# The boundary Harnack inequality for variable exponent \(p\)-Laplacian, Carleson estimates, barrier functions and \({p(\cdot )}\)-harmonic measures

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## Abstract

We investigate various boundary decay estimates for \({p(\cdot )}\)-harmonic functions. For domains in \({\mathbb {R}}^n, n\ge 2\) satisfying the ball condition (\(C^{1,1}\)-domains), we show the boundary Harnack inequality for \({p(\cdot )}\)-harmonic functions under the assumption that the variable exponent \(p\) is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for \({p(\cdot )}\)-harmonic functions in NTA domains in \({\mathbb {R}}^n\) and provide lower and upper growth estimates and a doubling property for a \({p(\cdot )}\)-harmonic measure.

### Keywords

Ball condition Boundary Harnack inequality Harmonic measure NTA domain Nonstandard growth equation p-harmonic### Mathematics Subject Classification

Primary 31B52 Secondary 35J92 35B09 31B25## 1 Introduction

The studies of boundary Harnack inequalities for solutions of differential equations have a long history. In the setting of harmonic functions on Lipschitz domains, such a result was first proposed by Kemper [41] and later studied by Ancona [11], Dahlberg [23] and Wu [60]. Subsequently, Kemper’s result was extended by Caffarelli et al. [21] to a class of elliptic equations, by Jerison and Kenig [40] to the setting of nontangentially accessible (NTA) domains, Bañuelos et al. [14] and Bass and Burdzy [15] studied the case of Hölder domains while Aikawa [6] the case of uniform domains. The extension of these results to the more general setting of \(p\)-harmonic operators turned out to be difficult, largely due to the nonlinearity of \(p\)-harmonic functions for \(p\not =2\). However, recently, there has been a substantial progress in studies of boundary Harnack inequalities for nonlinear Laplacians: Aikawa et al. [7] studied the case of \(p\)-harmonic functions in \(C^{1,1}\)-domains, while in the same time, Lewis and Nyström [45, 47, 48] began to develop a theory applicable in more general geometries such as Lipschitz and Reifenberg-flat domains. Lewis–Nyström results have been partially generalized to operators with variable coefficients, Avelin et al. [12], Avelin and Nyström [13], and to \(p\)-harmonic functions in the Heisenberg group, Nyström [55]. Moreover, in [52], the second author proved a boundary Harnack inequality for \(p\)-harmonic functions with \(n<p\le \infty \) vanishing on a \(m\)-dimensional hyperplane in \({\mathbb {R}}^n\) for \(0 \le m \le n-1\). We also refer to Bhattacharya [18] and Lundström and Nyström [53] for the case \(p = \infty \), where the latter investigated \(A\)-harmonic and Aronsson-type equations in planar uniform domains. Concerning the applications of boundary Harnack inequalities, we mention free boundary problems and studies of the Martin boundary.

Apart from interesting theoretical considerations, such equations arise in the applied sciences, for instance in fluid dynamics, see, e.g., Diening and Růžička [25], in the study of image processing, see, for example, Chen et al. [22] and electro-rheological fluids, see, e.g., Acerbi and Mingione [1, 2]; we also refer to Harjulehto et al. [35] for a recent survey and further references. In spite of the symbolic similarity to the constant exponent \(p\)-harmonic equation, various unexpected phenomena may occur when the exponent is a function, for instance the minimum of the \({p(\cdot )}\)-Dirichlet energy may not exist even in the one-dimensional case for smooth functions \(p\); also smooth functions need not be dense in the corresponding variable exponent Sobolev spaces. Although Eq. (1.1) is the Euler-Lagrange equation of the \({p(\cdot )}\)-Dirichlet energy and thus is natural to study, it has many disadvantages comparing to the \(p=\text {const}\) case. For instance, solutions of (1.1) are, in general, not scalable, also the Harnack inequality is nonhomogeneous with constant depending on solution. In a consequence, the analysis of nonstandard growth equation is often difficult and leads to technical and nontrivial estimates (nevertheless, see Adamowicz and Hästö [4, 5] for a variant of Eq. (1.1) that overcomes some of the aforementioned difficulties, the so-called strong \({p(\cdot )}\)-harmonic equation).

