+64\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\alpha}\mathcal{M}^{70\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}\right)\int_{2B_{h}}\mathcal{D}_{\textrm{c}}^{2}\,{\rm d}x\nonumber \\ &\le&c\mathcal{M}^{72\delta_{0}(\mu-1+\omega_{\mu})}\lvert h\rvert^{2(1-\beta_{0})}\int_{2B_{h}}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}^{2}\,{\rm d}x\nonumber \\ &&+c\mathcal{M}^{72\delta_{0}(\mu-1+\omega_{\mu})}\lvert h\rvert^{2(1-\beta_{0})}\int_{2B_{h}}\mathcal{A}_{1}^{2}+\mathcal{A}_{2}^{2}\,{\rm d}x\nonumber \\ &&+c\varrho^{2\alpha}\lvert h\rvert^{2\alpha\beta_{0}}\mathcal{M}^{12(\mu-1+\omega_{\mu})}\int_{2B_{h}}\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{2+2\vartheta_{*}}+1\,{\rm d}x\nonumber \\ &&+c\left(\mathcal{M}^{1+64\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\alpha}\mathcal{M}^{70\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}\right)\int_{2B_{h}}\mathcal{D}_{\textrm{c}}^{2}\,{\rm d}x\nonumber \\ &\le&c\mathcal{M}^{72\delta_{0}(\mu-1+\omega_{\mu})}\lvert h\rvert^{2(1-\beta_{0})}\int_{2B_{h}}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}^{2}\nonumber \\ &&+c\varrho^{2\alpha}\lvert h\rvert^{2\alpha\beta_{0}}\mathcal{M}^{84\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+\omega_{2}+1}\int_{2B_{h}}\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{1-\omega_{2}}+1\,{\rm d}x\nonumber \\ &&+c\left(\mathcal{M}^{1+136\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\alpha}\mathcal{M}^{142\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}\right)\mathcal{C}, \end{align}\tag{28}\] for \(c\equiv c(\textrm{\texttt{data}})\). To complete estimate 28 , we only need to distinguish two cases: \(n\ge 3\) or \(n=2\) and \(0<\alpha<2/3\), and \(n=2\) with \(\alpha\ge 2/3\). In the first case, by \(\eqref{d2.3}_{2}\), [rrr], [a.7.1.x]\(_{1,4}\), and [a12.x]\(_{1}\) we bound \[\begin{align} &\left(\mathcal{M}^{1+136\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\alpha}\mathcal{M}^{142\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}\right)\mathcal{C}\nonumber \\ &\qquad \qquad \le c\lvert h\rvert^{\alpha\beta_{0}}\left(\varrho^{\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}+\varrho^{2\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}A_{\varrho}(x,\lvert Du_{\varrho}\rvert)+1\,{\rm d}x\nonumber \\ &\qquad \qquad \quad +c\lvert h\rvert^{2\alpha\beta_{0}}\left(\varrho^{2\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}+\varrho^{3\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}A_{\varrho}(x,\lvert Du_{\varrho}\rvert)^{\vartheta_{*}}\ell_{\delta}(\lvert Du_{\varrho}\rvert)\,{\rm d}x\nonumber \\ &\qquad \qquad \le c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1+\omega_{1}}\varrho^{\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{1}}\,{\rm d}x\nonumber \\ &\qquad \qquad\quad + c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1+\omega_{2}}\varrho^{2\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{2}}\,{\rm d}x\nonumber \\ &\qquad \qquad\quad + c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma \delta_{0}(\mu-1+\omega_{\mu})+3\vartheta_{*}+1+\omega_{3}}\varrho^{3\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{3}}\,{\rm d}x, \end{align}\] for \(c\equiv c(\textrm{\texttt{data}})\), while in the second one, via \(\eqref{d2.3}_{2}\), [rrr], [a.7.1.x]\(_{1}\), [a12.x]\(_{1}\), and [a.5.3s] we have \[\begin{align} &\left(\mathcal{M}^{1+136\gamma\delta_{0}(\mu-1+\omega_{\mu})}+\varrho^{\alpha}\mathcal{M}^{142\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}\right)\mathcal{C}\nonumber \\ &\qquad \qquad \le c\lvert h\rvert^{\alpha\beta_{0}}\left(\varrho^{\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+1+\vartheta_{*}}+\varrho^{2\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}1+\tilde{A}_{\varrho}(x,\lvert Du_{\varrho}\rvert)\,{\rm d}x\nonumber \\ &\qquad \qquad \quad +c\lvert h\rvert^{\alpha\beta_{0}}\left(\varrho^{\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}+\varrho^{2\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}\tilde{A}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)+\sigma_{\varepsilon}\ell_{1}(\lvert Du_{\varrho}\rvert^{2})^{2\gamma}\nonumber \\ &\qquad \qquad \le c\lvert h\rvert^{\alpha\beta_{0}}\left(\varrho^{\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}+\varrho^{2\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}1+A_{\varrho}(x,\lvert Du_{\varrho}\rvert)\,{\rm d}x\nonumber \\ &\qquad \qquad \quad +c\lvert h\rvert^{\alpha\beta_{0}}\left(\varrho^{\alpha}\mathcal{M}^{140\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1}+\varrho^{2\alpha}\mathcal{M}^{146\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1}\right)\int_{8B_{h}}\tilde{A}_{\textrm{c}}(\lvert Du_{\varrho}\rvert)\,{\rm d}x\nonumber \\ &\qquad \qquad \le c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma\delta_{0}(\mu-1+\omega_{\mu})+\vartheta_{*}+1+\omega_{1}}\varrho^{\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{1}}\,{\rm d}x\nonumber \\ &\qquad \qquad\quad +c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1+\omega_{2}}\varrho^{2\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{2}}\,{\rm d}x\nonumber \\ &\qquad \qquad\quad +c\lvert h\rvert^{\alpha\beta_{0}}\mathcal{M}^{160\gamma\delta_{0}(\mu-1+\omega_{\mu})+2\vartheta_{*}+1+\tilde{\omega}}\varrho^{2\alpha}\int_{8B_{h}}\ell_{1}(\lvert Du_{\varrho}\rvert)^{\gamma-\tilde{\omega}}\,{\rm d}x, \end{align}\] with \(c\equiv c(\textrm{\texttt{data}})\). Plugging the content of the two previous displays into 28 and recalling 1 , [ab0]–[bbb], we obtain \[\begin{align} \label{64621} \int_{B_{h}}\lvert \tau_{h}(\textrm{\texttt{a}}_{\varrho}(\cdot,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x&\le&c\mathcal{M}^{2\mathcal{b}_{0}}\lvert h\rvert^{2\alpha_{0}}\int_{8B_{h}}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}^{2}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\sum_{i=1}^{3}\mathbb{1}_{i}\mathcal{M}^{2\mathcal{b}_{i}}\varrho^{i\alpha}\int_{8B_{h}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{i}}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\tilde{\mathbb{1}}\mathcal{M}^{2\tilde{b}}\varrho^{2\alpha}\int_{8B_{h}}\ell_{1}(\lvert