December 06, 2024
We give a sufficient condition for Hölder continuity at a boundary point for quasiminima of double-phase functionals of \(p,q\)-Laplace type, in the setting of metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We use a purely analytic variational approach based on De Giorgi-type conditions to give a pointwise estimate near a boundary point. The proofs rely on a careful phase analysis and estimates in the intrinsic geometries.
This paper aims to obtain boundary regularity of quasiminima for the double-phase integral defined on a complete metric measure space \((X, d, \mu)\), equipped with a metric \(d\) and a
doubling measure \(\mu\), which supports a weak \((1,p)\)-Poincaré inequality. Let \(\Omega\) be an open and bounded subset of \(X\). The double-phase integral is given by \[\label{J}
\int_{\Omega} H(x,g_u) \, \mathrm{d}\mu = \int_{\Omega} \big(g_u^p + a(x) g_u^q\big) \, \mathrm{d}\mu,\tag{1}\] where \(g_u\) is the minimal \(p\)-weak upper gradient of \(u\) and \(1<p<q<Q\), with \(Q\) generalization of the euclidean concept of space dimension (see Section 2). The modulating
coefficient function \(a(\cdot) \geq 0\) is assumed to meet certain standard regularity conditions, which are detailed in 4 below. The integral 1 shows an energy density that
alternates between two types of degenerate behaviour, depending on the modulating coefficient \(a(\cdot)\), which dictates the phase. Specifically, when \(a(x) = 0\), the variational
integral 1 simplifies to the classical \(p\)-growth problem. In contrast, when \(a(x) \geq c > 0\), it corresponds to the \((p,q)\)-problem. Additionally, the ratio of exponents \(q/p\) is subjected to the condition \[\label{pqcond}
\frac{q}{p} \leq 1 + \frac{\alpha}{Q},\tag{2}\] where \(0 < \alpha \leq 1\), and \(Q = \log_2 C_D\), with \(C_D\) being the doubling
constant of the measure. This condition is known to be sharp for ensuring regularity of minima already in the Euclidean setting, in this direction see the recent contributions of Baroni-Colombo-Mingione [1] and Colombo-Mingione [2], [3].
The main advantage of the notion of quasiminima of 1 is that it simultaneously covers a wide class of problems where the variational integrand \(F:\Omega\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}\)
satisfies the Carathéody conditions and \[\lambda H(x,z)\le F(x,u,z)\le\Lambda H(x,z),
\quad 0<\lambda<\Lambda<\infty,\] for every \(x \in \Omega\) and \(u,z\in \mathbb{R}\). Originally developed by Giaquinta and Giusti [4], [5], quasiminima offer a flexible framework that extends beyond traditional differentiable structures, thus providing a unified
treatment of variational integrals, elliptic equations and systems, obstacle problems and quasiregular mappings. When defined over a metric measure space, quasiminima allow the analysis of variational problems in settings where classical derivatives may
not be defined, relying instead on notions such as minimal weak upper gradients to understand how functions vary along paths. Those notions permit to overcome the difficulties due to working on general metric spaces, which provide a relaxed geometric
setting and lack a smooth structure. This adaptability has opened the application of regularity theory to non-Euclidean spaces and various areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds, and potential theory on graphs,
to name a few, see Björn-Björn [6], Franchi-Hajłasz-Koskela [7], Hajłasz
[8], Heinonen-Koskela-Shanmugalingam-Tyson [9], Kilpeläinen-Kinnunen-Martio
[10], Liu-Zhou-Shanmugalingam [11], Shanmugalingam [12] and references therein. The study of variational problems on metric measure spaces originated from independent proofs by Grigor’yan [13] and Saloff-Coste [14], which show that, on Riemannian manifolds, the doubling property and the Poincaré inequality
are equivalent to a specific Harnack-type inequality for solutions of the heat equation. However, our focus changes from Riemannian manifolds to more general spaces.
Recently, the study of Sobolev spaces without a differentiable structure, along with the variational theory of different energy functionals in metric measure spaces, has captured the interest of researchers. For example, Cheeger [15] studied differentiability properties, while Kinnunen-Martio [16] focused on aspects of potential
theory. Additionally, Kinnunen-Shanmugalingam [17] established local properties of quasiminima for the \(p\)-energy integral.
Björn-Björn-Shanmugalingam [18] explored the Dirichlet problem for \(p\)-harmonic functions. Furthermore, Kinnunen-Marola-Martio
[19] proved the Harnack principle. Recently, for the double-phase problems, Kinnunen-Nastasi-Pacchiano Camacho [20] established higher integrability results for the gradient of quasiminima and Nastasi-Pacchiano Camacho [21]
derived other types of local regularity results. On the other hand, the regularity theory of nonlinear parabolic problems in the metric space context has been developed and studied in Herán [22], [23], Ivert-Marola-Masson [24], Marola-Masson [25], Masson-Miranda Jr.-Paronetto-Parviainen [26] and Masson-Siljander [27]. We also report some recent contributions concearning the study of the fractional \(p\)-Laplacian in the context of general metric
measure spaces, see Capogna-Kline-Korte-Shanmugalingam-Snipes [28].
Double-phase integrals have been introduced by Zhikov in the context of Homogenization and Lavrentiev’s phenomenon [29]–[31]. Regularity theory for double-phase integrals has been extensively developed, starting from the seminal works of Marcellini [32]–[34]. In the Euclidean context, Colombo-Mingione [2], [3], Cupini-Marcellini-Mascolo [35], De Filippis-Mingione [36], Esposito-Leonetti-Mingione [37], Marcellini-Nastasi-Pacchiano Camacho [38], Mingione [39], Ok [40]
considered classes of nonuniformly elliptic problems and proved different regularity properties. For example, Hölder regularity results for certain class of double-phase problems with non-standard growth conditions can be found in Düzgün-Marcellini-Vespri
[41], [42], Eleuteri [43], Harjulehto-Hästö-Toivanen [44], Di Benedetto-Gianazza-Vespri [45]. Moreover, Di Benedetto-Trudinger [46] and Baroni-Colombo-Mingione [1] independently obtained Harnack inequalities for double-phase problems. See also Kinnunen-Lehrbäck-Vähäkangas [47] and Marcellini
[48], [49] for other results and literature reviews.
However, the analysis of boundary regularity, especially in metric measure spaces, remains relatively underdeveloped. Existing works include boundary regularity results for minimizers and quasiminima in the Euclidean case, such as the Wiener-type condition
proved by Ziemer [50], see also Tachikawa [51] for double-phase
functionals and Irving-Koch [52] for \((p,q)\)-growth functionals. For the generalization to the metric setting, we have the works of
Björn [53] and Nastasi-Pacchiano Camacho [54]. Further contributions include
boundary oscillation estimates by Björn, MacManus, and Shanmugalingam [55] for \(p\)-harmonic functions and \(p\)-energy minimizers. Despite these advances, the boundary behaviour of double-phase problems remains a relatively open problem. Addressing this gap, the present work studies quasiminima of 1 in the
Newtonian–Sobolev space \(N^{1,1}(X)\), with \(H(\cdot,g_u) \in L^1(X)\), for \(X\) a general metric measure space. The lack of classical derivatives in
metric measure spaces requires variational methods that focus on the modulus of the gradient, therefore avoiding the need for smoothness.
To the best of our knowledge, boundary regularity for double-phase problems remains largely unexplored, even in the Euclidean setting. Motivated by recent developments, such as the works of Tachikawa [51] and Björn [53], we aim to extend regularity results up to the boundary within the general framework of metric
measure spaces, where no smooth structure is assumed. A central aspect of our approach is the analysis of functionals with non-standard growth, particularly the transition between the \(p\)- and \((p,q)\)-phases governed by the coefficient function \(a(x)\). A key novelty of this work is the introduction of a Maz’ya-type estimate for frozen functionals (Theorem 4), inspired by classical pointwise capacitary estimates due to Maz’ya [56], [57]. As far as we know, such a result has not been previously established, even in the Euclidean case. This estimate enables us to generalize local regularity results, such as those in De Filippis–Mingione [36], to boundary points.
The present work is organized as follows: we first prove the local boundedness of quasiminima (Section 3), then examine the behavior of frozen functionals (Section 4), and finally
derive a pointwise boundary estimate (Proposition 3, Section 5). This leads to the main
result of the paper: a sufficient condition for the Hölder continuity of quasiminima at boundary points (Theorem 6, Section 6). Throughout, we aim to provide detailed and self-contained arguments that contribute to the regularity theory of variational integrals under double-phase growth.
This section collects definitions and results needed when working in the general context of metric measure spaces. Further details can be found in the book by Björn-Björn [6]. Let \((X, d, \mu)\) be a complete metric measure space with a metric \(d\) and a Borel regular measure \(\mu\). The measure \(\mu\) is supposed to be doubling, meaning that there exists a constant \(C_D \geq 1\) such that, for every ball \(B_r\) in \(X\), \[\label{doubling} 0<\mu(B_{2r})\leq C_D \mu(B_r)<\infty.\tag{3}\] We denote by \(B_r=B(x,r)=\{x\in X:d(y,x)<r\}\) an open ball centered in \(x\in X\) and with radius \(0<r<\infty\). The following lemma introduces \(Q\), which is a generalization of the usual concept of dimension in metric spaces supporting a doubling measure. Indeed, in the Euclidean \(n\)-space with the Lebesgue measure, we have \(Q=n\).
Lemma 1 ([6], Lemma 3.3). Let \((X, d, \mu)\) be a metric measure space with a doubling measure \(\mu\). Then \[\label{s} \frac{\mu(B(y,r))}{\mu(B(x,R))}\geq C\left(\frac{r}{R}\right)^Q,\qquad{(1)}\] for every \(0<r\le R<\infty\), \(x \in X\) and \(y \in B(x,R)\). Here \(Q=\log_2C_D\) and \(C=C_D^{-2}\).
Let \(a:X\to[0,\infty)\) be the coefficient function in 1 and let \(\delta_{\mu}\) be a quasi-distance defined by \[\delta_{\mu}(x,y)=\bigl(\mu(B(x,d(x,y)))+\mu(B(y,d(x,y)))\bigr)^{1/Q},\quad x,y \in X,\, x\ne y,\] with \(Q=\log_2C_D\) as in ?? and \(\delta_{\mu}(x,x)=0\). We suppose that there exists \(0<\alpha\le1\) such that \[\label{aalpha} [a]_{\alpha}= \sup_{x,y \in X, x\neq y} \dfrac{|a(x)-a(y)|}{\delta_{\mu}(x,y)^{\alpha}}<\infty.\tag{4}\] In the next remark, we point out that under certain regularity conditions on the measure, the quasi-distance \(\delta_{\mu}\) and the usual distance \(d\) are equivalent.
Remark 1. A measure is called Ahlfors–David regular, if there exist constants \(0<C_2\le C_1<\infty\) such that \[\label{ahlfors} C_2r^Q\le\mu(B(x,r))\le C_1r^Q,\qquad{(2)}\] for every \(x\in X\) and \(0<r\le\mathop{\mathrm{diam}}(X)\). If the measure \(\mu\) is Ahlfors–David regular, then \(\delta_{\mu}(x,y)\approx d(x,y)\) for every \(x,y\) and, consequently, \([a]_\alpha<\infty\) if and only if \(a\) is Hölder continuous with the exponent \(\alpha\).
Throughout this manuscript, we suppose that \(\mu\) is upper \(Q\)-Ahlfors regular, that is there exists a constant \(C_1>0\) such that our measure
satisfies the following inequality \[\label{upper32Q-Ahlfors}
\mu(B(x,r))\leq C_1 r^Q \quad for every x\in X and 0<r\leq {\rm diam(X)}.\tag{5}\] This assumption ensures some uniformity and regularity in the distribution of the measure. We note that this is a general assumption. For example,
self-similar fractals, metric measure spaces with controlled curvature, uniformly rectifiable sets and Carnot groups can exhibit upper Ahlfors regularity.
We discuss the notion of upper gradient as a way to generalize the modulus of the gradient in the Euclidean case to the metric setting.
Definition 1. A nonnegative Borel function \(g\) is said to be an upper gradient of function \(u: X \to [-\infty,\infty]\) if, for all paths \(\gamma\) connecting \(x\) and \(y\), we have \[|u(x)-u(y)|\leq \int_{\gamma}g\, \mathrm{d}s,\] whenever \(u(x)\) and \(u(y)\) are both finite and \(\int_{\gamma}g \, \mathrm{d}s= \infty\) otherwise. Here \(x\) and \(y\) are the endpoints of \(\gamma\). Moreover, if a nonnegative measurable function \(g\) satisfies the inequality above for \(p\)-almost every path, that is, with the exception of a path family of zero \(p\)-modulus, then \(g\) is called a \(p\)-weak upper gradient of \(u\).