The main goal of this paper is to show the boundary Harnack inequality for \({p(\cdot )}\)-harmonic functions on domains satisfying the ball condition (see Theorem 5.4 below). Let us briefly describe the main ingredients leading to this result, as it requires number of auxiliary lemmas and observations which are interesting per se and can be applied in other studies of variable exponent PDEs.

In Sect. 3, we study oscillations of \({p(\cdot )}\)-harmonic functions close to the boundary of a domain and prove, among other results, variable exponent Carleson estimates on NTA domains, cf. Theorem 3.7. Similar estimates play an important role, for instance in studies of the Laplace operator, in particular in relations between the topological boundary and the Martin boundary of the given domain, also in the \(p\)-harmonic analysis (see presentation in Sect. 3 for further details and references). The main tools used in the proof of Theorem 3.7 are Hölder continuity up to the boundary, Harnack’s inequality and an argument by Caffarelli et al. [21] which, in our situation, relies on various geometric concepts such as quasihyperbolic geodesics and related chaining arguments, also on characterizations of uniform and NTA domains.

Section 4 is devoted to introducing two types of barrier functions, called Wolanski-type and Bauman-type barrier functions, respectively. In the analysis of PDEs, barrier functions appear, for example, in comparison arguments and in establishing growth conditions for functions, see, e.g., Aikawa et al. [7], Lundström [52], Lundström and Vasilis [54] for the setting of \(p\)-harmonic functions. Furthermore, barriers can be applied in the solvability of the Dirichlet problem, especially in studies of regular points, see, e.g., Chapter 6 in Heinonen et al. [38] and Chapter 11 in Björn and Björn [19]. We would like to mention that our results on barriers enhance the existing results in variable exponent setting, see Remark 4.2.

Finally, in Sect. 6, we define and study lower and upper estimates for a \({p(\cdot )}\)-harmonic measure. We also prove a weak doubling property for such measures. In the constant exponent setting, similar results were obtained by Eremenko and Lewis [26], Kilpeläinen and Zhong [43] and Bennewitz and Lewis [17]. For \(p = \text {const}\), \(p\)-harmonic measures were employed to prove boundary Harnack inequalities, see, e.g., [17], Lewis and Nyström [46] and Lundström and Nyström [53]. The \(p\)-harmonic measure, defined as in the aforementioned papers, as well as boundary Harnack inequalities, have played a significant role when studying free boundary problems, see, e.g., Lewis and Nyström [48].

## 2 Preliminaries

We let \( \bar{\Omega }\) and \(\partial \Omega \) denote, respectively, the closure and the boundary of the set \(\Omega \subset {\mathbb {R}}^{n}\), for \(n\ge 2\). We define \(d(y, \Omega )\) to equal the Euclidean distance from \(y \in {\mathbb {R}}^{n} \) to \(\Omega \), while \( \langle \cdot , \cdot \rangle \) denotes the standard inner product on \( {\mathbb {R}}^{2} \) and \( | x | = \langle x, x \rangle ^{1/2} \) is the Euclidean norm of \(x\). Furthermore, by \(B(x, r)=\{y \in {\mathbb {R}}^{n} : |x - y|<r\}\), we denote a ball centered at point \(x\) with radius \(r>0\), and we let \(dx\) denote the \(n\)-dimensional Lebesgue measure on \({\mathbb {R}}^{n}\). If \(\Omega \subset {\mathbb {R}}^n \) is open and \(1 \le q < \infty \), then by \(W^{1 ,q} (\Omega )\), \(W^{1 ,q}_0 (\Omega )\) we denote the standard Sobolev space and the Sobolev space of functions with zero boundary values, respectively. Moreover, let \(\Delta (w, r) = B(w,r)\cap \partial \Omega \). By \(f_A\), we denote the integral average of \(f\) over a set \(A\).

For background on variable exponent function spaces, we refer to the monograph by Diening et al. [24].