Du_{\varrho}\rvert)^{\gamma-\tilde{\omega}}\,{\rm d}x, \end{align}\tag{29}\] where in particular the choice made in [ab0] yields equality \(\alpha\beta_{0}=2(1-\beta_{0})\). Let us glue estimates 29 via a dyadic covering argument. Specifically, we take a lattice \(\mathcal{L}_{\lvert h\rvert^{\beta_{0}}/\sqrt{n}}\) of open, disjoint cubes \(\{Q_{\lvert h\rvert^{\beta_{0}}/\sqrt{n}}(y)\}_{y\in (2\lvert h\rvert^{\beta_{0}}/\sqrt{n})\mathbb{Z}^{n}}\). From this lattice, we pick \(\textrm{\texttt{n}}\approx_{n}\lvert h\rvert^{-n\beta_{0}}\) cubes centered at points \(\{x_{\textrm{c}}\}_{\textrm{c}\le \textrm{\texttt{n}}}\subset (2\lvert h\rvert^{\beta_{0}}/\sqrt{n})\mathbb{Z}^{n}\) such that \(\lvert x_{\textrm{c}}\rvert\le 1/2+2\lvert h\rvert^{\beta_{0}}\), thus determining the corresponding family \(\{Q_{\textrm{c}}\}_{\textrm{c}\le \textrm{\texttt{n}}}\equiv \{Q_{\lvert h\rvert^{\beta_{0}}/\sqrt{n}}(x_{\textrm{c}})\}_{\textrm{c}\le \textrm{\texttt{n}}}\). Observe that, in general, if \(\lvert x\rvert>1/2+2\lvert h\rvert^{\beta_{0}}\), then \(Q_{\lvert h\rvert^{\beta_{0}}/\sqrt{n}}(x)\cap B_{1/2}=\emptyset\) as \(Q_{\lvert h\rvert^{\beta_{0}}/\sqrt{n}}(x)\subset B_{\lvert h\rvert^{\beta_{0}}}(x)\) and \(B_{\lvert h\rvert^{\beta_{0}}}(x)\cap B_{1/2}=\emptyset\). We indeed have \[\begin{align} \label{com4613} \left| \; B_{1/2}\setminus \bigcup_{\textrm{c}\le \textrm{\texttt{n}}}Q_{\textrm{c}} \;\right|=0,\qquad\qquad Q_{\textrm{c}_{1}}\cap Q_{\textrm{c}_{2}}=\emptyset \;\Longleftrightarrow \;\textrm{c}_{1}\not =\textrm{c}_{2}. \end{align}\tag{30}\] Such a family of cubes corresponds to a family of balls \(\{B_{\textrm{c}}\}_{\textrm{c}\le \textrm{\texttt{n}}}:=\{B_{\lvert h\rvert^{\beta_{0}}}(x_{\textrm{c}})\}_{\textrm{c}\le \textrm{\texttt{n}}}\) in the sense that \(Q_{\textrm{c}}\) is the largest hypercube concentric to \(B_{\textrm{c}}\), with sides parallel to the coordinate axes. By construction \(8B_{\textrm{c}}\Subset B_{1}\) for all \(\textrm{c}\le \textrm{\texttt{n}}\). Moreover, each of the dilated balls \(8B_{\textrm{c}_{t}}\) intersects the similar ones \(8B_{\textrm{c}_{s}}\), \(\textrm{c}_{t}\not =\textrm{c}_{s}\) a finite, quantified number of times, depending only on \(n\) (uniform finite intersection property). In fact, notice that the family of outer cubes \(\{Q_{\lvert h\rvert^{\beta_{0}}}(x_{\textrm{c}})\}_{\textrm{c}\le \textrm{\texttt{n}}}\) has the same property and \(B_{\textrm{c}}\subset Q_{\lvert h\rvert^{\beta_{0}}}(x_{\textrm{c}})\). This yields: \[\label com.13.1 \sum_\textrm{c}\le \textrm{\texttt{n}}\phi(8B_\textrm{c})\lesssim_n\phi(B_1),\] for every Borel measure \(\phi\) defined on \(B_{1}\). By 30 it turns out that also \(\{B_{\textrm{c}}\}_{\textrm{c}\le \textrm{\texttt{n}}}\) is a measure covering of \(B_{1/2}\), i.e.: \[\label com.13.2 \left| \; B_1/2\setminus \bigcup_\textrm{c}\le \textrm{\texttt{n}}B_\textrm{c} \;\right|=0.\] By [com.13.2], 29 and [com.13.1] we obtain \[\begin{align} \label{64622} \int_{B_{1/2}}\lvert \tau_{h}(\textrm{\texttt{a}}_{\varrho}(\cdot,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x&\stackrel{\eqref{com.13.2}}{\le}&\sum_{\textrm{c}\le \textrm{\texttt{n}}}\int_{B_{\textrm{c}}}\lvert \tau_{h}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}\rvert^{2}\,{\rm d}x\nonumber \\ &\stackrel{\eqref{64621}}{\le}& c\lvert h\rvert^{2\alpha_{0}}\mathcal{M}^{2\mathcal{b}_{0}}\sum_{\textrm{c}\le\textrm{\texttt{n}}}\int_{8B_{\textrm{c}}}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}^{2}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\sum_{i=1}^{3}\mathbb{1}_{i}\mathcal{M}^{2\mathcal{b}_{i}}\varrho^{i\alpha}\sum_{\textrm{c}\le\textrm{\texttt{n}}}\int_{8B_{\textrm{c}}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{i}}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\tilde{\mathbb{1}}\mathcal{M}^{2\tilde{\mathcal{b}}}\varrho^{2\alpha}\sum_{\textrm{c}\le \textrm{\texttt{n}}}\int_{8B_{\textrm{c}}}\ell_{1}(\lvert Du_{\varrho}\rvert)^{\gamma-\tilde{\omega}}\,{\rm d}x\nonumber \\ &\stackrel{\eqref{com.13.1}}{\le}& c\lvert h\rvert^{2\alpha_{0}}\mathcal{M}^{2\mathcal{b}_{0}}\int_{B_{1}}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert)-\,\kappa)_{+}^{2}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\sum_{i=1}^{3}\mathbb{1}_{i}\mathcal{M}^{2\mathcal{b}_{i}}\varrho^{i\alpha}\int_{B_{1}}\ell_{1}(\textrm{\texttt{a}}_{\varrho}(x,\lvert Du_{\varrho}\rvert))^{1-\omega_{i}}\,{\rm d}x\nonumber \\ &&+c\lvert h\rvert^{2\alpha_{0}}\tilde{\mathbb{1}}\mathcal{M}^{2\tilde{\mathcal{b}}}\varrho^{2\alpha}\int_{B_{1}}\ell_{1}(\lvert Du_{\varrho}\rvert)^{\gamma-\tilde{\omega}}\,{\rm d}x, \end{align}\tag{31}\] for \(c\equiv c(\textrm{\texttt{data}})\). The availability of 31 allows applying Lemma 3, so that, after scaling back to \(B_{\varrho}(x_{0})\), \[\begin{align} \label{66460} \mathpalette\@thickbar{\lVert} (\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)-\,\kappa)_{+} \rVert_{L^{2\chi}(B_{\varrho/2}(x_{0}))}&\le& c\mathcal{M}^{\mathcal{b}_{0}}\mathpalette\@thickbar{\lVert} (\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)-\,\kappa)_{+} \rVert_{L^{2}(B_{\varrho}(x_{0}))} \nonumber \\ &&+c\sum_{i=1}^{3}\mathbb{1}_{i}\mathcal{M}^{\mathcal{b}_{i}}\varrho^{\frac{i\alpha}{2}}\left(\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{B_{\varrho}(x_{0})}\ell_{1}(\textrm{\texttt{a}}(x,\lvert Du\rvert))^{1-\omega_{i}}\,{\rm d}x\right)^{\frac{1}{2}}\nonumber \\ &&+c\tilde{\mathbb{1}}\mathcal{M}^{\tilde{\mathcal{b}}}\varrho^{\alpha}\left(\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{B_{\varrho}(x_{0})}\ell_{1}(\lvert Du\rvert)^{\gamma-\tilde{\omega}}\,{\rm d}x\right)^{\frac{1}{2}}, \end{align}\tag{32}\] for all \(\,\kappa\ge 0\), with \(\chi:=n/(n-2\beta)>1\) for all \(\beta\in (0,\alpha_{0})\) — say \(\beta=\alpha_{0}/2\) to fix dependencies — and \(c\equiv c(\textrm{\texttt{data}})\).