We note that if \(u\) has an upper gradient \(g\in L^p(X)\), there exists a unique minimal \(p\)-weak upper gradient \(g_u\in
L^p(X)\) with \(g_u\le g\) \(\mu\)-almost everywhere for all \(p\)-weak upper gradients \(g\in L^p(X)\) of \(u\), see [6].
Let \(\Vert u\Vert_{N^{1,p}(X)}=\Vert u\Vert_{L^{p}(X)}+\Vert g_u\Vert_{L^{p}(X)}\). Consider the collection of functions \(u\in L^p(X)\) with an upper gradient \(g\in L^p(X)\) and let \[\widetilde{N}^{1,p}(X) =\lbrace u:\Vert u\Vert_{N^{1,p}(X)}<\infty\rbrace.\] The Newtonian space is defined by \[N^{1,p}(X)=\lbrace u:\Vert
u\Vert_{N^{1,p}(X)}<\infty\rbrace/\sim,\] where \(u\sim v\) if and only if \(\Vert u-v\Vert_{N^{1,p}(X)}=0\).
The corresponding local Newtonian space is defined by \(u\in N^{1,p}_{\mathrm{loc}}(X)\) if \(u\in N^{1,p}(\Omega')\) for all \(\Omega'\Subset X\),
see [6], where \(\Omega'\Subset X\) means that \(\overline{\Omega'}\) is a compact subset of
\(X\).
Let \(\Omega\) be an open subset of \(X\). We define \(N^{1,p}_0(\Omega)\) as the set of functions \(u\in N^{1,p}(X)\) that
are zero on \(X\setminus\Omega\) \(\mu\)-a.e. The space \(N_0^{1,p}(\Omega)\) is equipped with the norm \(\Vert\cdot\Vert_{N^{1,p}}\). Note also that if \(\mu(X\setminus\Omega) = 0\), then \(N^{1,p}_0(\Omega)=N^{1,p}(X)\). We shall therefore always assume that \(\mu(X \setminus \Omega) > 0\).
We assume that \(X\) supports the following Poincaré inequality.
Definition 2. Let \(1\le p<\infty\). A metric measure space \((X,d,\mu)\) supports a weak \((1, p)\)-Poincaré inequality if there exist a constant \(C_{PI}\) and a dilation factor \(\lambda \geq 1\) such that \[\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r} |u-u_{B_r}|\, \mathrm{d}\mu \leq C_{PI} r \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\lambda r}}g_u^p \, \mathrm{d}\mu\right)^{\frac{1}{p}},\] for every ball \(B_r\) in \(X\) and for every \(u\in L^1_{\mathrm{loc}}(X)\), where we denote the integral average with \[u_{B_r}=\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r} u\, \mathrm{d}\mu =\frac{1}{\mu(B_r)}\int_{B_r}u\,\mathrm{d}\mu.\]
As shown in [58] by Keith and Zhong, see also [6], the Poincaré inequality is a self-improving property.
Theorem 1. Let \((X, d, \mu)\) be a complete metric measure space with a doubling measure \(\mu\) and a weak \((1,p)\)-Poincaré inequality with \(p>1\). Then there exists \(\varepsilon>0\) such that \(X\) supports a weak \((1, s)\)-Poincaré inequality for every \(s>p-\varepsilon\). Here, \(\varepsilon\) and the constants associated with the \((1, s)\)-Poincaré inequality depend only on \(C_D\), \(C_{PI}\) and \(p\).
From now on and without further notice, we fix \(s=s(C_{PI},C_D,p,q)\) such that \(1 < s < p<q<s^*\) and for which \(X\) also admits a weak
\((1, s)\)-Poincaré inequality. Such \(s\) is given by Theorem 1 and will be used in various of our results.
The following result shows that the Poincaré inequality implies a Sobolev–Poincaré inequality, see [6].
Theorem 2. Assume that \(\mu\) is a doubling measure and \(X\) supports a weak \((1,p)\)-Poincaré inequality and let \(Q=\log_2C_D\) be as in ?? . Let \(1\le p^*\le\frac{Qp}{Q-p}\) for \(1\le p<Q\) and \(1\leq p^*<\infty\) for \(Q\leq p<\infty\). Then \(X\) supports a weak \((p^*,p)\)-Poincaré inequality, that is, there exist a constant \(C=C(C_D,C_{PI},p)\) such that \[\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r} |u-u_{B_r}|^{p^*} \,\mathrm{d}\mu\right)^{\frac{1}{p^*}} \leq Cr\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{2\lambda r}}g_u^p \,\mathrm{d}\mu\right)^{\frac{1}{p}},\] for every ball \(B_r\) in \(X\) and every \(u \in L^1_{\mathrm{loc}}(X)\).
Corollary 1 ([6], Corollary 4.26). If \(X\) supports a weak \((1,p)\)-Poincaré inequality and \(Q\) in (?? ) satisfies \(Q\leq p\), then \(X\) supports a weak \((t,p)\)-Poincaré inequality for all \(1\leq t<\infty\).
Remark 2. By the Hölder inequality we see that a weak \((p^*,p)\)-Poincaré inequality implies the same inequality for smaller values of \(p^*\). Meaning that \(X\) will then support a weak \((t,p)\)-Poincaré inequality for all \(1<t<p^*\).
Remark 3. The exponent \(Q\) in (?? ) is not uniquely determined. In particular, since \(\rho < R\), one can always choose a larger value for \(Q\). Therefore, the assumption \(Q > p\) in Theorem 2 can always be satisfied.
Definition 3 ([53], Definition 6.13). Let \(E\subset \Omega\). Then we define the variational capacity \[\textrm{cap}_p(E,\Omega)=\inf_{u}\int_{\Omega} g_u^p\mathrm{d}\mu,\] where the infimum is taken over all \(u\in N^{1,p}_0(\Omega)\) such that \(u\geq 1\) on E.
The variational capacity is also known as the relative capacity. We define the infimum to be \(\infty\) in the absence of any admissible functions \(u\).
The next result provides explicit estimates for the variational capacity, especially in the case of balls. A detailed proof is available in [6].
Proposition 1. Assume that \(\mu\) is a doubling measure with constant \(C_D\) and \(X\) supports a weak \((1,p)\)-Poincaré inequality. Then there exists \(C>0\) such that if \(E\subset B_r=B(x_0,r)\) with \(0<r<\frac{\textrm{diam}(X)}{8}\), then \[\frac{\mu(E)}{Cr^p}\leq\textrm{cap}_p(E,B_{2r})\leq\frac{C_D\mu(B_r)}{r^p}.\]
Remark 4. Notice that for \(s_2\leq s_1\) and a small radius \(r\leq 1\), we have that if \(u\in N^{1,s_1}_0(B_r)\) is such that \(u\geq 1\) on \(E\), it is then clear that \(u\in N^{1,s_2}_0(B_r)\) with \(u\geq 1\) on \(E\). Therefore by Hölder’s inequality we obtain \[\begin{align} \textrm{cap}_{s_2}(E,B_r)&\leq \int_{B_r}g_u^{s_2}\,\mathrm{d}\mu\\ &=\left(\int_{B_r}g_u^{s_1}\,\mathrm{d}\mu\right)^{\frac{s_2}{s_1}}\mu(B_r)^{\frac{s_1-s_2}{s_1}}. \end{align}\] So, \[\mu(B_r)^{\frac{s_2-s_1}{s_1}}\textrm{cap}_{s_2}(E,B_r)\leq \left(\int_{B_r}g_u^{s_1}\,\mathrm{d}\mu\right)^{\frac{s_2}{s_1}}.\] Therefore, \[\left(\mu(B_r)^{\frac{s_2-s_1}{s_1}}\textrm{cap}_{s_2}(E,B_r)\right)^{\frac{s_1}{s_2}}\leq \int_{B_r}g_u^{s_1}\,\mathrm{d}\mu,\qquad\textrm{for all }u\in N^{1,s_2}_0(B_r)\textrm{ with }u\geq 1\textrm{ on }E.\] Meaning, \[\mu(B_r)^{\frac{s_2-s_1}{s_1}}\textrm{cap}_{s_2}(E,B_r)\leq \textrm{cap}_{s_1}^{s_2/s_1}(E,B_r),\qquad\textrm{for }s_2\leq s_1.\] Furthermore, we have \[\frac{1}{\textrm{cap}_{s_1}(E,B_r)}\leq\left(\mu(B_r)^{\frac{s_1-s_2}{s_1}}\frac{1}{\textrm{cap}_{s_2}(E,B_r)}\right)^{\frac{s_1}{s_2}}.\] Finally, since we are asking for the measure \(\mu\) to be upper \(Q\)-Alhfors regular, by inequality 5 and the assumption \(r\leq 1\), we conclude \[\label{eq2465NEW} \frac{1}{\textrm{cap}_{s_1}(E,B_r)}\leq\frac{C}{\textrm{cap}_{s_2}^{s_1/s_2}(E,B_r)},\qquad{(3)}\] where \(C=C(C_1, s_1,s_2)\).
The following proposition is a capacity version of the Sobolev-Poincaré inequality, also referred to as Maz’ya type estimate. The proof is a straightforward generalization of the Euclidean case and can be found in [53].
Proposition 2 ([53], Proposition 3.2). Let \(X\) be a doubling metric measure space supporting a weak \((1,s)\)-Poincaré inequality. Then there exists \(C\) and \(\lambda\geq 1\) such that for all balls \(B_r\) in \(X\), \(u\in N^{1,s}(X)\) and \(S=\lbrace x\in B_{\frac{r}{2}}: u(x)=0\rbrace\), then \[\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\vert u\vert^t \, \mathrm{d}\mu\right)^{\frac{1}{t}}\leq\left(\frac{C}{{\rm cap}_s(S,B_r)}\int_{B_{\lambda r}}g^s \, \mathrm{d}\mu\right)^{\frac{1}{s}},\] for \(C\) depending on \(C_{PI}\) and \(t\) as in Corollary 1.
Throughout the paper, positive constants are denoted by \(C\) and the dependencies on parameters are listed in the parentheses. We denote \[C(\mathrm{data})=C(C_D,C_{PI},\lambda,p,q,\alpha,[a]_\alpha).\]
In this section, our main interest focuses on proving the so-called Caccioppoli inequality, Lemma 2, and local boundedness, Corollary 2, for quasiminima of the double-phase integral 1 . We begin by defining quasiminima with boundary values.
Definition 4.
Let \(w\in N^{1,1}(X)\) with \(H(\cdot,g_w)\in L^1(X)\). A function \(u\in N^{1,1}(X)\) with \(H(\cdot,g_u)\in L^1(X)\) is a quasiminimizer on \(\Omega\) with boundary data \(w\) if \(u-w\in N^{1,1}_0(\Omega)\) and there exists a constant \(K\geq 1\) such that \[\int_{\{u\ne v\}}H(x,g_u)\, \mathrm{d}\mu \leq K\int_{\{u\ne v\}}H(x,g_v)\, \mathrm{d}\mu,\] for every function \(v \in N^{1,1}(\Omega)\) with \(u-v\in N^{1,1}_0(\Omega)\).
Lemma 2 (Caccioppoli inequality). Assume that \(u\in N^{1,1}(X)\) with \(H(\cdot,g_u)\in L^1(X)\) is a quasiminimizer in \(\Omega\) with boundary data \(w\in N^{1,1}(X)\), \(H(\cdot,g_w)\in L^1(X)\). Let \(x_0 \in X\), \(B(x_0,r)=B_{r}\) for every \(r>0\), and \(S_{k,r}=\{x\in B_r:u(x)>k\}\). Then, for \(0<r<R\) and \(k\geq \mathop{\mathrm{ess\,sup}}_{B_R}w\) there exists \(C= C(K,q)>0\) such that the following inequality \[\begin{align} \label{CaccioppoliIneq}\int_{S_{k,r}}H(x,g_u)\, \mathrm{d}\mu & \leq C\int_{B_R} H\left(x,\frac{(u-k)_+}{R-r}\right)\, \mathrm{d}\mu \end{align}\qquad{(4)}\] is satisfied, where \((u-k)_+= \max\{u-k,0\}\).
Proof. The proof is a modification of the one originally given for local quasiminima of double-phase functionals in [20], which we furthermore specialize
to the case of quasiminimizer \(u\in N^{1,1}(X)\) with boundary data as in Definition 4.