*variable exponent*and we denote

*-Hölder continuous*if there is constant \(L>0\) such that

In this paper, we study only log-Hölder continuous or Lipschitz continuous variable exponents. Both types of exponents can be extended to the whole \({\mathbb {R}}^n\) with their constants unchanged, see [24, Proposition 4.1.7] and McShane-type extension result in Heinonen [37, Theorem 6.2], respectively. Therefore, without loss of generality, we assume below that variable exponents are defined in the whole \({\mathbb {R}}^n\).

*(semi)modular*on the set of measurable functions by setting

*variable exponent Lebesgue space*\(L^{p(\cdot )}(\Omega )\) consists of all measurable functions \(u:\Omega \rightarrow {\mathbb {R}}\) for which the modular \(\varrho _{L^{p(\cdot )}(\Omega )}(u/\mu )\) is finite for some \(\mu > 0\). The Luxemburg norm on this space is defined as

*unit ball property*:

*conjugate exponent*, \(1/p(x) +1/p'(x)\equiv 1\).

*variable exponent Sobolev space*\(W^{1,p(\cdot )}(\Omega )\) consists of functions \(u\in L^{p(\cdot )}(\Omega )\) whose distributional gradient \(\nabla u\) belongs to \(L^{p(\cdot )}(\Omega )\). The variable exponent Sobolev space \(W^{1,p(\cdot )}(\Omega )\) is a Banach space with the norm

*the Sobolev space with zero boundary values*, \(W_0^{1,p(\cdot )}(\Omega )\), as the closure of \(C_0^\infty (\Omega )\) in \(W^{1,{p(\cdot )}}(\Omega )\).

**Definition 2.1**

*Sobolev*\({p(\cdot )}\)

*-capacity*of a set \(\Omega \subset {\mathbb {R}}^n\) is defined as

The properties of \({p(\cdot )}\)-capacity are similar to those in the constant case, see Theorem 10.1.2 in [24]. In particular, \(C_{{p(\cdot )}}\) is an outer measure, see Theorem 10.1.1 in [24].

Another type of capacity used in the paper is the so-called *relative* \({p(\cdot )}\) *-capacity* which appears for instance in the context of uniform \({p(\cdot )}\)-fatness (see next section and Chapter 10.2 in [24] for more details).

**Definition 2.2**

*relative*\({p(\cdot )}\)

*-capacity*of a compact set \(K\subset \Omega \) is a number defined by

**Definition 2.3**

In what follows, we will exchangeably be using terms (sub)solution and \({p(\cdot )}\)-(sub)solution. Similarly, we say that \(u\) is a *supersolution* (\({p(\cdot )}\) *-supersolution)* if \(-u\) is a subsolution. A function which is both a subsolution and a supersolution is called a (weak) solution to the \({p(\cdot )}\)-harmonic equation. A continuous weak solution is called a \({p(\cdot )}\) *-harmonic function*.

Among properties of \({p(\cdot )}\)-harmonic functions, let us mention that they are locally \(C^{1,\alpha }\), see, e.g., Acerbi and Mingione [1] or Fan [27, Theorem 1.1]. Another tool, crucial from our point of view, is the comparison principle.

**Lemma 2.4**

(cf. Lemma 3.5 in Harjulehto et al. [32]) Let \(u\) be a supersolution and \(v\) a subsolution such that \(u>v\) on \(\partial \Omega \) in the Sobolev sense. Then, \(u>v\) a.e. in \(\Omega \).

By the standard reasoning, the comparison principle implies the following maximum principle: *If* \(u\in W^{1,{p(\cdot )}}(\Omega )\cap C({\overline{\Omega }})\) *is a* \({p(\cdot )}\) *-subsolution in* \(\Omega \), *then the maximum of* \(u\) *is attained at the boundary of* \(\Omega \). For further discussion on comparison principles in the variable exponent setting, we refer, e.g., to Section 3 in Adamowicz et al. [3].

We close our discussion of basic definitions and results with a presentation of the geometric concepts used in the paper.