Step 6: Lipschitz bounds↩︎

Let \(B_{r}\subset B_{2r}\Subset B\) be any ball with radius \(r\in (0,1]\), consider concentric balls \(B_{r/2}\subseteq B_{\tau_{2}}\Subset B_{\tau_{1}}\subseteq B_{3r/4}\) and notice that there is no loss of generality in assuming that \(\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{r/2})}\ge \textrm{\texttt{t}}\), otherwise there would be nothing to prove. By [a12.x] and [areg], all \(x_{0}\in B_{\tau_{2}}\) are Lebesgue points for \(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)\), see [38], so we set \(r_{0}:=(\tau_{1}-\tau_{2})/8\) so that \(B_{2r_{0}}(x_{0})\Subset B_{\tau_{1}}\), and, via 32 applied on \(B_{r_{0}}(x_{0})\) with \(\mathcal{M}:=\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}\) (that satisfies [mm]) we can apply Lemma 2 choosing \(k=4\), \(\,\kappa_{0}=0\), \(w(x)=\textrm{\texttt{a}}(x,\lvert Du(x)\rvert)\), \(M_{0}= \mathcal{M}^{b_{0}}\), \(M_{i}=\mathbb{1}_{i}M^{\mathcal{b}_{i}}\) for \(i\in \{1,2,3\}\), \(M_{4}:=\tilde{\mathbb{1}}\mathcal{M}^{\tilde{\mathcal{b}}}\), \(f_{i}=\ell_{1}(\textrm{\texttt{a}}(x,\lvert Du\rvert))^{1-\omega_{i}}\) as \(i\in \{1,2,3\}\), \(f_{4}:=\ell_{1}(\lvert Du\rvert)^{\gamma-\tilde{\omega}}\), \(\sigma_{i}=i\alpha/2\) if \(i\in \{1,2,3\}\), \(\sigma_{4}=\alpha\), and \(\vartheta_{i}=1/2\) for all \(i\in \{1,\cdots,4\}\), to get \[\begin{align} \textrm{\texttt{a}}(x_{0},\lvert Du(x_{0})\rvert)&\le&c\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}\chi}{\chi-1}}\left(\mathop{\int\hskip -1,05em -\, \!\!\!}\nolimits_{B_{r_{0}}(x_{0})}\textrm{\texttt{a}}(x,\lvert Du\rvert)^{2}\,{\rm d}x\right)^{\frac{1}{2}}+c\nonumber \\ &&+c\sum_{i=1}^{3}\mathbb{1}_{i}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\mathcal{b}_{i}}\mathbf{P}^{\frac{1}{2}}_{\frac{i\alpha}{2}}\left(\ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert))^{1-\omega_{i}};x_{0} ,\frac{\tau_{1}-\tau_{2}}{4}\right)\nonumber \\ &&+c\tilde{\mathbb{1}}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\tilde{\mathcal{b}}}\mathbf{P}^{\frac{1}{2}}_{\alpha}\left(\ell_{1}(\lvert Du\rvert)^{\gamma-\tilde{\omega}};x_{0} ,\frac{\tau_{1}-\tau_{2}}{4}\right), \end{align}\] with \(c\equiv c(\textrm{\texttt{data}})\). The arbitrariness of \(x_{0}\in B_{\tau_{2}}\) grants \[\begin{align} \label{14460} \lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{2}})}&\le&\frac{c\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}\chi}{\chi-1}+\frac{1}{2}}}{(\tau_{1}-\tau_{2})^{n/2}}\left(\int_{B_{\tau_{1}}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{1}{2}}+c\nonumber \\ &&+c\sum_{i=1}^{3}\mathbb{1}_{i}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\mathcal{b}_{i}}\left\|\mathbf{P}^{\frac{1}{2}}_{\frac{i\alpha}{2}}\left(\ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert))^{1-\omega_{i}};\;\cdot \;,\frac{\tau_{1}-\tau_{2}}{4}\right)\right\|_{L^{\infty}(B_{\tau_{2}})}\nonumber \\ &&+c\tilde{\mathbb{1}}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\tilde{\mathcal{b}}}\left\|\mathbf{P}^{\frac{1}{2}}_{\alpha}\left(\ell_{1}(\lvert Du\rvert)^{\gamma-\tilde{\omega}};\;\cdot \;,\frac{\tau_{1}-\tau_{2}}{4}\right)\right\|_{L^{\infty}(B_{\tau_{2}})}, \end{align}\tag{33}\] for \(c\equiv c(\textrm{\texttt{data}})\). At this stage, we need to reabsorbe the \(L^{\infty}\)-norms of \(\textrm{\texttt{a}}\) on the right-hand side of 33 and, simultaneously, to keep under control the nonlinear potentials. To do so, let us premise that, since all the constraints in [a.3], [a.5.2s] are strict, there is no loss of generality in assuming \(\alpha\in (0,1)\) — otherwise one could just replace it in estimates 29 , 31 with an \(\tilde{\alpha}\in (0,\alpha)\) arbitrarily close to \(\alpha\). Keeping this in mind, let us treat separately the subcritical and the supercritical cases.