Let \(\eta\) be a \((R-r)^{-1}\)-Lipschitz cutoff function such that \(0\leq \eta \leq1\), \(\eta=1\) on \(B_r\) and \(\eta=0\) in \(X\setminus B_R\). Let \(v=u- \eta(u-k)_+\). Then, as \(u=w\leq k\) on
\(B_R\setminus\Omega\) and \(\eta=0\) outside of \(B_R\), we have \(u-v\in N^{1,1}_0(\Omega)\). Also, \(v=(1-\eta)(u-k)_+ +k\) on \(S_{k,R}\) and \(v=u\) outside \(S_{k,R}\).
By the Leibniz rule for the upper gradients [6], we have \[g_{v}\leq (u-k)_+g_{\eta}+(1-\eta) g_u +g_u \chi_{X\setminus S_{k,R}}.\]
Since \(u\) is a quasiminimizer, by Definition 4 we obtain \[\label{5467}
\begin{align}
\int_{S_{k,r}}H(x,g_u)\, \mathrm{d}\mu
&\leq\int_{\{u\neq v\}}H(x,g_u)\, \mathrm{d}\mu
\leq K\int_{\{u\neq v\}}H(x,g_v)\, \mathrm{d}\mu \\
& \leq K\left(\int_{S_{k,R}} H(x, (u-k)_+g_{\eta}+(1-\eta) g_u) \, \mathrm{d}\mu\right)\\
& \leq 2^qK\left(\int_{B_{R}}H\left(x,\frac{(u-k)_+}{R-r}\right)\,\mathrm{d}\mu
+\int_{S_{k,R}\setminus S_{k,r}}H(x,g_u)\, \mathrm{d}\mu\right).
\end{align}\tag{6}\] By adding \(K 2^{q}\int_{S_{k,r}}H(x,g_u)\, \mathrm{d}\mu\) to the both sides of (6 ), we get \[(1+K 2^{q}) \int_{S_{k,r}}H(x,g_u)\,
\mathrm{d}\mu
\leq K2^{q}\left(\int_{B_{R}}H\left(x,\frac{(u-k)_+}{R-r}\right)\, \mathrm{d}\mu
+ \int_{S_{k,R}}H(x,g_u)\, \mathrm{d}\mu\right).\] This implies \[\begin{align}
\int_{S_{k,r}}H(x,g_u)\, \mathrm{d}\mu
&\leq \theta\left(\int_{B_{R}}H\left(x,\frac{(u-k)_+}{R-r}\right)\, \mathrm{d}\mu
+ \int_{S_{k,R}}H(x,g_u)\, \mathrm{d}\mu\right)\\
&\le(R-r)^{-p}\int_{B_{R}}(u-k)_+^p\, \mathrm{d}\mu
+(R-r)^{-q}\int_{B_{R}}a(u-k)_+^q\, \mathrm{d}\mu
+\theta\int_{S_{k,R}}H(x,g_u)\, \mathrm{d}\mu,
\end{align}\] with \(\theta= \frac{K 2^{q}}{1+K 2^{q}}<1\). We apply a standard iteration lemma, see [59], to obtain \[\int_{S_{k,r}}H(x,g_u)\, \mathrm{d}\mu
\leq C\int_{B_{R}}H\left(x,\frac{(u-k)_+}{R-r}\right)\, \mathrm{d}\mu,\] where \(C=C(q,K)\). ◻
Remark 5. Notice that if \(u\) is a quasiminimizer then \(-u\) is also a quasiminimizer. Therefore, by Lemma 2, we get that \(-u\) satisfies ?? .
The goal now is to prove that a quasiminimizer, as Definition 4, is locally bounded. We first prove an auxiliary lemma. We note that a version of these results was first proved for local quasminimizers in [21]. The main difference is that the results proven in the present work hold for any ball in \(X\) and not only for compactly contained balls in \(\Omega\).
Lemma 3. Let \(u\in N^{1,1}(X)\) with \(H(\cdot, g_u)\in L^1(X)\), \(x_0\in X\), \(B(x_0,R)=B_{R}\), \(0<\frac{R}{2}<\rho<s\leq R\leq \min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{6}\rbrace\), and concentric balls \(B_{\rho}\subset B_s\subseteq B_R\). Assume \(u\) satisfies the double-phase Caccioppoli inequality ?? for all \(k\geq k^*\) and \(0<r<R\). Then, for any positive numbers \(k^*\leq h<k\), there exists a constant \(C= C(\mathop{\mathrm{data}}, \Vert u\Vert_{N^{1,p}(X)})\), and an exponent \(0<\theta= \theta(data)\), such that \[\int_{S_{k,\rho}}H(x,u-k) \mathrm{d}\mu\leq \frac{C}{(s-\rho)^{q-p}}\left(\dfrac{\mu(S_{k, s})}{\mu(B_s)}\right)^{\theta} \int_{S_{h,s}} H\left(x,\frac{u-h}{s-\rho}\right) \mathrm{d}\mu.\]
Proof. Let \(0<\frac{R}{2}<\rho<s\leq R\leq \min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{6}\rbrace\). We define \(t=\frac{s+\rho}{2}\) to simplify notation. Notice that, by definition, \(\rho<t<s\). By the double-phase Caccioppoli inequality ?? , for \(k\geq k^*\) we have \[\begin{align} \label{ks2} \int_{B_t} H(x, g_{(u-k)_+}) \mathrm{d}\mu&\leq C \int_{B_s} H\left(x,\frac{(u-k)_+}{s-t}\right)\, \mathrm{d}\mu \nonumber\\ & \leq C 2^{q} \int_{B_s} H\left(x,\frac{(u-k)_+}{s-\rho}\right)\, \mathrm{d}\mu \nonumber\\ &= C \int_{B_s} H\left(x,\frac{(u-k)_+}{s-\rho}\right)\, \mathrm{d}\mu, \end{align}\tag{7}\] where \(C=C(K,q)\). Let \(\tau\) be a \(\dfrac{1}{s-\rho}\)-Lipschitz cutoff function so that \(0\leq \tau \leq1\), \(\tau=1\) on \(B_\rho\) and the support of \(\tau\) is contained in \(B_t\). Let \(w=\tau (u-k)_+\in N_0^{1,1}(B_t)\). By Leibniz rule, [6], we have \[g_w \leq g_{(u-k)_+}\tau +(u-k)_+ g_{\tau}\leq g_{(u-k)_+}+ \frac{1}{s-\rho}(u-k)_+,\qquad\textrm{on }B_t.\] Using inequality 7 , we get \[\begin{align} \label{15notes} \int_{B_t} H(x,g_w) \mathrm{d}\mu&\leq 2^{q-1}\int_{B_t} H(x,g_{(u-k)_+}) \mathrm{d}\mu+2^{q-1}\int_{B_{t}} H\left(x,\frac{(u-k)_+}{s-\rho}\right)\mathrm{d}\mu \nonumber \\ &\leq C \, 2^{q-1} \int_{B_s} H\left(x,\frac{(u-k)_+}{s-\rho}\right)\mathrm{d}\mu +2^{q-1} \int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu \nonumber \\ &= C \int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu. \end{align}\tag{8}\]
By Hölder inequality, the doubling property, the definition of \(w\) and [21], there is a constant \(C=C({\rm data})\) and exponents \(0<d_2<1\leq d_1<\infty\), such that
\[\begin{align} \label{ks40} \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\rho}}H\left(x,(u-k)_+\right) \mathrm{d}\mu&\leq \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\rho}}H\left(x,\frac{(u-k)_+}{t}\right) \mathrm{d}\mu\nonumber\\ &\leq \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\rho}}H\left(x,\frac{(u-k)_+}{t}\right) ^{d_1} \mathrm{d}\mu\right)^{\frac{1}{d_1}}\nonumber\\ &=\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\rho}}H\left(x,\frac{\tau(u-k)_+}{t}\right) ^{d_1} \mathrm{d}\mu\right)^{\frac{1}{d_1}}\nonumber\\ &= \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{\rho}}H\left(x,\frac{w}{t}\right) ^{d_1} \mathrm{d}\mu\right)^{\frac{1}{d_1}}\nonumber\\ &\leq \left(\dfrac{\mu\big(B_{t}\big)}{\mu(B_{\rho})}\right)^{\frac{1}{d_1}}\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{t}} H\left(x,\frac{w}{t}\right) ^{d_1} \mathrm{d}\mu\right)^{\frac{1}{d_1}} \nonumber\\ &\leq C\left(1+\|g_w\|^{q-p}_{L^p(B_{t})} \mu(B_{t})^{\frac{\alpha}{Q}-\frac{q-p}{p}}\right)\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{t}}H\left(x,g_w\right) ^{d_2} \mathrm{d}\mu\right)^{\frac{1}{d_2}}. \end{align}\tag{9}\] In the second to last inequality, we used the doubling property to estimate \[\label{starextra} \left(\dfrac{\mu\big(B_{t}\big)}{\mu(B_{\rho})}\right)^{\frac{1}{d_1}}\leq C,\tag{10}\] where \(C\) depends on the exponent \(Q\) in ?? . By Hölder inequality, we have \[\begin{align} \label{ks4first} \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_{t}}H\left(x,g_w\right) ^{d_2} \mathrm{d}\mu\right)^{\frac{1}{d_2}}&=\mu(B_{t})^{-\frac{1}{d_2}}\left(\int_{S_{k, t}}H\left(x,g_w\right)^{d_2} \mathrm{d}\mu\right)^{\frac{1}{d_2}}\nonumber\\ &\leq \mu(B_{t})^{-\frac{1}{d_2}}\mu(S_{k, t})^{\frac{1}{d_2}-1}\int_{S_{k, t}}H\left(x,g_w\right) \mathrm{d}\mu. \end{align}\tag{11}\] By 9 , 11 and 8 , we get \[\begin{align} \label{ks4} &\int_{B_{\rho}}H\left(x,(u-k)_+\right) \mathrm{d}\mu\nonumber\\ &\leq C\left(1+\|g_w\|^{q-p}_{L^p(B_{t})} \mu(B_{t})^{\frac{\alpha}{Q}-\frac{q-p}{p}}\right)\left(\dfrac{\mu(S_{k, t})}{\mu(B_{t})}\right)^{\frac{1}{d_2}-1} \int_{S_{k, t}}H\left(x,g_w\right) \mathrm{d}\mu \nonumber\\ &\leq C\left(1+\|g_w\|^{q-p}_{L^p(B_t)} \mu(B_R)^{\frac{\alpha}{Q}-\frac{q-p}{p}}\right)\left(\dfrac{\mu(S_{k, t})}{\mu(B_{t})}\right)^{\frac{1}{d_2}-1}\int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu. \end{align}\tag{12}\] On the other hand, notice that \[\begin{align} \Vert g_w\Vert_{L^p(B_t)}^p&=\int_{B_t}g_w^p\mathrm{d}\mu\nonumber\\ &\leq \int_{B_t}\left(g_{(u-k)_+}+\frac{(u-k)_+}{s-\rho}\right)^p\mathrm{d}\mu\nonumber\\ &\leq 2^{p-1}\left(\int_{B_t}g_{(u-k)_+}^p\mathrm{d}\mu+\int_{B_t}\left(\frac{(u-k)_+}{s-\rho}\right)^p\mathrm{d}\mu\right)\\ &\leq C\left(\Vert g_u\Vert_{L^p(B_t)}^{p}+\frac{\Vert u\Vert^p_{L^p(B_t)}}{(s-\rho)^p}\right)\\ &\leq \frac{C}{(s-\rho)^p}\left(\Vert g_u\Vert_{L^p(B_t)}^{p}+\Vert u\Vert^p_{L^p(B_t)}\right). \end{align}\] Since \(u\in N^{1,1}(X)\), \(H(\cdot,g_u)\in L^{1}(X)\), \(B_t\subset B_R\subset X\) and the nature of the double-phase functional, then \(u\in N^{1,p}(X)\). Therefore, \[\begin{align} \Vert g_w\Vert_{L^p(B_t)}^{q-p}&\leq\frac{C}{(s-\rho)^{q-p}}\left(\Vert g_u\Vert_{L^p(B_t)}^{p}+\Vert u\Vert^p_{L^p(B_t)}\right)^{\frac{q-p}{p}}\\ &\leq\frac{C}{(s-\rho)^{q-p}}\left(\Vert g_u\Vert_{L^p(B_t)}+\Vert u\Vert_{L^p(B_t)}\right)^{q-p}\\ &=\frac{C}{(s-\rho)^{q-p}}\Vert u\Vert_{N^{1,p}(B_t)}^{q-p}\\ &\leq\frac{C}{(s-\rho)^{q-p}}\Vert u\Vert_{N^{1,p}(X)}^{q-p}. \end{align}\] By 12 , the assumption that our measure is upper \(Q\)-Alhfors regular 5 , \(r\leq 1\) and the last inequality, we then obtain
\[\begin{align}
\label{new1}
\int_{B_{\rho}}&H\left(x,(u-k)_+\right) \mathrm{d}\mu\nonumber\\
&\leq \frac{C}{(s-\rho)^{q-p}}\left(1+\Vert u\Vert_{N^{1,p}(X)}^{q-p}\mu(B_R)^{\frac{\alpha}{Q}-\frac{q-p}{p}}\right)\left(\dfrac{\mu(S_{k, t})}{\mu(B_{t})}\right)^{\frac{1}{d_2}-1}\int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu
\nonumber\\
&\leq \frac{C}{(s-\rho)^{q-p}}\left(1+\Vert u\Vert_{N^{1,p}(X)}^{q-p}\right)\left(\dfrac{\mu(S_{k, t})}{\mu(B_{t})}\right)^{\frac{1}{d_2}-1}\int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu \nonumber\\
&= \frac{C}{(s-\rho)^{q-p}}\left(\dfrac{\mu(S_{k, t})}{\mu(B_{t})}\right)^{\frac{1}{d_2}-1}\int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu,
\end{align}\tag{13}\] where \(C=C(\mathop{\mathrm{data}}, C_1, \Vert u\Vert_{N^{1,p}(X)})\).