**Definition 2.5**

*uniform domain*if there exists a constant \(M_\Omega \ge 1\), called a

*uniform constant*, such that whenever \(x,y\in \Omega \) there is a rectifiable curve \(\gamma :[0,l(\gamma )]\rightarrow \Omega \), parameterized by arc length, connecting \(x\) to \(y\) and satisfying the following two conditions:

**Definition 2.6**

A uniform domain \(\Omega \subset {\mathbb {R}}^n\) with constant \(M_{\Omega }\) is called a *nontangentially accessible (NTA) domain* if \(\Omega \) and its complement \({\mathbb {R}}^n\setminus \Omega \) satisfy, additionally, the so-called *corkscrew condition*:

We note that in fact the (interior) corkscrew condition is implied by a uniform domain, see Bennewitz and Lewis [17] and Gehring [30]. Among examples of NTA domains, we mention quasidisks, bounded Lipschitz domains and domains with fractal boundary such as the von Koch snowflake. A domain with the internal power-type cusp is an example of a uniform domain which fails to be NTA domain. Uniform domains are necessarily John domains, the latter one enclosing, e.g., bounded domains satisfying the interior cone condition. See Näkki and Väisälä [56] and Väisälä [58] for further information on uniform and John domains.

*quasihyperbolic distance*\(k_{\Omega }\) between points \(x,y\) in a domain \(\Omega \subsetneq {\mathbb {R}}^n\) is defined as follows

*quasihyperbolic geodesic*, i.e., a curve for which the above infimum can be achieved. See Bonk et al. [20, Section 2] and Gehring and Osgood [31] for more information.

We end this section by recalling the following geometric definition.

**Definition 2.7**

A domain \(\Omega \subset {\mathbb {R}}^n\) is said to satisfy the *interior ball condition* with radius \(r_i > 0\) if for every \(w \in \partial \Omega \) there exists \(\eta ^i \in \Omega \) such that \(B(\eta ^i, r_i) \subset \Omega \) and \(\partial B(\eta ^i, r_i) \cap \partial \Omega = \{w\}\). Similarly, a domain \(\Omega \subset {\mathbb {R}}^n\) is said to satisfy the *exterior ball condition* with radius \(r_e > 0\) if for every \(w \in \partial \Omega \) there exists \(\eta ^e \in {\mathbb {R}}^n \setminus \Omega \) such that \(B(\eta ^e, r_e) \subset {\mathbb {R}}^n \setminus \Omega \) and \(\partial B(\eta ^e, r_e) \cap \partial \Omega = \{w\}\). A domain \(\Omega \subset {\mathbb {R}}^n\) is said to satisfy the *ball condition* with radius \(r_b\) if it satisfies both the interior ball condition and the exterior ball conditions with radius \(r_b\).

It is well known that \(\Omega \subset {\mathbb {R}}^n\) satisfies the ball condition if and only if \(\Omega \) is a \(C^{1,1}\)-domain. See Aikawa et al. [7, Lemma 2.2] for a proof. We also note that if \(\Omega \subset {\mathbb {R}}^n\) satisfies the ball condition then \(\Omega \) is a NTA domain and hence also a uniform domain.

Throughout the paper, unless otherwise stated, \(c\) and \(C\) will denote constants whose values may vary at each occurrence. If \(c\) depends on the parameters \(a_1, \ldots , a_n\), we sometimes write \(c(a_1, \ldots , a_n)\). When constants depend on the variable exponent \({p(\cdot )}\), we write “depending on \(p^-, p^+, c_\mathrm{log}\)” in place of “depending on \(p\)” whenever dependence on \(p\) easily reduces to \(p^-, p^+, c_\mathrm{log}\).