Step 7: case \(n\ge 3\) or \(n=2\) and \(0<\alpha<2/3\)↩︎

Looking back at 1 , here \(\mathbb{1}_{3}=1\) and \(\tilde{\mathbb{1}}=0\), so 33 becomes \[\begin{align} \label{14461} \lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{2}})}&\le&\frac{c\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}\chi}{\chi-1}+\frac{1}{2}}}{(\tau_{1}-\tau_{2})^{n/2}}\left(\int_{B_{\tau_{1}}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{1}{2}}+c\nonumber \\ &&+c\sum_{i=1}^{3}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\mathcal{b}_{i}}\left\|\mathbf{P}^{\frac{1}{2}}_{\frac{i\alpha}{2}}\left(\ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert))^{1-\omega_{i}};\;\cdot \;,\frac{\tau_{1}-\tau_{2}}{4}\right)\right\|_{L^{\infty}(B_{\tau_{2}})}. \end{align}\tag{34}\] Thanks to [atat] and to the choices made in [omega]–[mu*], we immediately see that \[\label 14.3 \frac{\mathcal{b}}{_}{0}\chi\chi-1+\frac{1}{2}<1,\qquad\quad \frac{\mathcal{b}}{_}{0}\chi-1+\mathcal{b}_i<1,\qquad\quad \frac{n}{i}\alpha>1,\qquad\quad 0<\frac{n}{(}1-\omega_{i})i\alpha<1,\] so we can apply Lemma 1 with \(m=(1-\omega_{i})^{-1}\), \(i\in \{1,2,3\}\), to bound \[\label 14.4 \left\|\mathbf{P}^\frac{1}{2}_\frac{i\alpha}{2}\left(\ell_1(\textrm{\texttt{a}}(\cdot,\lvert D\rvert u))^1-\omega_{i};\;\cdot \;,\frac{\tau}{_}{1}-\tau_{2}4\right)\right\|_L^{\infty}(B_{\tau_{2}})\le c\lVert \ell \rVert_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert))_L^{1}(B_{r})^\frac{1-\omega_{i}}{2},\] for \(c\equiv c(n,\alpha,\vartheta_{*})\), use Young’s inequality with conjugate exponents \[\label 14.6 \left(\frac{2}{(}\chi-1)2\mathcal{b}_{0}\chi+\chi-1,\frac{2}{(}\chi-1)\chi-1-2\mathcal{b}_{0}\chi\right),\qquad \quad \left(\frac{\chi}{-}1\mathcal{b}_{0}+\mathcal{b}_{i}(\chi-1),\frac{\chi}{-}1(\chi-1)(1-\mathcal{b}_{i})-\mathcal{b}_{0}\right),\] with \(i\in \{1,2,3\}\), and conclude with \[\begin{align} \lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{2}})}&\le&\frac{1}{4}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}+\frac{c}{(\tau_{1}-\tau_{2})^{\frac{n(\chi-1)}{\chi-1-2\mathcal{b}_{0}\chi}}}\left(\int_{B_{r}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{\chi-1}{\chi-1-2\mathcal{b}_{0}\chi}}\nonumber \\ &&+c\sum_{i=1}^{3}\lVert \ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)) \rVert_{L^{1}(B_{r})}^{\frac{(1-\omega_{i})(\chi-1)}{2((\chi-1)(1-\mathcal{b}_{i})-\mathcal{b}_{0})}}+c, \end{align}\] for \(c\equiv c(\textrm{\texttt{data}})\). Lemma 5 eventually yields ?? below, and we are done.