Furthermore, we observe that \(\mu(S_{k, t})\leq \mu(S_{k, s})\) and \(\frac{1}{d_2}-1>0\). Thus, by 13 and the doubling property of the measure (as we did in 10 ) we get
\[\begin{align}
\label{algo}
\int_{B_{\rho}}H\left(x,(u-k)_+\right) \mathrm{d}\mu \leq \frac{C}{(s-\rho)^{q-p}}\left(\dfrac{\mu(S_{k, s})}{\mu(B_{s})}\right)^{\frac{1}{d_2}-1}\int_{B_s}H\left(x,\frac{(u-k)_+}{s-\rho}\right) \mathrm{d}\mu.
\end{align}\tag{14}\] with \(C= C(\mathop{\mathrm{data}}, C_1, \Vert u\Vert_{N^{1,p}(X)})\).
Let \(k^*\leq h<k\), then \((u-k)_+\leq (u-h)_+\). Therefore, we have \[\begin{align}
\int_{S_{k,\rho}}H(x,u-k) \mathrm{d}\mu &\leq \frac{C}{(s-\rho)^{q-p}}\left(\dfrac{\mu(S_{k, s})}{\mu(B_{s})}\right)^{\frac{1}{d_2}-1} \int_{S_{h,s}} H\left(x,\frac{u-h}{s-\rho}\right) \mathrm{d}\mu.
\end{align}\] That is \[\int_{S_{k,\rho}}H(x,u-k) \mathrm{d}\mu\leq \frac{C}{(s-\rho)^{q-p}}\left(\dfrac{\mu(S_{k, s})}{\mu(B_s)}\right)^{\theta} \int_{S_{h,s}} H\left(x,\frac{u-h}{s-\rho}\right) \mathrm{d}\mu,\]
where \(\theta=\frac{1}{d_2}-1>0\). ◻
As a consequence of the previous lemma, we obtain the following weak Harnack inequality type result.
Theorem 3. Let \(X\) be a doubling metric measure space admitting a weak \((1,p)\)-Poincaré inequality. Then, there exists \(C>0\), \(C=C(\mathop{\mathrm{data}}, C_1, R, \Vert u\Vert_{N^{1,p}(X)})\), such that if \(u\in N^{1,1}(X)\) with \(H(\cdot, g_u)\in L^1(X)\) and the condition ?? holds for all \(k\geq k^*\) and \(0<r<R<\min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{6}\rbrace\), then for all \(k_0\geq k^*\) \[\label{weakHarnackBoundary} \mathop{\mathrm{ess\,sup}}_{B_{R/2}}u\leq k_0+C\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_R}H\left(x,(u-k_0)_+\right)\mathrm{d}\mu\right)^{\frac{1}{p}}\qquad{(5)}\]
Proof. This proof is similar to the one of [21], with few modifications. Let \(k_0\geq k^*\). For \(n\in \mathbb{N}\cup \{0\}\), let \(\rho_{n}=\frac{R}{2}\left(1+\frac{1}{2^{n}}\right)\leq R\) and \(k_{n}=k_{0}+d\left(1-\frac{1}{2^n}\right)\), where \(d>0\) will be chosen later. Then, \(\rho_{0}=R\), \(\rho_{n}\searrow \frac{R}{2}\) and \(k_n\nearrow k_0+d\). We apply Lemma 3 with \(\rho=\rho_{i+1}\), \(R=\rho_i\), \(k=k_{i+1}\) and \(h=k_i\) and we get \[\begin{align} \label{8notes} \int_{S_{k_{i+1},\rho_{i+1}}}H(x,u-k_{i+1}) \mathrm{d}\mu&\leq \frac{C}{(\rho_i-\rho_{i+1})^{q-p}}\left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_{R})}\right)^{\theta} \int_{S_{k_i,\rho_i}} H\left(x,\frac{u-k_i}{\rho_i-\rho_{i+1}}\right) \mathrm{d}\mu\nonumber \\ & = \frac{C}{(R 2^{-i}2^{-2})^{q-p}}\left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_{R})}\right)^{\theta} \int_{S_{k_i,\rho_i}} H\left(x,\frac{u-k_i}{R 2^{-i}2^{-2}}\right) \mathrm{d}\mu\nonumber \\ & = \frac{C2^{i(2q-p)}}{R^{q-p}}\left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_{R})}\right)^{\theta} \int_{S_{k_i,\rho_i}} H\left(x,\frac{u-k_i}{R}\right) \mathrm{d}\mu \nonumber\\ & \leq \frac{C4^{iq}}{R^{2q-p}}\left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_R)}\right)^{\theta} \int_{S_{k_i,\rho_i}} H(x,u-k_i) \mathrm{d}\mu \nonumber\\ & \leq \frac{C4^{iq}}{R^{2q}}\left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_R)}\right)^{\theta} \int_{S_{k_i,\rho_i}} H(x,u-k_i) \mathrm{d}\mu. \end{align}\tag{15}\] We observe that \[\begin{align} d^{-p}(k_{i+1}-k_i)^p \mu(S_{k_{i+1}, \rho_i})&= d^{-p}\int_{S_{k_{i+1}, \rho_i}}(k_{i+1}-k_i)^p \mathrm{d}\mu\\ &\leq d^{-p}\int_{S_{k_{i+1}, \rho_i}}H(x, u-k_i) \mathrm{d}\mu\\ &\leq d^{-p}\int_{S_{k_i, \rho_i}}H(x, u-k_i) \mathrm{d}\mu. \end{align}\] So, we have \[\psi_i= d^{-p}\int_{S_{k_i, \rho_i}}H(x, u-k_i) \mathrm{d}\mu \geq d^{-p}(k_{i+1}-k_i)^p \mu(S_{k_{i+1}, \rho_i}).\] This implies \[\label{9notes} \mu(S_{k_{i+1}, \rho_i})\leq \psi_i d^{p}(k_{i+1}-k_i)^{-p}.\tag{16}\] Therefore, by 15 and 16 we obtain \[\begin{align} d^p \psi_{i+1}&\leq \frac{C4^{iq}}{R^{2q}} \left(\dfrac{\mu(S_{k_{i+1}, \rho_i})}{\mu(B_R)}\right)^{\theta} d^p\psi_i\nonumber\\ &\leq \frac{C4^{iq}}{R^{2q}} \left(\psi_i d^{p}(k_{i+1}-k_i)^{-p}\right)^{\theta}d^p\psi_i \mu(B_R)^{-\theta}\nonumber\\ &= \frac{C4^{iq}}{R^{2q}} (d2^{-1-i})^{-p\theta} d^{p\theta}\psi_i^{1+\theta} d^p \mu(B_R)^{-\theta}\nonumber\\ &\leq \frac{C4^{(1+\theta)qi}}{R^{2q}} \psi_i^{1+\theta}d^p \mu(B_R)^{-\theta}. \end{align}\] That is, \[\psi_{i+1} \leq\frac{C4^{(1+\theta)qi}}{R^{2q}} \psi_i^{1+\theta}\mu(B_R)^{-\theta},\] for every \(i\geq0\), where \(C=C({\rm data},C_1, \Vert u\Vert_{N^{1,p}(X)})\). By using a standard iteration lemma [59] we get \[\lim_{i\to \infty}\psi_i=\lim_{i\to \infty}d^{-p}\int_{S_{k_i, \rho_i}}H(x, u-k_i) \mathrm{d}\mu=0,\] provided that \(d=CR^{\frac{-2q}{p\theta}} \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_R}H(x, (u-k_0)_+) \mathrm{d}\mu\right)^{\frac{1}{p}}>0\). As a consequence, \[\int_{B_{\frac{R}{2}}} H(x, (u-(k_0+d))_+ \mathrm{d}\mu=0\] and so \[\label{quasiboundedness} u\leq k_0+d \quadalmost everywhere in B_{\frac{R}{2}} and for all k_0 \geq k^*.\tag{17}\] We conclude that \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B_{\frac{R}{2}}}u &\leq k_0 +d= k_0+ C \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_R}H\left(x,(u-k_0)_+\right) \, \mathrm{d}\mu\right)^{\frac{1}{p}}, \end{align}\] where \(C=C(\mathop{\mathrm{data}}, C_1, R, \Vert u\Vert_{N^{1,p}(X)})\). ◻
As a corollary of Theorem 3, we obtain local boundedness for quasiminima of 1 . We emphasize that these results are a generalization of those initially obtained in [21], since here the conclusions are valid for any ball contained in \(X\) and not only for balls compactly contained in the domain \(\Omega\).
Corollary 2. Let \(u\in N^{1,1}(X)\) with \(H(\cdot, g_u)\in L^1(X)\). Assume \(u\) satisfies the double-phase Caccioppoli inequality ?? for all \(k\geq k^*\) and \(0<r<R<\min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{6}\rbrace\). Then, \[\Vert u\Vert_{L^{\infty}(B_{R/2})}<\infty\] holds. In particular, if \(u\) is a quasiminimizer in \(\Omega\) with boundary data \(w\in N^{1,1}(X)\), \(H(\cdot,g_w)\in L^1(X)\), then \(u\) is locally bounded.
Proof. Since \(u\) satisfies the double-phase Caccioppoli inequality ?? , then by Theorem 3 and, in particular, by inequality 17 , we obtain the desiered result.
In case \(u\) is a quasiminimizer, by Lemma 2, \(u\) satisfies the double-phase Caccioppoli inequality
?? for all \(k\geq \mathop{\mathrm{ess\,sup}}_{B(x_0,R)}w\). Without loss of generality, we can assume \(\mathop{\mathrm{ess\,sup}}_{B(x_0,R)}w<\infty\), otherwise by inequality 17 the result trivially holds true. Therefore, the result follows as before. ◻
Once we have that quasiminima are locally bounded in any ball \(B_r\subset X\), we can prove the following almost standard Caccioppoli’s inequality. This states that, in the \(p\)-regime case, double-phase quasiminima satisfy a Caccioppoli type inequality analogous to the one satisfied for quasiminima of functionals with just \(p\)-growth plus some extra controllable terms. This result was proved in [21] for local quasiminima of double-phase problems. In the Euclidean case this result has been proven by Colombo-Mingione [3].