## 3 Oscillation and Carleson estimates for \({p(\cdot )}\)-harmonic functions

This section is devoted to discussing some important auxiliary results used throughout the rest of the paper. Namely, in Lemmas 3.4, 3.5 and 3.6, we study oscillations of \({p(\cdot )}\)-harmonic functions over the balls intersecting the boundary of the underlying domain. We also employ geometric concepts such as NTA and uniform domains, quasihyperbolic geodesics and distance together with the Harnack inequality to obtain a supremum estimate for a \({p(\cdot )}\)-harmonic function over a chain of balls. Such estimates, discussed in \(p=\text {const}\) setting for instance in Aikawa and Shanmugalingam [8] or Holopainen et al. [39], require extra attention for variable exponent \({p(\cdot )}\) as now constant in the Harnack inequality depends on a \({p(\cdot )}\)-harmonic function and the inequality is nonhomogeneous. In Theorem 3.7, we show the main result of this section, namely the variable exponent Carleson estimate. Such estimates play a crucial role in studies of positive \(p\)-harmonic functions, see, e.g., Aikawa and Shanmugalingam [8], also Garofalo [29] for an application of Carleson estimates for a class of parabolic equations. According to our best knowledge, Carleson estimates in the setting of equations with nonstandard growth have not been known so far in the literature. We apply Lemma 3.7 in the studies of \({p(\cdot )}\)-harmonic measures in Sect. 5. Moreover, the geometry of the underlying domain turns out to be important in our investigations, in particular properties of NTA domains and uniform \({p(\cdot )}\)-fatness of the complement come into play.

We begin with recalling the Harnack estimate for \({p(\cdot )}\)-harmonic functions.

**Lemma 3.1**

*Remark 3.2*

The variable Harnack inequality in the above form was proved by Alkhutov [9] (see also Alkhutov and Krasheninnikova [10]) and subsequently improved to embrace the case of unbounded solutions by Harjulehto et al. [36, Theorem 3.9]. There, \(c_H\) depends only on \(n, p\) and the \(L^{q's}(B(w,4r))\)-norm of \(u\) for \(1<q<\frac{n}{n-1}\) and \(s > p^+_{B(w,4r)} - p^-_{B(w,4r)}\).

- 1.
\(B_i\cap B_{i+1}\not = \emptyset \) for each \(i\),

- 2.
\(2B_i\subset B(w, 4r)\cap \Omega \),

- 3.
\(N\le 3k_{\Omega }(x,y)\).

In some results of this section, we appeal to notion of *uniform* \({p(\cdot )}\) *-fatness*. For the sake of completeness of the presentation, we recall necessary definitions, cf. Lukkari [50, Sections 3 and 4] and Holopainen et al. [39, Section 3].

**Definition 3.3**

*uniformly*\({p(\cdot )}\)

*-fat complement*, if there exist a radius \(r_0>0\) and a constant \(c_0>0\) such that

The next lemma provides an oscillation estimate. Similar result was proven by Lukkari in [50, Proposition 4.2]. However, here, we adapt the discussion from [50] to our case; for instance, we do not require the boundary data to be Hölder continuous.

**Lemma 3.4**

*Proof*

Assume now that \(p_0 \le n\). To prove the lemma in this case, we will follow the steps and notation of the proof of Proposition 4.2 in Lukkari [50]. In the applications of Lemma 3.4, we will need to understand the exact dependence on constants, and therefore, we repeat parts of the proof from [50].

To prove Hölder continuity up to the boundary, we will also use the following oscillation estimate which follows from Theorem 4.2, Lemma 2.8 in Fan and Zhao [28] and Lemma 4.8 in Ladyzhenskaya and Ural’tseva [44]. The careful scrutiny of the presentation in [28] reveals the dependance of \(c\) and \(\kappa \) on \(\sup _{\Omega } u\) and structure constants (cf. Lemma 3.5). A similar result is given by Theorem 2.2 in Lukkari [50], but under the assumption that \(p^+ \le n\).

**Lemma 3.5**

We are now ready to formulate the version of Hölder continuity up to the boundary which will be needed in this paper.

**Lemma 3.6**

*Proof*

Let \(x,y \in B(w, r)\cap \Omega \) and let \(x_0\in \partial \Omega \) be such that \(d(x, \partial \Omega )=|x-x_0|\). We distinguish two cases.