Step 8: case \(n=2\) and \(\alpha\ge 2/3\)↩︎

Now \(\mathbb{1}_{3}=0\), \(\tilde{\mathbb{1}}=1\), and 33 reads as \[\begin{align} \label{14462} \lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{2}})}&\le&\frac{c\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}\chi}{\chi-1}+\frac{1}{2}}}{(\tau_{1}-\tau_{2})^{n/2}}\left(\int_{B_{\tau_{1}}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{1}{2}}+c\nonumber \\ &&+c\sum_{i=1}^{2}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\mathcal{b}_{i}}\left\|\mathbf{P}^{\frac{1}{2}}_{\frac{i\alpha}{2}}\left(\ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert))^{1-\omega_{i}};\;\cdot \;,\frac{\tau_{1}-\tau_{2}}{4}\right)\right\|_{L^{\infty}(B_{\tau_{2}})}\nonumber \\ &&+c\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}^{\frac{\mathcal{b}_{0}}{\chi-1}+\tilde{\mathcal{b}}}\left\|\mathbf{P}^{\frac{1}{2}}_{\alpha}\left(\ell_{1}(\lvert Du\rvert)^{\gamma-\tilde{\omega}};\;\cdot \;,\frac{\tau_{1}-\tau_{2}}{4}\right)\right\|_{L^{\infty}(B_{\tau_{2}})}. \end{align}\tag{35}\] The terms in the first two lines of 35 can be controlled via [14.3]–[14.4] with \(i\in \{1,2\}\), so we only need to take care of the last one. Notice that \[\eqref{a.5.2s}\;\;and\;\;\alpha\ge \frac{2}{3} \;\stackrel{\vartheta_{*}\le \gamma-1}{\Longrightarrow} \;\frac{\mathcal{b}_{0}}{\chi-1}+\tilde{\mathcal{b}}\le \frac{\mathcal{b}_{0}}{\chi-1}+80\gamma\delta_{0}(\mu-1+\omega_{\mu})+\gamma-1+\frac{1+\tilde{\omega}}{2}\stackrel{\eqref{omega},\eqref{bbb}}{<}1,\] so the exponents \[\label 14.4.2 \left(\frac{\chi}{-}1\mathcal{b}_{0}+\tilde{\mathcal{b}}(\chi-1),\frac{\chi}{-}1(\chi-1)(1-\tilde{\mathcal{b}})-\mathcal{b}_{0}\right)\] are both finite and larger than one. Moreover, \[\label 14.4.1 \begin cases \displaystyle \;\eqref a.5.2s\;\; and\;\;\alpha\ge \frac{2}{3} \;\Longrightarrow \;\frac{1}{\alpha}<\frac{2}{\gamma}-1\gamma-\tilde{\omega} 1.5mm\\ \displaystyle \;\eqref a.5.2s\;\; and\;\;\eqref muma \;\Longrightarrow \;\gamma<2-\mu+\frac{\alpha}{2} 1.5mm\\ \displaystyle \;\eqref a.5.2s \;\; and\;\;\eqref muma \;\Longrightarrow \;2\gamma-1<\frac{2}{-}\mu 1-\alpha, \end cases\] and Lemma 1 yields \[\label 14.5 \left\|\mathbf{P}^\frac{1}{2}_\alpha\left(\ell_1(\lvert D\rvert u)^\gamma-\tilde{\omega};\;\cdot \;,\frac{\tau}{_}{1}-\tau_{2}4\right)\right\|_L^{\infty}(B_{\tau_{2}})\le c\lVert \ell \rVert_{1}(\lvert Du\rvert)_L^{2\gamma-1}(B_{r})^\frac{\gamma-\tilde{\omega}}{2},\] for \(c\equiv c(n,\alpha,\gamma)\). In 35 , we can then apply Young’s inequality with conjugate exponents [14.6]\(_{i\in \{1,2\}}\) and [14.4.2], and plug in [14.5] and [12.2] — recall that by \(\eqref{14.4.1}_{2,3}\), Theorem 13 applies — to obtain \[\begin{align} \lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{2}})}&\le&\frac{1}{4}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}+\frac{c}{(\tau_{1}-\tau_{2})^{\frac{n(\chi-1)}{\chi-1-2\mathcal{b}_{0}\chi}}}\left(\int_{B_{r}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{\chi-1}{\chi-1-2\mathcal{b}_{0}\chi}}\nonumber \\ &&+c\sum_{i=1}^{2}\lVert \ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)) \rVert_{L^{1}(B_{r})}^{\frac{(1-\omega_{i})(\chi-1)}{2((\chi-1)(1-\mathcal{b}_{i})-\mathcal{b}_{0})}}+c\lVert \ell_{1}(\lvert Du\rvert) \rVert_{L^{2\gamma-1}(B_{r})}^{\frac{(\gamma-\tilde{\omega})(\chi-1)}{2((\chi-1)(1-\tilde{\mathcal{b}})-\mathcal{b}_{0})}}+c\nonumber \\ &\le&\frac{1}{4}\lVert \textrm{\texttt{a}}(\cdot,\lvert Du\rvert) \rVert_{L^{\infty}(B_{\tau_{1}})}+\frac{c}{(\tau_{1}-\tau_{2})^{\frac{n(\chi-1)}{\chi-1-2\mathcal{b}_{0}\chi}}}\left(\int_{B_{r}}\textrm{\texttt{a}}(x,\lvert Du\rvert)\,{\rm d}x\right)^{\frac{\chi-1}{\chi-1-2\mathcal{b}_{0}\chi}}\nonumber \\ &&+\frac{c}{r^{\textrm{\texttt{d}}}}\left(\lVert Du \rVert_{L^{1}(B_{2r})}+\sqrt{\sigma_{\varepsilon}}\lVert Du \rVert_{L^{4\gamma}(B_{2r})}^{2\gamma}+1\right)^{\textrm{\texttt{d}}}\nonumber \\ &&+c\sum_{i=1}^{2}\lVert \ell_{1}(\textrm{\texttt{a}}(\cdot,\lvert Du\rvert)) \rVert_{L^{1}(B_{r})}^{\frac{(1-\omega_{i})(\chi-1)}{2((\chi-1)(1-\mathcal{b}_{i})-\mathcal{b}_{0})}}+c, \end{align}\] with \(c\equiv c(\textrm{\texttt{data}})\) and \(\textrm{\texttt{d}}\equiv \textrm{\texttt{d}}(n,\alpha,\gamma,\mu)\). Recalling that by \(\eqref{14.4.1}_{2}\), Theorem 13 applies, we achieve again ?? via Lemma 5, [d2.3]\(_{2}\), and \(\eqref{a.7.1.x}_{1}\). Restoring the original notation, cf. Remark 12, we have shown the validity of the following proposition.

A standard covering argument, ?? , \(\eqref{a.7.1.x}_{1}\), 6 , [oe] and the minimality of \(u_{\delta}^{\varepsilon}\) in the Dirichlet class \(\tilde{u}_{\varepsilon}+W^{1,4\gamma}_{0}(B,\mathbb{R}^{N})\) gives for any open ball \(\tilde{B}\Subset B\), \[\begin{align} \label{lipbc} \lVert \textrm{\texttt{a}}_{\delta}^{\varepsilon}(\cdot,\lvert Du_{\delta}^{\varepsilon}\rvert) \rVert_{L^{\infty}(\tilde{B})}\le c\left(\lVert A_{\delta}^{\varepsilon}(\cdot,\lvert Du_{\delta}^{\varepsilon}\rvert) \rVert_{L^{1}(B)}+1\right)^{\textrm{\texttt{d}}}\le c\left(\lVert A_{\delta}(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}+\texttt{o}(\varepsilon)+1\right)^{\textrm{\texttt{d}}} \end{align}\tag{36}\] with \(c\equiv c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B))\) and \(\textrm{\texttt{d}}\equiv \textrm{\texttt{d}}(n,\gamma,\mu,\vartheta_{*})\). This will be helpful to prove gradient Hölder continuity in the next section.