Lemma 4 (Almost standard Caccioppoli’s inequality). Assume that \(u\in N^{1,1}(X)\) with \(H(\cdot,g_u)\in L^1(X)\) satisfies the double-phase Caccioppoli inequality ?? for all \(k\geq k^*\) and \(0<r<R<\min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{6}\rbrace\). Furthermore, assume that \[\label{p-regime} \sup_{B(x_0,R)} a(x) \leq C [a]_{\alpha} \mu(B(x_0,R))^{\frac{\alpha}{Q}},\qquad{(6)}\] holds for \(x_0\in X\), \(0\leq R\leq 1\). Then there exists \(C= C(C_1, p, q,\alpha, Q ,[a]_{\alpha}, \|u\|_{L^{\infty}(B(x_0,R))}, K)>0\) such that for any choice of concentric balls \(B(x_0,s)\subset B(x_0,t) \subset B(x_0,R)\subset X\), with \(0<t<s\leq R\leq 1\) and \(2\Vert u\Vert_{L^{\infty}(B(x_0,R))}\geq \vert k\vert\geq k\geq\mathop{\mathrm{ess\,sup}}_{B(x_0,R)}w\), the following inequality \[\begin{align} \label{ASCaccioppoli} \int_{B(x_0,t)}g_{(u-k)_+}^p\, \mathrm{d}\mu & \leq C \left(\left(\frac{R}{s-t}\right)^q\int_{B(x_0,s)} \left|\frac{(u-k)_+}{R}\right|^p\, \mathrm{d}\mu\right) \end{align}\qquad{(7)}\] is satisfied, where \((u-k)_+= \max\{u-k,0\}\). In particular, if \(u\) is a quasiminimizer in \(\Omega\) with boundary data \(w\in N^{1,1}(X)\), \(H(\cdot, g_w)\in L^1(X)\), then ?? holds as well.
Proof. For this proof, we treat the case when \(u\) is a quasiminimizer. By Lemma 2, since \(u\) is a quasiminimizer with boundary values \(w\), then for all \(k\geq\mathop{\mathrm{ess\,sup}}_{B(x_0,R)}w\), \(u\)
satisfies the double-phase Caccioppoli’s inequality ?? . Therefore, we have \[\begin{align}
\label{asc1}
\int_{B(x_0,t)}g_{(u-k)_+}^p\, \mathrm{d}\mu &\leq \int_{B(x_0,t)}H(x,g_{(u-k)_+})\, \mathrm{d}\mu \leq C \int_{B(x_0,s)} H\left(x,\frac{(u-k)_+}{s-t}\right)\, \mathrm{d}\mu \nonumber\\
& = C \left(\int_{B(x_0,s)} \left|\frac{(u-k)_+}{s-t}\right|^p\, \mathrm{d}\mu + \int_{B(x_0,s)} a(x)\left|\frac{(u-k)_+}{s-t}\right|^q\, \mathrm{d}\mu\right)\nonumber\\
& = C \Bigg(\left(\frac{R}{s-t}\right)^p\int_{B(x_0,s)} \left|\frac{(u-k)_+}{R}\right|^p\, \mathrm{d}\mu \nonumber\\
&\qquad\qquad\qquad+\left(\frac{R}{s-t}\right)^q \int_{B(x_0,s)} a(x)\left|\frac{(u-k)_+}{R}\right|^q\, \mathrm{d}\mu\Bigg),
\end{align}\tag{18}\] here \(C=C(K,q)\).
Now, we estimate the integrand function in the last term of the previous inequality. Using ?? and 5 , we get \[\begin{align}
a(x)\left|\frac{(u-k)_+}{R}\right|^q &\leq C[a]_{\alpha}\frac{\mu(B(x_0,R))^{\frac{\alpha}{Q}}}{R^q} \|u\|^{q-p}_{L^{\infty}(B(x_0,s))} |(u-k)_+|^p\\
& \leq C[a]_{\alpha}\frac{C_1 R^{\alpha}}{R^q} \|u\|^{q-p}_{L^{\infty}(B(x_0,R))} |(u-k)_+|^p\\
&= \frac{C[a]_{\alpha}}{R^{q-\alpha}} \|u\|^{q-p}_{L^{\infty}(B(x_0,R))} |(u-k)_+|^p\\
&\leq \frac{C [a]_{\alpha}}{R^{p}}\|u\|^{q-p}_{L^{\infty}(B(x_0,R))} |(u-k)_+|^p= C \left|\frac{(u-k)_+}{R}\right|^p,
\end{align}\] where \(C=C(C_1, p, q,\alpha, Q ,[a]_{\alpha}, \|u\|_{L^{\infty}(B(x_0,R))})\). The last inequality holds true by 2 and Remark 3. Thus, 18 becomes \[\begin{align}
\int_{B(x_0,t)}g_{(u-k)_+}^p\, \mathrm{d}\mu &\leq C \left(\left(\frac{R}{s-t}\right)^p\int_{B(x_0,s)} \left|\frac{(u-k)_+}{R}\right|^p\, \mathrm{d}\mu +C\left(\frac{R}{s-t}\right)^q\int_{B(x_0,s)} \left|\frac{(u-k)_+}{R}\right|^p\,
\mathrm{d}\mu\right)\\
& \leq C \left(\frac{R}{s-t}\right)^q \int_{B(x_0,s)} \left|\frac{(u-k)_+}{R}\right|^p\, \mathrm{d}\mu.
\end{align}\] ◻
In this section, we collect the necessary regularity results for quasiminima of the so-called frozen functionals used in subsequent sections. We consider functionals of the type
\[\label{eq1frozen}
\int_{\Omega}H_0(g_u)\mathrm{d}\mu=\int_{\Omega}(g_u^p+a_0g_u^q)\mathrm{d}\mu,\tag{19}\] where \(a_0\geq 0\) is a constant. Frozen functionals belong to the class of functionals introduced by the seminal work
of Lieberman [60]. One of the main difficulties when doing the careful analysis of the different phases of the double-phase functional can be solved by
using frozen functionals theory, which is based on the fact that under appropriate conditions a quasiminimizer \(u\) of functional 1 can be seen as a quasiminimizer of a certain frozen functional.
The related theory for this notion due to Lieberman [60] can be used to obtain the needed estimates for regularity properties. For example, local
regularity results were obtained in the Euclidean case in [3] and in the general context of a metric measure space in [21]. Here we prove the following generalization of Maz’ya’s estimate, Proposition 2 (see also [6]), for frozen functionals. As far as we know this result is new even in the Euclidean case.
Theorem 4 (Maz’ya’s type estimate for frozen functionals and small radii). Let \(u\in N^{1,1}(X)\) with \(H_0(g_u)\in L^1(X)\), \(B_r\subset X\) any ball with radius \(r\leq \min\lbrace 1, \frac{\mathop{\mathrm{diam}}(X)}{8}\rbrace\) and \(S=\lbrace x\in B_{\frac{r}{2}}: u(x)=0\rbrace\). Then, there exists \(C=C(C_D,C_{\textrm{PI}}, C_1,p,q)>0\) and exponents \(d_1\geq 1>d_2>0\), with \(d_1=d_1(C_D,C_{\textrm{PI}},p,q)\) and \(d_2=d_2(C_D,C_{\textrm{PI}},p, q)\), such that the following inequality holds \[\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r} H_0\left(u\right)^{d_1}\mathrm{d}\mu\right)^{\frac{1}{d_1}}\leq C\left(\frac{1}{\textrm{cap}_{pd_2}^{q/p}(S,B_r)}\int_{B_{2\lambda r}} H_0(g_u)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}},\] where \(\lambda\) is the dilation constant in the \((1,p)\)-Poincaré inequality.
Proof. By splitting \(u\) into its positive and negative parts and considering them separately, we can assume that \(u\geq 0\) in \({B_r}\).
Recall that we are assuming that our space supports a \((1,s)\)-Poincaré inequality with \(1<s<p<q<s^*\). By Remark 3 and Theorem 2, \(X\) supports a \((s^*,s)\)-Poincaré inequality.
Let \(\frac{s}{p}<d_2<1\), \(d_2=d_2(C_{\textrm{PI}}, C_D,p,q)\), and \(\frac{s^*}{q}\geq d_1\geq 1\), \(d_1=d_1(C_{\textrm{PI}}, C_D,p,q)\). By Proposition 2, the following two inequalities hold at the same time
\[\label{eq2463NEW} \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle-
}\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\left\vert u\right\vert^{p
d_1}\mathrm{d}\mu\right)^{\frac{d_2}{d_1}}\leq \frac{C}{\textrm{cap}_{p d_2}(S, B_r)}\int_{B_{2\lambda r}}g_u^{p d_2}\mathrm{d}\mu\tag{20}\] and \[\label{eq2464NEW}
\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox
0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\left\vert u\right\vert^{q
d_1}\mathrm{d}\mu\right)^{\frac{d_2}{d_1}}\leq \frac{C}{\textrm{cap}_{q d_2}(S, B_r)}\int_{B_{2\lambda r}}g_u^{q d_2}\mathrm{d}\mu\tag{21}\]
Therefore, by 21 and ?? in Remark 4, we have \[\begin{align} \label{eq2466NEW} \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\left\vert u\right\vert^{q d_1}\mathrm{d}\mu\right)^{\frac{d_2}{d_1}}&\leq \frac{C}{\textrm{cap}_{q d_2}(S, B_r)}\int_{B_{2\lambda r}}g_u^{q d_2}\mathrm{d}\mu \nonumber\\ &\leq \frac{C}{\textrm{cap}_{p d_2}^{q/p}(S, B_r)}\int_{B_{2\lambda r}}g_u^{q d_2}\mathrm{d}\mu, \end{align}\tag{22}\] where \(C=C(C_{\textrm{PI}}, C_1,p,q)\). Now, by 20 , 22 , Proposition 1, \(\mu\) being upper \(Q\)-Alhfors regular and \(r\leq 1\), we obtain the next chain of inequalities
\[\begin{align} \Bigg(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}H_0^{d_1}\left(u\right)\mathrm{d}\mu\Bigg)^{\frac{1}{d_1}}&\leq \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}2^{d_1-1}\left(\left\vert u\right\vert^{pd_1}+a_0^{d_1}\left\vert u\right\vert^{qd_1}\right)\mathrm{d}\mu\right)^{\frac{1}{d_1}}\\ &\leq 2\left(\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\left\vert u\right\vert^{pd_1}\mathrm{d}\mu\right)^{\frac{1}{d_1}}+a_0\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}\left\vert u\right\vert^{qd_1}\mathrm{d}\mu\right)^{\frac{1}{d_1}}\right)\\ &\leq C\Bigg( \left(\frac{1}{\textrm{cap}_{p d_2}(S,B_r)}\int_{B_{2\lambda r}}g_u^{pd_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}+\left(\frac{1}{\textrm{cap}_{p d_2}^{q/p}(S,B_r)}\int_{B_{2\lambda r}}(a_0g_u^q)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\Bigg)\\ &\leq C\Bigg( \left(\left(1+\frac{1}{\textrm{cap}_{p d_2}(S,B_r)}\right)\int_{B_{2\lambda r}}g_u^{pd_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\qquad \qquad+\left(\left(1+\frac{1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{q/p}\int_{B_{2\lambda r}}(a_0g_u^q)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\Bigg)\\ &\leq C\left(1+\frac{1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{\frac{q}{pd_2}}\left(\left(\int_{B_{2\lambda r}}g_u^{pd_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}+\left(\int_{B_{2\lambda r}}(a_0g_u^q)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\right)\\ &\leq C\left(\left(\frac{\textrm{cap}_{p d_2}(S,B_r)+1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{\frac{q}{pd_2}}\left(\int_{B_{2\lambda r}}g_u^{pd_2}\mathrm{d}\mu+\int_{B_{2\lambda r}}(a_0g_u^q)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\right)\\ &\leq C\left(\left(\frac{\left(\frac{C_D\mu(B_{\frac{r}{2}})}{\left(\frac{r}{2}\right)^{pd_2}}\right)+1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{q/p}\int_{B_{2\lambda r}}\left(g_u^{pd_2}+(a_0g_u^q)^{d_2}\right)\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\leq C\left(\left(\frac{C_DC_1\left(\frac{r}{2}\right)^{Q-pd_2}+1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{q/p}\int_{B_{2\lambda r}}\left(g_u^{pd_2}+(a_0g_u^q)^{d_2}\right)\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\leq C\left(\left(\frac{C+1}{\textrm{cap}_{p d_2}(S,B_r)}\right)^{q/p}\int_{B_{2\lambda r}}\left(g_u^{pd_2}+(a_0g_u^q)^{d_2}\right)\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\leq C\left(\frac{1}{\textrm{cap}_{pd_2}^{q/p}(S,B_R)}\int_{B_{2\lambda r}}H_0(g_u)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}. \end{align}\] Therefore, \[\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B_r}H_0^{d_1}\left(u\right)\mathrm{d}\mu\right)^{\frac{1}{d_1}}\leq C\left(\frac{1}{\textrm{cap}_{pd_2}^{q/p}(S,B_R)}\int_{B_{2\lambda r}}H_0(g_u)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}},\] where \(C=C(C_{\textrm{PI}}, C_D, C_1,p,q)\). ◻
This section is devoted to the proof of a pointwise estimate near a boundary point which has a key role in obtaining the sufficient condition for Hölder continuity (Theorem 5, Section 6).