*Case 1.*\(|x-y|<\frac{1}{2} d(x,\partial \Omega )\). Lemma 3.5 applied with \(\rho =|x-y|\) and \(r=\tau /2\) for \(\tau = d(x,\partial \Omega )\) together with Lemma 3.4 imply the following inequalities:

*Case 2.*\(|x-y|\ge \frac{1}{2} d(x,\partial \Omega )\). Since \(u(x_0)=0\), we have by Lemma 3.4 that

Following the proof of Theorem 6.31 in Heinonen et al. [38], one can show that *if the complement of* \(\Omega \) *satisfies the corkscrew condition at* \(w\in \partial \Omega \), *then* \({\mathbb {R}}^n\setminus \Omega \) *is* \({p(\cdot )}\) *-fat at* \(w\). Indeed, using the elementary properties of the relative \({p(\cdot )}\)-capacity (see Section 10.2 in Dieninig et al. [24], in particular Lemma 10.2.9 in [24] and the discussion following it), one shows that (3.3) holds at \(w\). Here, the log-Hölder continuity of \({p(\cdot )}\) plays an important role as one also employs property (2.2). Hence, the complement of a NTA domain is uniformly \({p(\cdot )}\)-fat, see Definition 2.6.

We are now in a position to prove the main result of this section, the Carleson-type estimate.

**Theorem 3.7**

*Proof*

## 4 Constructions of \({p(\cdot )}\)-barriers

Below, we present two types of barrier functions. The first type is based on a work of Wolanski [59]; however, our Lemma 4.1 improves result of [59], see Remark 4.2. We employ Wolanski-type barriers in the upper and lower boundary Harnack estimates, see Sect. 5. The second type of barriers has been inspired by a work of Bauman [16] who uses barriers in studies of a boundary Harnack inequality for uniformly elliptic equations with bounded coefficients. Both approaches have advantages. On one hand, a radius of a ball for which a Wolanski-type barrier exists, depends on less number of parameters then a radius of a corresponding ball for a Bauman-type barrier, but on the other hand, exponents in Wolanski-type barriers depend on larger number of parameters than exponents in Bauman-type barriers, cf. Lemmas 4.1 and 4.3. Therefore, both types of barriers are useful in applications.

### 4.1 Upper and lower \({p(\cdot )}\)-barriers of Wolanski-type

**Lemma 4.1**

*Remark 4.2*

We would like to point out that the above theorem improves substantially some results on barrier functions in variable exponent setting, see Corollary 4.1 in Wolanski [59]. Namely in [59], the radius \(r\) depends also on \(M\), whereas here, we manage to avoid such a dependence [see (4.7) and (4.8) for details]. This plays a role in the proof of Lemma 5.1.

*Proof*

### 4.2 Upper and lower \({p(\cdot )}\)-barriers of Bauman-type

**Lemma 4.3**

*Proof*

## 5 Upper and lower boundary growth estimates: The boundary Harnack inequality

This section contains main result of the paper, namely the proof of the boundary Harnack inequality for positive \({p(\cdot )}\)-harmonic functions on domains satisfying the ball condition, see Theorem 5.4. The proof relies on Lemmas 5.1 and 5.3, where we show the lower and, respectively, the upper estimates for a rate of decay of a \({p(\cdot )}\)-harmonic function close to a boundary of the underlying domain. In particular, Lemmas 5.1 and 5.3 imply stronger result than the usual boundary Harnack inequality, namely that a \({p(\cdot )}\)-harmonic function vanishes at the same rate as the distance function. Moreover, Lemma 5.1 illustrates the following phenomenon: the geometry of the domain effects the sets of parameters on which the rate of decay depends. Indeed, it turns out that constants in our lower estimate depend whether domain satisfies the interior ball condition or the ball condition, cf. parts (i) and (ii) of Lemma 5.1. As a corollary, we also obtain a decay estimate for supersolutions (a counterpart of Proposition 6.1 in Aikawa et al. [7]).