7 Vectorial Schauder↩︎

Let \(\varepsilon,\delta\in (0,1/4)\) be the small parameters introduced at the beginning of Section 4. Fix now \(\varepsilon\in (0,1/4)\), and take \(\delta\in (0,\sigma_{\varepsilon})\), cf. [oe]. We can then bound \[\begin{align} \label{goodb} \lVert A_{\delta}(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}&\stackrel{\eqref{difdif}}{\le}&\lVert A(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}+c\delta\lVert \ell_{1}(\lvert D\tilde{u}_{\varepsilon}\rvert)^{\gamma-1} \rVert_{L^{1}(B)}\nonumber \\ &\le&\lVert A(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}+c\delta\lVert \ell_{1}(\lvert D\tilde{u}_{\varepsilon}\rvert^{2})^{2\gamma} \rVert_{L^{1}(B)}\nonumber \\ &\stackrel{\eqref{conv}}{\le}&\mathcal{F}(u;B)+\frac{c\delta}{\sigma_{\varepsilon}}+\texttt{o}(\varepsilon), \end{align}\tag{37}\] with \(c\equiv c(A,\gamma)\). Next, let \(\tilde{B}\Subset B\) be a ball and notice that, whenever \(B_{r}\subseteq \tilde{B}\), \[\begin{align} \label{gh461} \lVert Du_{\delta}^{\varepsilon} \rVert_{L^{\infty}(B_{r})}&\stackrel{\eqref{a.7.1.x}_{4}}{\le}&\lVert \textrm{\texttt{a}}_{\delta}^{\varepsilon}(\cdot,\lvert Du_{\delta}^{\varepsilon}\rvert) \rVert_{L^{\infty}(B_{r})}^{\frac{1}{2-\mu}}\nonumber \\ &\stackrel{\eqref{lipbc}}{\le}&c\left(\lVert A_{\delta}(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}+1+\texttt{o}(\varepsilon)\right)^{\frac{\textrm{\texttt{d}}}{2-\mu}}\nonumber \\ &\stackrel{\eqref{goodb}}{\le}&c\left(\mathcal{F}(u;B)+\texttt{o}(\varepsilon)+1\right)^{\frac{\textrm{\texttt{d}}}{2-\mu}}\nonumber \\ &\stackrel{\eqref{oe}}{\le}&c\left(\mathcal{F}(u;B)+1\right)^{\frac{\textrm{\texttt{d}}}{2-\mu}}=:\textrm{\texttt{c}}_{B}\equiv \textrm{\texttt{c}}_{B}(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B)). \end{align}\tag{38}\] for \(\textrm{\texttt{d}}\equiv \textrm{\texttt{d}}(n,\gamma,\mu,\vartheta_{*})\). In the next lines, the constant \(\textrm{\texttt{c}}_{B}\) will possibly be magnified by a multiplicative factor depending on \(\textrm{\texttt{data}}\), or raised to a positive power depending at most on \((n,\mu,\gamma,\vartheta_{*})\), but it will maintain the same increasing monotonicity with respect to \(\mathcal{F}(u;B)\) — we shall keep on denoting it \(\textrm{\texttt{c}}_{B}\). We then take a ball \(B_{2\sigma}(\equiv B_{2\sigma}(x_{\textrm{c}}))\subseteq \tilde{B}/4\), and let \(v\in u_{\delta}^{\varepsilon}+W^{1,4\gamma}_{0}(B_{2\sigma},\mathbb{R}^{N})\) be the solution to the Dirichlet problem \[\label pd00 u_\delta^\varepsilon+W^1,4\gamma_0(B_2\sigma,\mathbb{R}^N)\ni w\mapsto \min_u_{\delta}^{\varepsilon}+W^{1,4\gamma}_{0}(B_{2\sigma},\mathbb{R}^{N})\int_B_{2\sigma}\textrm{\texttt{f}}_\textrm{c}(Dw)\,{\rm d}x,\] solving by minimality \[\label el00 \int_B_{2\sigma}\langle\partial \textrm{\texttt{f}}_\textrm{c}(Dv ),Dw \rangle \,{\rm d}x=0\qquadfor all \;\;w\in W^1,4\gamma_0(B_2\sigma,\mathbb{R}^N),\] where, as before, \(\textrm{\texttt{f}}_{\textrm{c}}(z)=\textrm{\texttt{f}}_{\delta}^{\varepsilon}(x_{\textrm{c}},z)\) for all \(z\in \mathbb{R}^{N\times n}\). By Theorem 15 and 38 it follows that \[\label hh.0 \textrm{\texttt{a}}_\textrm{c}(\lvert D\rvert v)\in W^1,2(B_\sigma,\mathbb{R}^N\times n)\qquadand\qquad \lVert D \rVert v_L^{\infty}(B_{3\sigma/2})+\lVert D \rVert u_{\delta}^{\varepsilon}_L^{\infty}(B_{2\sigma})\le \textrm{\texttt{c}}_B.\] Let \(M_{*}\equiv M_{*}(A,\gamma)\) be the constant from Corollary 1, set \(M:=\max\{\textrm{\texttt{c}}_{B},M_{*}\}+1\) and let \(\textrm{\texttt{f}}_{M}(z):=A_{M}(x_{\textrm{c}},\lvert z\rvert)\) be the integrand constructed in [amam]. By [h'']–[[h''']](#h’’’){reference-type=“eqref” reference=“h’’’”}, [hh.0] and the definition of \(M\equiv M(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\), we have that \(v\in W^{1,\infty}(B_{3\sigma/2},\mathbb{R}^{N})\) solves \[\label hh.8 \int_B_{3\sigma/2}\langle\partial\textrm{\texttt{f}}_M(Dv),Dw\rangle\,{\rm d}x\stackrel M> \textrm{\texttt{c}}_{B}=\int_B_{3\sigma/2}\langle\partial\textrm{\texttt{f}}_\textrm{c}(Dv),Dw\rangle\,{\rm d}x\stackrel\eqref{el00}=0\qquadfor all \;\;w\in W^1,8\gamma_0(B_3\sigma/2,\mathbb{R}^N).\] The convexity of \(z\mapsto \textrm{\texttt{f}}_{M}(z)\) and [hh.8] imply that \(v\in W^{1,\infty}(B_{3\sigma/2},\mathbb{R}^{N})\) is a local minimizer of integral \[W^{1,8\gamma}(B_{3\sigma/2},\mathbb{R}^{N})\ni w\mapsto \mathcal{F}_{M}(w;B_{3\sigma/2}):=\int_{B_{3\sigma/2}}\textrm{\texttt{f}}_{M}(Dw)\,{\rm d}x.\] By [h''], \(\textrm{\texttt{f}}_{M}\) satisfies the assumptions of [84], so there exists \(\tilde{\beta}\equiv \tilde{\beta}(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\in (0,1)\) such that \(Dv\in C^{0,\tilde{\beta}}_{\operatorname{loc}}(B_{\sigma},\mathbb{R}^{N\times n})\) with \[\label oscv \mathop{\mathrm{osc}}_B_{\theta\sigma}Dv\le c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\theta^\tilde{\beta}\qquadfor all \;\;\theta\in (0,1),\] cf. [84]. Furthermore, estimate 27 with \(B_{2\sigma}\) replacing \(8B_{h}\) yields \[\begin{align} \label{hh4648} \mathpalette\@thickbar{\lVert} Du_{\delta}^{\varepsilon}-Dv \rVert_{L^{2}(B_{\sigma})}&\stackrel{\eqref{hh.0}_{2}}{\le}&\textrm{\texttt{c}}_{B}\mathpalette\@thickbar{\lVert} \ell_{1}(\lvert Du_{\delta}^{\varepsilon}\rvert+\lvert Dv\rvert)^{-\mu/2}\lvert Du_{\delta}^{\varepsilon}-Dv\rvert \rVert_{L^{2}(B_{\sigma})}\nonumber \\ &\stackrel{\eqref{a.7.1.x}_{3}}{\le}&\textrm{\texttt{c}}_{B}\mathpalette\@thickbar{\lVert} \sqrt{\lambda_{0}(\lvert Du_{\delta}^{\varepsilon}\rvert+\lvert Dv\rvert)}\lvert Du_{\delta}^{\varepsilon}-Dv\rvert \rVert_{L^{2}(B_{2\sigma})}\nonumber \\ &\stackrel{\eqref{com}}{\le}&\textrm{\texttt{c}}_{B}\sigma^{\frac{\alpha}{2}}\mathpalette\@thickbar{\lVert} 1+A_{\delta}(\cdot,\lvert Du_{\delta}^{\varepsilon}\rvert)+A_{\textrm{c}}(\lvert Du_{\delta}^{\varepsilon}\rvert) \rVert_{L^{1}(B_{2\sigma})}^{\frac{1}{2}}\stackrel{\eqref{hh.0}_{2}}{\le}\textrm{\texttt{c}}_{B}\sigma^{\frac{\alpha}{2}}. \end{align}\tag{39}\] We then bound, for every ball \(B_{\varrho}\Subset B_{\sigma}\), \(\varrho\in (0,\sigma)\), \[\begin{align} \mathpalette\@thickbar{\lVert} Du_{\delta}^{\varepsilon}-(Du_{\delta}^{\varepsilon})_{B_{\varrho}} \rVert_{L^{2}(B_{\varrho})}&\le& 2\mathpalette\@thickbar{\lVert} Du_{\delta}^{\varepsilon}-Dv \rVert_{L^{2}(B_{\varrho})}+2\mathpalette\@thickbar{\lVert} Dv-(Dv)_{B_{\varrho}} \rVert_{L^{2}(B_{\varrho})}\nonumber \\ &\stackrel{\eqref{oscv}}{\le}&c\left(\sigma/\varrho\right)^{\frac{n}{2}}\mathpalette\@thickbar{\lVert} Du_{\delta}^{\varepsilon}-Dv \rVert_{L^{2}(B_{\sigma})}+c\left(\varrho/\sigma\right)^{\tilde{\beta}}\stackrel{\eqref{hh4648}}{\le}c\left(\sigma/\varrho\right)^{\frac{n}{2}}\sigma^{\frac{\alpha}{2}}+c\left(\varrho/\sigma\right)^{\tilde{\beta}}, \end{align}\] for \(c\equiv c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\). Let us equalize the powers by choosing \(\varrho=(\sigma/2)^{\frac{n+2\tilde{\beta}+\alpha}{n+2\tilde{\beta}}}\) to gain \[\label hol.1 \mathpalette\@thickbar{\lVert} D \rVert u_{\delta}^{\varepsilon}-(Du_{\delta}^{\varepsilon})_{B_{\varrho}}_L^{2}(B_{\varrho})\le c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\varrho^\frac{\alpha\tilde{\beta}}{n+2\tilde{\beta}+\alpha}.\] We point out that if \(\sigma\le \varrho\le 2\sigma\), estimate [hol.1] trivially follows. Set \(\beta_{*}:=\alpha\tilde{\beta}/(n+2\tilde{\beta}+\alpha)\), so that \(\beta_{*}\equiv \beta_{*}(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\in (0,1)\). A standard covering argument and Campanato-Meyers characterization of Hölder continuity yield that \(Du_{\delta}^{\varepsilon}\) is locally Hölder continuous in \(\tilde{B}\), and, thanks to the arbitrariness of \(\tilde{B}\Subset B\) we further deduce the local Hölder continuity of \(Du_{\delta}^{\varepsilon}\) in \(B\). Specifically, given any ball \(\tilde{B}\Subset B\), \([Du_{\delta}^{\varepsilon}]_{0,\beta_{*};\tilde{B}}\le c(\textrm{\texttt{data}}(B),\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\). Overall, we have just proven the following proposition.