Let \(w\in N^{1,1}(X)\) with \(H(\cdot, g_w)\in L^1(X)\), and such that \(w - u \in N^{1,1}_0(\Omega)\). We shall use the notation \[M(r,r_{0})=\Big(\mathop{\mathrm{ess\,sup}}_{B(x_{0},r)} u -\mathop{\mathrm{ess\,sup}}_{B(x_{0},r_{0})} w\Big)_{+},\] where \(0<r\leq r_{0}\), \(a_{+}=\max\lbrace
a,0\rbrace\), and \(u \in N^{1,1}(X)\), with \(H(\cdot, g_u)\in L^1(X)\) is a quasiminimizer on \(\Omega\), with boundary values \(w\). Let also \[\gamma_{p,q}(s,r)=\frac{r^{-sq/p}\mu(B(x_{0},r))}{{\rm cap}_{s}^{q/p}(B(x_{0},r)\setminus\Omega, B(x_{0},2r))}.\]
Now we state the main result of this section, a pointwise estimate for quasiminima near a boundary point. The proof of this estimate is based on a careful analysis of the phases.
Proposition 3 (Pointwise estimate). Let \(w\in N^{1,1}(X)\) with \(H(\cdot, g_w)\in L^1(X)\). Let \(u \in N^{1,1}(X)\), with \(H(\cdot, g_u)\in L^1(X)\) a quasiminimizer on \(\Omega\) with boundary values \(w\). Then there exist \(\lambda\geq 1\), \(0<d_2<1\) and \(C>0\) such that, for all \(x_{0}\in\partial\Omega\) and \(0<4\lambda r\leq r_{0}<\min\lbrace 1, \frac{\textrm{diam}(X)}{12 \lambda}\rbrace\), the next inequality holds \[M\left(\frac{r}{2}, r_{0}\right)\leq (1-2^{-n(r)-1})M(4\lambda r, r_{0}),\] where \(n(r)\) is a sufficiently large integer such that \[n(r)\geq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}.\]
Proof. We denote \(M=M(4\lambda r, r_0)\), where \(x_0\in\partial\Omega\) and \(r_0>0\) are fixed. Without loss of generality, we can assume \(0<M<+\infty\), otherwise the proof is finished.
We define \[k_j=\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-j}),\] and \[v_j=(u-k_j)_+-(u-k_{j+1})_+.\] Notice that in \(B(x_0,2\lambda
r)\setminus\Omega\) we have \(v_j=0\), where \(\lambda\) is given by Theorem 4. Define \(T(k,l,r)=S_{k,r}\setminus S_{l,r}\), then \(g_uX_{T(k_j,k_{j+1},2\lambda r)}\) is a \(p\)-weak upper gradient of \(v_j\) in
\(B(x_0,2\lambda r)\).
We define \[a_0=\inf_{x\in B(x_0, 8\lambda r)}a(x).\]
Case 1: First assume that \[\label{case1INFIMUM} a_0>C[a]_{\alpha}\mu(B(x_0,8\lambda r))^{\frac{\alpha}{Q}},\tag{23}\] Note that for every \(x,y\in B(x_0, 8\lambda r)\), we have \[\begin{align} \delta_{\mu}(x,y) &=\left(\mu(B(x,d(x,y)))+\mu(B(y,d(x,y)))\right)^{1/Q}\\ &\leq \left(\mu(B(x,16\lambda r))+\mu(B(y,16\lambda r))\right)^{1/Q}\\ &\leq \left(2\mu (B(x_0,24\lambda r))\right)^{1/Q} \leq (2C_D^2\mu(B(x_0, 8\lambda r)))^{1/Q}\\ &= C\mu(B(x_0, 8\lambda r))^{1/Q}. \end{align}\] By 23 we obtain \[\begin{align} 2a_0&= 2a(x)-2\left(a(x)-a_0\right)\geq a(x)+a_0-2\left(a(x)-a_0\right)\\ &\geq a(x)+2[a]_{\alpha}(2C_D^2\mu(B(x_0, 8\lambda r)))^{\alpha/Q}-2\left(a(x)-a_0\right)\\ &\geq a(x)+2\sup_{\substack{x,y \in B(x_0, 8\lambda r)\\ x\neq y}} \frac{|a(x)-a(y)|}{\delta_{\mu}(x,y)^{\alpha}}(2C_D^2\mu(B(x_0, 8\lambda r)))^{\alpha/Q}-2\left(a(x)-a_0\right)\\ &\geq a(x)+2\sup_{x,y \in B(x_0, 8\lambda r)}|a(x)-a(y)|-2\left(a(x)-a_0\right)\\ &\geq a(x)+2\sup_{x,y \in B(x_0, 8\lambda r)}(a(x)-a(y))-2\left(a(x)-a_0\right)\\ &\geq a(x)+2a(x)-2\inf_{y \in B(x_0, 8\lambda r)}a(y)-2\left(a(x)-a_0\right) =a(x), \end{align}\] for every \(x \in B(x_0, 8\lambda r)\). On the other hand, we have \(a(x)\geq\inf_{x\in B(x_0, 8\lambda r)} a(x) \geq a_0\) for every \(x\in B(x_0, 8\lambda r)\). This implies that \[\label{eq2pointwiseProof} a_0\le a(x)\le 2a_0\qquad \textrm{for every }x \in B(x_0, 8\lambda r).\tag{24}\] We define the Frozen functional \[H_0(x)=\vert z\vert^p+a_0\vert z\vert^q.\]
By Theorem 4 applied in \(B(x_0,8\lambda r)\) and with \(v_j\), there exist \(d_1\geq 1> d_2>0\), with \(d_1=d_1(C_{\textrm{PI}},C_D,p,q)\) and \(d_2=d_2(C_{\textrm{PI}},C_D,p,q)\), such that by Hölder inequality we have \[\begin{align} \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}H_0\left(v_j\right)\mathrm{d}\mu&\leq \left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}H_0\left(v_j\right)^{d_1}\mathrm{d}\mu\right)^{\frac{1}{d_1}}\\ &\leq C\left(\frac{1}{\textrm{cap}_{pd_2}^{q/p}(B\left(x_0,\frac{r}{2}\right)\setminus\Omega, B(x_0,r))}\int_{B(x_0, 2\lambda r)}H_0(g_{v_j})^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &= C \left(\frac{r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\int_{B(x_0,2\lambda r)}H_0(g_{v_j})^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\leq C \left(\frac{r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\int_{T(k_j,k_{j+1},2\lambda r)}H_0(g_{u})^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &=C\left(\frac{r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\mu(T(k_j,k_{j+1},2\lambda r))}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\right)^{\frac{1}{d_2}}\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{T(k_j,k_{j+1},2\lambda r)}H_0(g_u)^{d_2}\mathrm{d}\mu\right)^{\frac{1}{d_2}}\\ &\leq \frac{Cr^q\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\mu(T(k_j,k_{j+1},2\lambda r)^{\frac{1}{d_2}-1}}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)^{\frac{1}{d_2}}}\int_{T(k_j,k_{j+1},2\lambda r)}H_0(g_u)\mathrm{d}\mu\\ &\leq C\left(\frac{r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\mu(T(k_j,k_{j+1},2\lambda r)^{1-d_2}}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\right)^{\frac{1}{d_2}}\int_{S_{k_j,2\lambda r}}H_0(g_u)\mathrm{d}\mu. \end{align}\] Therefore, by the doubling property we obtain \[\label{1Prop2} \int_{B(x_0,r)}H_0\left(v_j\right)\mathrm{d}\mu\leq C\left(\frac{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}{\mu(T(k_j,k_{j+1},2\lambda r)}\right)^{1-\frac{1}{d_2}}\left(r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{d_2}}\int_{S_{k_j,2\lambda r}}H_0(g_u)\mathrm{d}\mu.\tag{25}\] For the measure of the set \(S_{k_{j+1},r}\) we have the following estimate \[\begin{align} \int_{B(x_0,r)}H_0\left(v_j\right)\mathrm{d}\mu&\geq H_0\left(k_{j+1}-k_j\right)\mu(S_{k_{j+1},r})\\ &= H_0\left(2^{-j-1}M\right)\mu(S_{k_{j+1},r})\\ &=H_0\left(\frac{M}{2^{j+1}}\right)\mu(S_{k_{j+1},r}) \end{align}\] Therefore, \[\label{2Prop2} \int_{B(x_0,r)}H_0\left(v_j\right)\mathrm{d}\mu\geq H_0\left(\frac{M}{2^{j+1}}\right)\mu(S_{k_{j+1},r})\tag{26}\] Now, by 24 , Lemma 2, because the functional \(H_0\) is increasing and \(r\leq 1\), we achieve \[\begin{align} \int_{S_{k_j,2\lambda r}} H_0(g_u)\mathrm{d}\mu&\leq \int_{S_{k_j,2\lambda r}} H(x,g_u)\mathrm{d}\mu\leq \int_{B(x_0,2\lambda r)}H(x,g_{(u-k_j)_+})\mathrm{d}\mu\\ &\leq C \int_{B(x_0,4\lambda r)}H\left(x,\frac{(u-k_j)_+}{2\lambda r}\right)\mathrm{d}\mu\leq 2C \int_{B(x_0,4\lambda r)}H_0\left(\frac{(u-k_j)_+}{ r}\right)\mathrm{d}\mu\\ &\leq C \int_{B(x_0,4\lambda r)}H_0\left(\frac{\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,4\lambda r)}u-k_j\right)_+}{ r}\right)\mathrm{d}\mu\\ &\leq C\int_{B(x_0,4\lambda r)}H_0\left(\frac{M}{r2^j}\right)\mathrm{d}\mu= C \mu(B(x_0,4\lambda r))H_0\left(\frac{M}{r2^j}\right)\\ &= C\mu(B(x_0,4\lambda r))\left(\left\vert\frac{M}{r2^j}\right\vert^p+a_0\left\vert\frac{M}{r2^j}\right\vert^q\right)\\ &\leq C\mu(B(x_0,4\lambda r))\frac{1}{r^q}H_0\left(\frac{M}{2^j}\right) . \end{align}\] So, \[\label{3Prop2} \int_{S_{k_j,2\lambda r}} H_0(g_u)\mathrm{d}\mu\leq C\mu(B(x_0,4\lambda r))\frac{1}{r^q}H_0\left(\frac{M}{2^j}\right),\tag{27}\] where \(C=C(K,q)\).
By 26 , 25 and 27 , and the doubling property of the measure, we have \[\begin{align} H_0\left(\frac{M}{2^{j+1}}\right)\mu(S_{j+1,r})&\leq
\int_{B(x_0,r)}H_0\left(v_j\right)\mathrm{d}\mu\\ &\leq C\left(\frac{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}{\mu(T(k_j,k_{j+1},2\lambda
r)}\right)^{1-\frac{1}{d_2}}\left(r^{qd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{d_2}}\int_{S_{k_j,2\lambda r}}H_0(g_u)\mathrm{d}\mu\\ &\leq C\left(\frac{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}{\mu(T(k_j,k_{j+1},2\lambda
r)}\right)^{1-\frac{1}{d_2}}r^{q}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\mu(B(x_0,4\lambda r))\frac{1}{r^q}H_0\left(\frac{M}{2^j}\right)\\ &\leq C\left(\frac{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}{\mu(T(k_j,k_{j+1},2\lambda
r)}\right)^{1-\frac{1}{d_2}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\mu(B(x_0, r))H_0\left(\frac{M}{2^j}\right).
\end{align}\] By inequality (5.9) in [21], we get \[\frac{1}{2^qq}H_0\left(\frac{M}{2^j}\right)\leq
H_0\left(\frac{M}{2^{j+1}}\right).\] So, \[\frac{1}{2^qq}H_0\left(\frac{M}{2^j}\right)\mu(S_{j+1,r})\leq C\left(\frac{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}{\mu(T(k_j,k_{j+1},2\lambda
r)}\right)^{1-\frac{1}{d_2}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\mu(B(x_0, r))H_0\left(\frac{M}{2^j}\right).\] Therefore, \[\label{4Prop2}
\frac{\mu(S_{k_{j+1},r})}{\mu(B(x_0,r))}\leq C\left(\frac{\mu(T(k_j,k_{j+1},2\lambda r)}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\right)^{\frac{1}{d_2}-1} \gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}},\tag{28}\] where \(C=C(C_{\textrm{PI}}, C_D,C
_1,K,p,q)\) and \(0<d_2<1\), \(d_2=d_2(C_{\textrm{PI}},C_D,p,q)\).