For \(w\in \partial \Omega \), we denote by \(A_r(w)\) a point satisfying \(d(A_{r}(w), \partial \Omega ) = r\) and \(|A_{r}(w) - w| = r\). Existence of such a point is guaranteed by the interior ball condition (with radius \(r_i\)) for \(r \le r_i/2\). Recall also that by \(c_H\), we denote the constant from the Harnack inequality, Lemma 3.1.

**Lemma 5.1**

- (i)There exist constants \(c\) and \(\tilde{c}\) such that if \(\tilde{r}: = r / \tilde{c}\) thenThe constant \(\tilde{c}\) depends only on \(r_i\) and \(p^-, \Vert \nabla p\Vert _{L^\infty }\), while \(c\) depends on \(\inf _{\Gamma _{w,\tilde{r}}} u, r_i\) and \(p^+, p^-, n, \Vert \nabla p\Vert _{L^\infty }\), where \(\Gamma _{w,\tilde{r}} = \{x \in \Omega | \tilde{r} < d(x, \partial \Omega ) < 3 \tilde{r} \} \cap B(w,r)\). Moreover, \(c\) is decreasing in \(\inf _{\Gamma _{w,\tilde{r}}} u\).$$\begin{aligned} c\, u(x) \ge \frac{d(x, \partial \Omega )}{r} \quad \text {for} \quad x \in \Omega \cap B(w, \tilde{r}). \end{aligned}$$

- (ii)Then, there exist constants \(c_L\) and \(\tilde{c}_L\) such that if \(\tilde{r}: = r / \tilde{c}_L\) thenThe constant \(\tilde{c}_L\) depends only on \(r_b\) and \(p^-, \Vert \nabla p\Vert _{L^\infty }\), while \(c_L\) depends on \(\sup _{\Omega \cap B(w, r)} u, u(A_{2\tilde{r}}(w)), r_b\) and \(p^+, p^-, n, \Vert \nabla p\Vert _{L^\infty }\). Moreover, \(c_L\) is decreasing in \(u(A_{2\tilde{r}}(w))\) and increasing in \(\sup _{\Omega \cap B(w,r)} u\).$$\begin{aligned} c_L \, u(x) \ge \frac{d(x, \partial \Omega )}{r} \quad \text {for} \quad x \in \Omega \cap B(w, \tilde{r}). \end{aligned}$$

*Proof*

To prove \((i)\), we start by applying Lemma 4.1 to obtain \(r_*\), depending only on \(\Vert \nabla p\Vert _{L^\infty }, p^{-}\), such that we can construct barriers in an annulus with radius less than \(r_*\). Assume \(\tilde{c}\) to be so large that \(\tilde{r} \le \min \{ r_*, r/6 \}\) and note that so far \(\tilde{c} \ge 6\) depends only on \(\Vert \nabla p\Vert _{L^\infty }, p^{-}\) and \(r_i\).

Denote \(u^*\) the lsc-regularization of a supersolution \(u\) (see, e.g., Adamowicz et al. [3, Theorem 3.5] and discussion therein).

**Corollary 5.2**

*Proof*

We now show the upper boundary growth estimates.

**Lemma 5.3**

*Proof*

We are now in a position to state and prove the main result of the paper.

**Theorem 5.4**

*Proof*

(Proof of Theorem 5.4) We observe that \(\sup _{B(w,4 \tilde{r})\cap \Omega } u\) in Lemma 5.3 can be replaced by \(\sup _{B(w, r)\cap \Omega } u\). Then, Lemma 5.1 and Lemma 5.3 immediately imply the assertion of the theorem. \(\square \)

*Remark 5.5*

*Remark 5.6*

*Remark 5.7*

Let us discuss how our main results correspond to those for constant \(p\). In such a case the Harnack inequality (Lemma 3.1) does not reduce to the usual one, when \(p=\text {const}\): the Harnack constant still depends on \(u\) and the estimate remains nonhomogeneous. Moreover, the upper and lower estimates (Lemmas 5.1, 5.3) still depend on \(u(A_{2\tilde{r}}(w))\) and \(\sup _{\Omega \cap B(w,r)} u\).