8 Approximation scheme and proof of Theorem 4↩︎

In this section we show that the sequence of approximating minimizers constructed in Section 4.1 converges in a uniform fashion to our original minimum \(u\in W^{1,1}_{\operatorname{loc}}(\Omega,\mathbb{R}^{N})\), thus transferring to \(u\) the Lipschitz and Hölder bounds obtained in Propositions 16 and 17, respectively.

Convergence of approximating minimizers↩︎

Keeping in mind the restrictions imposed at the beginning of Section 7, by the minimality of \(u_{\delta}^{\varepsilon}\) in the Dirichlet class \(\tilde{u}_{\varepsilon}+W^{1,4\gamma}_{0}(B,\mathbb{R}^{N})\), 37 and [oe] we obtain \[\begin{align} \label{enesed} \mathcal{F}_{\delta}^{\varepsilon}(u_{\delta}^{\varepsilon};B)\le \lVert A_{\delta}(\cdot,\lvert D\tilde{u}_{\varepsilon}\rvert) \rVert_{L^{1}(B)}+\sigma_{\varepsilon}\lVert \ell_{1}(\lvert D\tilde{u}_{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}\le \mathcal{F}(u;B)+\frac{c\delta}{\sigma_{\varepsilon}}+\texttt{o}(\varepsilon), \end{align}\tag{40}\] for \(c\equiv c(A,\gamma)\). Moreover, \[\begin{align} \label{enesed461} \mathcal{F}(u_{\delta}^{\varepsilon};B)+\sigma_{\varepsilon}\lVert \ell_{1}(\lvert Du_{\delta}^{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}&\stackrel{\eqref{difdif}}{\le}& \mathcal{F}_{\delta}^{\varepsilon}(u_{\delta}^{\varepsilon};B)+c\delta\lVert \ell_{1}(\lvert Du_{\delta}^{\varepsilon}\rvert)^{\gamma-1} \rVert_{L^{1}(B)}\nonumber \\ &\le&\mathcal{F}_{\delta}^{\varepsilon}(u_{\delta}^{\varepsilon};B)+c\delta\lVert \ell_{1}(\lvert Du_{\delta}^{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}\nonumber \\ &\le&\mathcal{F}_{\delta}^{\varepsilon}(u_{\delta}^{\varepsilon};B)\left(1+\frac{c\delta}{\sigma_{\varepsilon}}\right)\stackrel{\eqref{enesed}}{\le}\mathcal{F}(u;B)\left(1+\frac{c\delta}{\sigma_{\varepsilon}}\right)+\frac{c\delta}{\sigma_{\varepsilon}}+\texttt{o}(\varepsilon), \end{align}\tag{41}\] with \(c\equiv c(A,\gamma)\). Estimate 41 implies that, for fixed \(\varepsilon>0\), the sequence \(\{u_{\delta}^{\varepsilon}\}_{\delta>0}\) converges to some \(u^{\varepsilon}\in \tilde{u}_{\varepsilon}+W^{1,4\gamma}_{0}(B,\mathbb{R}^{N})\) weakly in \(W^{1,4\gamma}(B,\mathbb{R}^{N})\). As \(\delta\to 0\), by weak lower semicontinuity we derive \[\begin{align} \label{enesed462} \mathcal{F}(u^{\varepsilon};B)+\sigma_{\varepsilon}\lVert \ell_{1}(\lvert Du^{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}&\le& \liminf_{\delta\to 0}\left(\mathcal{F}(u_{\delta}^{\varepsilon};B)+\sigma_{\varepsilon}\lVert \ell_{1}(\lvert Du_{\delta}^{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}\right)\nonumber \\ &\stackrel{\eqref{enesed461}}{\le}&\mathcal{F}(u;B)+\texttt{o}(\varepsilon). \end{align}\tag{42}\] Next, notice that \[c\lVert b(\lvert Du^{\varepsilon}\rvert) \rVert\stackrel{\eqref{a.4}_{4}}{\le}\mathcal{F}(u^{\varepsilon};B)+\lvert B\rvert\stackrel{\eqref{enesed462}}{\le}\mathcal{F}(u;B)+\lvert B\rvert+\texttt{o}(\varepsilon),\] for \(c\equiv c(A)\), so, recalling [binf], via Dunford & Pettis and de la Vallée Poussin theorems we deduce that, up to subsequences, \(u^{\varepsilon}\rightharpoonup \tilde{u}\) weakly in \(W^{1,1}(B,\mathbb{R}^{N})\) for some \(\tilde{u}\in u+W^{1,1}(B,\mathbb{R}^{N})\). Letting \(\varepsilon\to 0\) in 42 and using weak lower semicontinuity we obtain \[\mathcal{F}(\tilde{u};B)\le \liminf_{\varepsilon\to 0}\left(\mathcal{F}(u^{\varepsilon};B)+\sigma_{\varepsilon}\lVert \ell_{1}(\lvert Du^{\varepsilon}\rvert^{2}) \rVert_{L^{2\gamma}(B)}^{2\gamma}\right)\stackrel{\eqref{enesed462}}{\le}\mathcal{F}(u;B).\] The content of the previous display, together with the minimality of \(u\), the fact that \(\left.\tilde{u}\right|_{\partial B}=\left.u\right|_{\partial B}\) and the strict convexity of \(\textrm{\texttt{f}}\) yield that \(\tilde{u}=u\) on \(B\). Summarizing, we have just proven that, up to (nonrelabelled) subsequences, \[\label convconv u_\delta^\varepsilon\rightharpoonup u^\varepsilon \;\; weakly in \;\;W^1,4\gamma(B,\mathbb{R}^N)\quadand\quad u^\varepsilon\rightharpoonup u\;\; weakly in \;\;W^1,1(B,\mathbb{R}^N).\]