If \(n\geq j+1\) then \(S_{k_{j+1},r}\) on the left-hand side of 28 can be replaced by \(S_{k_n,r}\) and the inequality remains
true. We get
\[\left(\frac{\mu(S_{k_{n},r})}{\mu(B(x_0,r))}\right)^{\frac{d_2}{1-d_2}}\leq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}\frac{\mu(T(k_j,k_{j+1},2\lambda r)}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)},\] summing up over \(j=0,1,\cdots n-1\) and Lemma 1, yields \[\begin{align} n\left(\frac{\mu(S_{k_{n},r})}{\mu(B(x_0,r))}\right)^{\frac{d_2}{1-d_2}}&\leq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}\frac{\mu(T(k_1,k_{n},2\lambda r)}{\mu\left(B\left(x_0,\frac{r}{2}\right)\right)}\\ &\leq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}\frac{\mu(B(x_0,2\lambda r))}{\mu\left(B\left(x_0,r\right)\right)}\\ &\leq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}\left(\frac{2\lambda r}{r}\right)^Q\\ &\leq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}. \end{align}\]
Therefore,
\[\label{5Prop2} \frac{\mu(S_{k_{n},r})}{\mu(B(x_0,r))}\leq\frac{C}{n^{\frac{1}{d_2}-1}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}},\tag{29}\] with \(C=C(C_{\textrm{PI}}, C_D, C_1, K,\lambda, p,q)\) and \(0<d_2<1\), \(d_2=d_2(C_{\textrm{PI}},C_D,p,q)\).
Since \(u\) is a quasiminimizer with boundary data \(w\) on \(\Omega\), the Caccioppoli inequality \[\int_{S_{k,\rho_1}}H(x,g_u)\mathrm{d}\mu\leq C\int_{B(x_0,\rho_2)}H\left(x,\frac{(u-k)_+}{\rho_2-\rho_1}\right)\mathrm{d}\mu,\] holds for all \(0<\rho_1<\rho_2\) and \(k\geq \mathop{\mathrm{ess\,sup}}_{B(x_0,\rho_2)}w\). In particular, by 24 , for any \(0<\rho_1<\rho_2\leq 8\lambda r\) and \(k\geq\mathop{\mathrm{ess\,sup}}_{B(x_0,8\lambda r)}w\), we have
\[\begin{align} \int_{B(x_0,\rho_1)}H_0(g_{(u-k)_+})\mathrm{d}\mu&\leq \int_{S_{k,\rho_1}}H(x,g_u)\mathrm{d}\mu\\ &\leq C\int_{B(x_0,\rho_2)}H\left(x,\frac{(u-k)_+}{\rho_2-\rho_1}\right)\mathrm{d}\mu\\ &\leq 2C \int_{B(x_0,\rho_2)}H_0\left(\frac{(u-k)_+}{\rho_2-\rho_1}\right)\mathrm{d}\mu. \end{align}\] Furthermore, notice that \((u-k_n)\) also satisfies the previous inequality in the ball \(B(x_0, 8\lambda r)\). Therefore, by an application of [21] with \(\Omega=B(x_0,8\lambda r)\), \(u=(u-k_n)_+\) and \(t_+=p\), there exist \(C=C(C_{\textrm{PI}}, C_D, p,q)\) such that
\[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}(u-k_n)&\leq \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}(u-k_n)_+\\
&\leq\frac{C}{\left(1-\frac{r/2}{r}\right)^{\frac{Q}{p}}}\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox
0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}(u-k_n)_+^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\
&\leq C\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox
0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}(u-k_n)_+^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\
&=C\left(\frac{1}{\mu(B(x_0,r))}\int_{S_{k_n,r}}(u-k_n)^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\
&\leq C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-k_n\right).
\end{align}\] Therefore, by 29 \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}u&\leq
k_n+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-k_n\right)\\
&=\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w-M(1-2^{-n})\right)\\ &\leq
\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}M2^{-n}\\
&\leq\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{1}{n^{\frac{1}{d_2}-1}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\right)^{\frac{1}{p}}M2^{-n},
\end{align}\] with \(C=C(C_{\textrm{PI}}, C_D,C_1, K, \lambda, p,q)\).
If \(n\geq \left(2^{pd_2}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{1-d_2}}\), then \(\left(\frac{1}{n^{\frac{1}{d_2}-1}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{d_2}}\right)^{\frac{1}{p}}\leq\frac{1}{2}\). Therefore, the last term on the right-hand side of the previous inequality is at most \(2^{-n-1}M\).
Thus, for
\[\label{condition95n} n\geq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}},\tag{30}\] we have \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}u&\leq \mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M-2^{-n}M+2^{-n-1}M\\ &= \mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w +M(1-2^{-n-1}). \end{align}\]
Finally, we get \[\label{conclusion1}
M\left(\frac{r}{2}, r\right)\leq (1-2^{-n-1})M,\tag{31}\] for \(n\) satisfying 30 . End Case 1.
Case 2: Next we consider the case which is complementary to 23 , that is, \[\label{3C}
a_0=\inf_{x\in B(x_0, 8\lambda r)}a(x)\leq C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q}.\tag{32}\] Notice that, for every \(x\in B(x_0, 8\lambda r)\) and \(y \in B(x_0,
r)\), with \(y\neq x\), we have \[\begin{align}
\label{notice} a(y)-a(x)&\leq \vert a(x)-a(y)\vert=\frac{\vert a(x)-a(y)\vert}{\delta_{\mu}(x,y)^{\alpha}}\delta_{\mu}(x,y)^{\alpha}\leq [a]_{\alpha}\delta_{\mu}(x,y)^{\alpha}.
\end{align}\tag{33}\] Note that, for every \(x\in B(x_0, 8\lambda r)\) and \(y \in B(x_0, r)\), with \(y\neq x\), we have \[\begin{align}
\delta_{\mu}(x,y)
&=\left(\mu(B(x,d(x,y)))+\mu(B(y,d(x,y)))\right)^{1/Q}\\
&\leq \left(\mu(B(x,9\lambda r))+\mu(B(y,9\lambda r))\right)^{1/Q}\\
&\leq \left(2\mu (B(x_0,10\lambda r))\right)^{1/Q}
\leq C\mu(B(x_0, 8\lambda r))^{1/Q},
\end{align}\] where \(C=C(C_D)\). By 33 , we get \[a(y)\leq a(x)+C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q},\] where \(C=C(C_D,\alpha)\). By taking infimum over all \(x\in 8\lambda B(x_0, r)\), we obtain \[\begin{align} a(y) &\leq \inf_{x\in B(x_0, 8\lambda
r)}a(x)+C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q}\\ &\leq 2[a]_{\alpha}C(\mu(B(x_0, 8\lambda r)))^{\alpha/Q}+C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q}\\\ &\leq C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q},
\end{align}\] where \(C=C(C_D,\alpha)\). By taking supremum over \(y\in B(x_0, r)\), we conclude that \[\label{eq11POINTWISE}
\sup_{y\in B(x_0, r)}a(y)
\leq C[a]_{\alpha}\mu(B(x_0, 8\lambda r))^{\alpha/Q}.\tag{34}\] Since \(1<s<pd_2<p\), as a consequence of Theorem 1 and Remark 2, \(X\) supports a weak \((pd_2,pd_2)\)-Poincaré inequality. Therefore, by the weak \((pd_2,pd_2)\)-Poincaré inequality, Proposition 1, the upper \(Q\)-Ahlfors inequality 5 , \(r\leq 1\), Remark 3, the doubling property of \(\mu\) and Hölder inequality,
we get \[\begin{align} \int_{B(x_0,r)}v_j^{pd_2}\mathrm{d}\mu&\leq \frac{C\mu(B(x_0,r))}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &\leq
C\mu(B(x_0,r))\left(1+\frac{1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\right)\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &\leq C\mu(B(x_0,r))\left(1+\frac{1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega,
B(x_0,r))}\right)^{\frac{q}{p}}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &=C\mu(B(x_0,r))\left(\frac{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))+1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega,
B(x_0,r))}\right)^{\frac{q}{p}}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &\leq
C\mu(B(x_0,r))\left(\frac{\left(\frac{C_D\mu(B(x_0,\frac{r}{2}))}{\left(\frac{r}{2}\right)^{pd_2}}\right)+1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\right)^{\frac{q}{p}}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\
&\leq C\mu(B(x_0,r))\left(\frac{C_DC_1\left(\frac{r}{2}\right)^{Q-pd_2}+1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\right)^{\frac{q}{p}}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &\leq
C\mu(B(x_0,r))\left(\frac{C+1}{\textrm{cap}_{pd_2}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\right)^{\frac{q}{p}}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &=\frac{
C\mu(B(x_0,r))}{\textrm{cap}_{pd_2}^{q/p}(B(x_0,\frac{r}{2})\setminus\Omega, B(x_0,r))}\int_{B(x_0,2\lambda r)}g_{v_j}^{pd_2}\mathrm{d}\mu\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})r^{qd_2}\int_{S_{k_j,2\lambda r}}g_u^{pd_2}\chi_{T(k_j,k_{j+1},2\lambda
r)}\mathrm{d}\mu\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})r^{pd_2}\left(\int_{S_{k_j,2\lambda r}}g_u^{p}\mathrm{d}\mu\right)^{d_2}\mu(T(k_j,k_{j+1},2\lambda r))^{1-d_2},
\end{align}\] with \(C\) depending on \(C_{\textrm{PI}}\), \(C_D\) and \(C_1\), therefore
\[\label{eq12POINTWISE} \int_{B(x_0,r)}v_j^{pd_2}\mathrm{d}\mu\leq C\gamma_{p,q}(pd_2,\frac{r}{2})r^{pd_2}\left(\int_{S_{k_j,2\lambda
r}}g_u^{p}\mathrm{d}\mu\right)^{d_2}\mu(T(k_j,k_{j+1},2\lambda r))^{1-d_2}.\tag{35}\] For the measure of the set \(S_{k_{j+1},r}\) we have the following estimate \[\label{eq13POINTWISE} \int_{B(x_0,r)}v_j^{pd_2}\mathrm{d}\mu\geq(k_{j+1}-k_j)^{pd_2}\mu(S_{k_{j+1}, r})=\frac{M^{pd_2}}{2^{pd_2(j+1)}}\mu(S_{k_{j+1},r}) .\tag{36}\] Since \(u\) is a
quasiminimizer with boundary data \(w\) in \(\Omega\), by 34 , Lemma 4, Hölder inequality and the doubling property, we have for \(R=8\lambda r\), \(s=2\lambda r\), \(t=\lambda r\) \[\begin{align} \int_{S_{k_j,\lambda r}}g_u^p\mathrm{d}\mu&=\int_{B(x_0,\lambda r)}g_{(u-k_j)_+}^p\mathrm{d}\mu\leq C\left(\left(\frac{8\lambda r}{\lambda
r}\right)\int_{B(x_0,2\lambda r)}\left\vert\frac{(u-k_j)_+}{8\lambda r}\right\vert^p\mathrm{d}\mu\right)\\ &\leq \frac{C}{r^p}\int_{B(x_0,2\lambda r)}(u-k_j)_+^p\mathrm{d}\mu\leq \frac{C}{r^p}\int_{B(x_0,2\lambda
r)}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,2\lambda r)}u-k_j\right)_+^p\mathrm{d}\mu\\ &\leq C\left(\frac{M}{r2^j}\right)^p\mu(B(x_0,2\lambda r)).
\end{align}\] Therefore, \[\label{eq14POINTWISE}
\int_{S_{k_j,\lambda r}}g_u^p\mathrm{d}\mu\leq C\left(\frac{M}{r2^j}\right)^p\mu(B(x_0,2\lambda r)).\tag{37}\] By 36 , 35 and 37 , we obtain \[\begin{align} \mu(S_{k_{j+1},r})\frac{M^{pd_2}}{2^{pd_2(j+1)}}&\leq \int_{B(x_0,r)}v_j^{pd_2}\mathrm{d}\mu\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})r^{pd_2}\left(\int_{S_{k_j,2\lambda
r}}g_u^{pd_2}\mathrm{d}\mu\right)^{d_2}\mu(T(k_j,k_{j+1},2\lambda r))^{1-d_2}\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})r^{pd_2}\frac{CM^{pd_2}}{r^{pd_2}2^{jpd_2}}\mu(B(x_0,2\lambda r))^{d_2}\mu(T(k_j,k_{j+1},2\lambda r))^{1-d_2}\\ &=
C\gamma_{p,q}(pd_2,\frac{r}{2})\frac{CM^{pd_2}}{2^{jpd_2}}\mu(B(x_0,2\lambda r))^{d_2}\mu(T(k_j,k_{j+1},2\lambda r))^{1-d_2}.