Proof of Theorem 4↩︎

Let \(\tilde{B}\Subset B\) be a ball. The bounds from Propositions 16 and 17 now read as \[\label bb.1 \lVert D \rVert u_{\delta}^{\varepsilon}_L^{\infty}(\tilde{B})+[Du_\delta^\varepsilon]_0,\beta_{*};\tilde{B}\le c,\] where \(c\equiv c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\) and \(\beta_{*}\equiv \beta_{*}(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\). We can then update (locally) the limits in [convconv] as \(u_{\delta}^{\varepsilon}\to u^{\varepsilon}\) weak* in \(W^{1,\infty}(\tilde{B},\mathbb{R}^{N})\), and uniformly in \(C^{1,\sigma}(\tilde{B},\mathbb{R}^{N})\) for all \(\sigma\in (0,\beta_{*})\) by [bb.1], so we can pass to the limit as \(\delta\to 0\) in [bb.1] to gain \[\label bb.2 \lVert D \rVert u^{\varepsilon}_L^{\infty}(\tilde{B})+[Du^\varepsilon]_0,\beta_{*};\tilde{B}\le c,\] with \(c\equiv c(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\) and \(\beta_{*}\equiv \beta_{*}(\textrm{\texttt{data}},\textrm{\texttt{l}}(\tilde{B},B),\mathcal{F}(u;B))\). We can then send \(\varepsilon\to 0\) in [bb.2] and recall [convconv] to complete the proof.

References↩︎

---

1. Acknowledgements. C. De Filippis is supported by the European Research Council, through the ERC StG project NEW, nr. 101220121, and by the University of Parma through the action "Bando di Ateneo 2024 per la ricerca". This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/PAT1850524]. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.↩︎

2. Solutions to \(\Delta u=0\).↩︎

3. Weak solutions to quasilinear elliptic PDEs with Hölder continuous coefficients have Hölder continuous gradients, but implication [i.0] may fail for degenerate operators, see [1].↩︎

4. Assume that field \(\textrm{\texttt{B}}\) is smooth and \(z\mapsto \partial \textrm{\texttt{B}}(\cdot,z)\) is symmetric for simplicity.↩︎

5. As several nonuniformly elliptic models feature \(p\)-Laplacian type degeneracy in zero, see e.g. [33], [34], the expected maximal regularity is gradient Hölder continuity of solutions [1], therefore it is useless to impose more regularity than Hölder continuity on coefficients.↩︎

6. See the comments in [59].↩︎

7. Formally, take \(p=1\) there.↩︎

8. Here \(A'(x,t)=\frac{\,{\rm d}}{\,{\rm d}t}A(x,t)\).↩︎

9. Condition \(\eqref{a.2.x}_{2}\) implies that \(A\) satisfies the \(\Delta_{2}\)-condition, that is \(A(x,\textrm{\texttt{d}}t)\le c(A,\gamma,\textrm{\texttt{d}})(A(x,t)+1)\) for all constants \(\textrm{\texttt{d}}\in [0,\infty)\), see [82].↩︎

10. If \(t=0\), \(A(x_{1},0)=A(x_{2},0)=0\) and there is nothing to prove.↩︎

11. Theorem 4 does not cover [72], which deals with bounded minima of integral \(\eqref{dp}\) within the sharp maximal nonuniformity range \(q<1+\alpha\), and requires a completely different strategy.↩︎

12. Since we are bounding \(A_{M}''(x,t)t\) and \(A_{M}'(x,t)\) below, there is no loss of generality in taking constant \(c\) appearing in displays 9 –10 less than one.↩︎

13. In [48], quantitatively superlinear integrands are considered, that is \(t^{p}-1\lesssim A(x,t)\le 1+t^{q}\) for some \(1<p\le q\) and all \((x,t)\in \Omega\times [0,\infty)\). Here, we do not control below any power larger than one, however, thanks to [a.4]–[binf] Dunford & Pettis and de la Vallée Poussin theorem assure the compactness of approximating sequences in \(W^{1,1}\), see also [70] and [72].↩︎

14. Only the structural properties of the (regularized) original integrand are listed; the perturbation term \(z\mapsto \sigma_{\varepsilon}\ell_{1}(\lvert z\rvert^{2})^{2\gamma}\) is omitted, cf. Remark 12.↩︎

15. Here \(\mathcal{w}\) represents the regularized integrand or the auxiliary functions defined at the beginning of Section 4.↩︎

16. Here we fixed \(\varepsilon\in (0,1/4)\), and imposed restriction \(\delta\in (0,\sigma_{\varepsilon})\), see the beginning of Section 7.↩︎