\end{align}\] So, \[\label{eq15POINTWISE} \frac{\mu(S_{k_{j+1},r})}{\mu(B(x_0,r))}\leq C\gamma_{p,q}(pd_2,\frac{r}{2})\left(\frac{\mu(T(k_j,k_{j+1},2\lambda
r))}{\mu(B(x_0,r))}\right)^{1-d_2}.\tag{38}\] As before, if \(n\geq j+1\), then 38 holds true when the set \(S_{k_{j+1},r}\) on the left part of
the inequality is replaced by \(S_{k_n,r}\). Thus, by Lemma 1
\[\begin{align} n\left( \frac{\mu(S_{k_{n},r})}{\mu(B(x_0,r))}\right)^{\frac{1}{1-d_2}}&\leq C\gamma_{p,q}(pd_2,\frac{r}{2})^{\frac{1}{1-d_2}}\left(\frac{\mu(T(k_j,k_{j+1},2\lambda r))}{\mu(B(x_0,r))}\right)\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})^{\frac{1}{1-d_2}}\left(\frac{\mu(B(x_0,2\lambda r))}{\mu(B(x_0,r))}\right)\\ &\leq C\gamma_{p,q}(pd_2,\frac{r}{2})^{\frac{1}{1-d_2}}\left(\frac{2\lambda r}{r}\right)^{Q}. \end{align}\] Meaning, \[\label{eq16POINTWISE} \frac{\mu(S_{k_{n},r})}{\mu(B(x_0,r))}\leq\frac{C}{n^{1-d_2}}\gamma_{p,q}(pd_2,\frac{r}{2}).\tag{39}\] Again, since \(u\) is a quasiminimizer with boundary data \(w\) in \(\Omega\), then by Lemma 2, in particular, \(u\) satisfies the Caccioppoli inequality in the ball \(B(x_0,8\lambda r)\). Since 34 holds, by an application of [21] with \(\Omega=B(x_0,8\lambda r)\), \(u=(u-k_n)_+\) and \(t_+=p\), there exists \(C=C(\mathop{\mathrm{data}}, C_1, \Vert u\Vert_{L^\infty(B(x_0,r))})\) such that
\[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}(u-k_n)&\leq \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}(u-k_n)_+\\ &\leq\frac{C}{\left(1-\frac{r/2}{r}\right)^{\frac{Q}{p}}}\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}(u-k_n)_+^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\ &\leq C\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}} \!\int_{B(x_0,r)}(u-k_n)_+^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\ &=C\left(\frac{1}{\mu(B(x_0,r))}\int_{S_{k_n,r}}(u-k_n)^p\mathrm{d}\mu\right)^{\frac{1}{p}}\\ &\leq C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-k_n\right). \end{align}\]
Therefore, by 39 \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}u&\leq k_n+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-k_n\right)\\ &=\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}\left(\mathop{\mathrm{ess\,sup}}_{B(x_0,r)}u-\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w-M(1-2^{-n})\right)\\ &\leq \mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{\mu(S_{k_n,r})}{\mu(B(x_0,r))}\right)^{\frac{1}{p}}M2^{-n}\\ &\leq\mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M(1-2^{-n})+C\left(\frac{1}{n^{1-d_2}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{p}}M2^{-n}, \end{align}\]
If \(n\geq \left(2^p\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{1-d_2}}\), then \(\left(\frac{1}{n^{1-d_2}}\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)\right)^{\frac{1}{p}}\leq\frac{1}{2}\). Therefore, the last term on the right-hand side of the previous inequality is at most \(2^{-n-1}M\).
Thus, for
\[\label{condition95n2} n\geq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}},\tag{40}\] we have \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B\left(x_0,\frac{r}{2}\right)}u&\leq \mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w+M-2^{-n}M+2^{-n-1}M\\ &= \mathop{\mathrm{ess\,sup}}_{B(x_0,r_0)}w +M(1-2^{-n-1}). \end{align}\]
Finally, we get \[\label{conclusion2}
M\left(\frac{r}{2}, r\right)\leq (1-2^{-n-1})M,\tag{41}\] for \(n\) satisfying 40 . End Case 2.
Notice that equations 30 and 40 are the same, and we can uniform conditions 31 and 41 .
Therefore, combining both cases, we obtain that by choosing the smallest integer such that \(n\geq C\gamma_{p,q}\left(pd_2,\frac{r}{2}\right)^{\frac{1}{1-d_2}}\), we finish the proof. ◻
This last section contains the main result of the present work, Theorem 6. It is worth mentioning that after obtaining the pointwise estimate on a boundary point, see Proposition 3, the arguments used in this section are inspired by techniques from the literature, particularly [53], [54], but adapted to our setting. For completeness, we provide full proofs below. We start with the following Theorem, a pointwise estimate for quasiminima near a boundary point, see [53], [54], [61].
Theorem 5. Let \(w\in N^{1,1}(X)\) with \(H(x,g_w)\in L^1(X)\). Let \(u \in N^{1,1}(X)\), with \(H(x,g_u)\in L^1(X)\) a quasiminimizer on \(\Omega\) with boundary values \(w\). Then there exist \(C_{0},C_{1}>0\) such that \[M(\rho,r_{0})\leq C_{1}M(r_{0},r_{0})\exp\left(-\frac{1}{4}\int_{\rho}^{r_{0}}\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{\, d r}{r}\right).\]
Proof. Let \(M(r)=M(r,r_{0})\), where \(r_{0}>0\) is fixed. It is not restrictive to suppose that \(0< M(r_{0}) < +\infty\). We consider \(C\) and \(n(r)\) as in Proposition 3, \(C_{0}=C\log 2\) and \[\zeta(r)=\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)=2^{-n(2r)}.\] We partition \((0,r_{0})=I_{1}\cup I_{2}\) with \(I_{1}\cap I_{2}=\emptyset\), where \[I_{m}=\bigcup_{i=1}^{+\infty}\left[(8\lambda)^{m-2i-1}r_{0}, (8\lambda)^{m-2i}r_{0}\right), \quad for m=1,2.\] It follows, \[\label{eq1theo24611} \int_{\rho}^{r_{0}}\zeta(r)\frac{dr}{r}\leq 2\int_{(\rho,r_{0})\cap I_{m}}\zeta(r)\frac{dr}{r},\tag{42}\] for \(m=1\) or \(m=2\). For each \(i \in \mathbb{N}\), we define \(r_{i}\in \left[(8\lambda)^{m-2i-1}r_{0}, (8\lambda)^{m-2i}r_{0}\right)\) satisfying \[\zeta(r_{i})\geq\frac{1}{(8\lambda)^{m-2i-1}r_{0}}\int_{(8\lambda)^{m-2i-1}r_{0}}^{(8\lambda)^{m-2i}r_{0}}\zeta(r)dr\geq \int_{(8\lambda)^{m-2i-1}r_{0}}^{(8\lambda)^{m-2i}r_{0}}\zeta(r)\frac{dr}{r}.\] Since \(\zeta(r)\leq 1\) for all \(r\), we obtain \[\label{eq2theo24611} \int_{(\rho,r_{0})\cap I_{m}}\zeta(r)\frac{dr}{r}\leq\sum_{\rho\leq r_{i}<\frac{r_{0}}{8\lambda}}\zeta(r_{i})+C.\tag{43}\] By Proposition 3, we get \[\begin{align} M((8\lambda)^{m-2i-1}r_{0})&\leq M(r_{i})\leq M(8\lambda r_{i})(1-2^{-n(2r_{i})-1})\\ &\leq M((8\lambda)^{m-2i+1}r_{0})\left(1-\frac{\zeta(r_{i})}{2}\right), \quad for i\geq 1. \end{align}\] By an iteration process and the trivial inequality \(1-t\leq \exp(-t)\), we deduce \[M(\rho)\leq M(r_{0})\exp\left(-\frac{1}{2}\sum_{\rho\leq r_{i}<\frac{r_{0}}{8\lambda}}\zeta(r_{i})\right), \quad for 0<\rho<r_{0}.\] Furthermore, \[-\frac{1}{2}\sum_{\rho\leq r_{i}<\frac{r_{0}}{8\lambda}}\zeta(r_{i})\leq C-\frac{1}{2}\int_{(\rho,r_{0})\cap I_{m}}\zeta(r)\frac{dr}{r}\leq C-\frac{1}{4}\int_{\rho}^{r_{0}}\zeta(r)\frac{dr}{r}.\] Therefore, \[\exp\left(-\frac{1}{2}\sum_{\rho\leq r_{i}<\frac{r_{0}}{8\lambda}}\zeta(r_{i})\right)\leq C_{1}\exp\left(-\frac{1}{4}\int_{\rho}^{r_{0}}\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{dr}{r}\right).\] ◻
Finally, Theorem 5 implies the sufficient condition for Hölder continuity of quasiminima of double-phase functionals at a boundary point.
Theorem 6. Let \(w\in N^{1,1}(X)\) with \(H(x,g_w)\in L^1(X)\). Let \(u \in N^{1,1}(X)\), with \(H(x,g_u)\in L^1(X)\) a quasiminimizer on \(\Omega\) with boundary values \(w\). If \(w\) is a Hölder continuous function at \(x_{0}\in\partial\Omega\), then there exists \(C_0>0\) such that \[\liminf_{\rho\rightarrow 0}\frac{1}{\vert \log \rho\vert}\int_{\rho}^{1}\exp\left(-C_0 \gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{dr}{r}>0.\] Thus \(u\) is Hölder continuous at \(x_{0}\).
Proof. We notice that \(-u\) is a quasiminimizer (see Remark 5). Furthermore, by hypothesis \(-u-(-w) \in N^{1,1}_{0}(\Omega)\), so it is enough to work with the positive part \((u(x)-w(x_{0}))_{+}\). Without loss of generality, we can make \(w(x_{0})=0\). Being \(w\) continuous at \(x_{0}\), Corollary 2 yields to \[\label{eq1theo24612} M(r_{0},r_{0})\leq M=\mathop{\mathrm{ess\,sup}}_{B(x_{0},R)}u_{+}<+\infty,\tag{44}\] for some \(R>0\) and all \(0<r_{0}<R\).
For \(0<\rho<r_{0}\), using Theorem 5 we obtain \[\begin{align} \label{eq2theo24612} \nonumber \mathop{\mathrm{ess\,sup}}_{B(x_{0},\rho)}u_{+}&\leq \mathop{\mathrm{ess\,sup}}_{B(x_{0},r_{0})}w_{+}+M(\rho,r_{0})\\ &\leq \mathop{\mathrm{ess\,sup}}_{B(x_{0},r_{0})}w_{+}+C_{1}M\exp\left(-\frac{1}{4}\int_{\rho}^{r_{0}}\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{dr}{r}\right). \end{align}\tag{45}\] So there are \(C, h,k>0\) satisfying \(\mathop{\mathrm{ess\,sup}}_{B(x_{0},r_{0})}w_{+}\leq Cr_{0}^{h}\) and, for all sufficiently small \(\rho\) and \(r_{0}\), \[\int_{\rho}^{1}\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{dr}{r}\geq k\vert\log\rho\vert.\]
We observe that \[\int_{r_{0}}^{1}\exp\left(-C_{0}\gamma_{p,q}(pd_2,r)^{\frac{1}{1-d_2}}\right)\frac{dr}{r}\leq \int_{r_{0}}^{1}\frac{dr}{r}=\vert\log r_{0}\vert,\]for all \(0<r_{0}<1\). For sufficiently small \(\rho\) and \(r_{0}\), using inequality (45 ), we derive \[\mathop{\mathrm{ess\,sup}}_{B(x_{0},\rho)}u_{+}\leq Cr_{0}^{h}+C_{1}M\rho^{\frac{k}{4}}r_{0}^{-\frac{1}{4}}.\]
The proof of the Hölder continuity of \(u\) at \(x_0\) is completed by choosing \(r_{0}=\rho^{k '}\) with \(0<k '<k\). ◻
A. Nastasi is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and was partly supported by GNAMPA-INdAM Project Regolarità per
problemi ellittici e parabolici con crescite non standard, CUP E53C22001930001.
This research was partly conducted while C. Pacchiano Camacho was at the Okinawa Institute of Science and Technology (OIST) through the Theoretical Sciences Visiting Program (TSVP). C. Pacchiano Camacho was supported by a grant from Simons Foundation
International SFI-MPS-T-Institutes-00011977 JS.