Nevertheless, one can slightly modify our proofs and retrieve results from [7]. Indeed, we observe that for constant \(p\) barriers used in proofs of Lemmas 5.1 and 5.3 depend on \(p\) and \(n\) only, while \(r_*\) does not arise. One may as well replace barriers by fundamental solution for the \(p\)-harmonic equation. Moreover, the claim (5.3) improves to \(u(A_{2\tilde{r}}(w)) \le c u(x)\) for a constant \(c\) depending only on \(r_b, p\) and \(n\) by trivial application of the stronger Harnack inequality available for constant \(p\) (see, e.g., Avelin et al. [12, Lemma 2.1]) This observation, together with using the stronger variant of the Carleson estimate for constant \(p\) (see, e.g., [12, Lemma 2.5]) in Lemma 5.3, in a similar way as described in (5.6), implies the Boundary Harnack inequality for constant \(p\) in [7].

## 6 \({p(\cdot )}\)-Harmonic measure

In this section, we study \({p(\cdot )}\)-harmonic measures. In Lemma 6.2, we show the existence of a \({p(\cdot )}\)-harmonic measure, and in Theorem 6.3, we provide our main results of this section: lower and upper growth estimates for such measures. Finally, using these growth estimates and the Carleson estimate (Theorem 3.7), we conclude in Corollary 6.5 a weak doubling property of the \({p(\cdot )}\)-harmonic measure. Let us now explain motivations for our studies.

Harmonic measures were employed to prove a Boundary Harnack inequality in the setting of harmonic functions, see Dahlberg [23] and Jerison and Kenig [40]. When studying boundary behavior of \(p\)-harmonic type functions, various versions of generalizations of harmonic measures have been introduced and studied for \(p \not = 2\), see, e.g., Llorente et al. [49]. In the case of constant \(p\) (\(p \not = 2\)), Bennewitz and Lewis employed the doubling property of a \(p\)-harmonic measure, first proved in Eremenko and Lewis [26], to obtain a Boundary Harnack inequality for \(p\)-harmonic functions in the plane, see Bennewitz and Lewis [17]. This result has been generalized to the setting of Aronsson-type equations by Lewis and Nyström [46] and Lundström and Nyström [53]. The \(p\)-harmonic measure, defined as in the aforementioned papers, as well as Boundary Harnack inequalities, have played a significant role when studying free boundary problems, see, for example, Lewis and Nyström [48]. The \(p\)-harmonic measure was also used to find the optimal Hölder exponent of \(p\)-harmonic functions vanishing near the boundary, see Kilpeläinen and Zhong [42] and Lundström [52]. Moreover, a work of Peres and Sheffield [57] provides discussion of connections between \(p\)-harmonic measures, defined in a different way though, and tug-of-war games. As for the equations with nonstandard growth, we mention paper by Lukkari et al. [51], where some upper estimates for \({p(\cdot )}\)-harmonic measures were studied in the context of Wolff potentials.

To prove our results concerning \({p(\cdot )}\)-measures, we begin by stating a Caccioppoli-type estimate.

**Lemma 6.1**

*Proof*

The proof goes the same lines as for the \(p=\text {const}\) case, namely one uses in (2.8) a test function \(\phi =u\eta ^{p^+}\), cf. Lemma 5.3 in Harjulehto et al. [33] for the proof of Caccioppoli estimate in the case of slightly modified \({p(\cdot )}\)-Laplace operator \(\mathrm{div}({p(\cdot )}|\nabla u|^{{p(\cdot )}-2}\nabla u)\)). \(\square \)

The following existence lemma is probably known to experts in the variable exponent analysis, but to our best knowledge has not appeared earlier in the literature. Therefore, we include its proof for the readers convenience.

**Lemma 6.2**

*Proof*

\(\square \)

The following theorem is the main result of this section. In the constant exponent setting similar results are well known, see for example Eremenko and Lewis [26], Kilpeläinen and Zhong [43] and Lundström and Nyström [53]. Our result in the variable exponent setting extends partially [53]. Indeed, by taking \(p=p^+=p^-\) in Theorem 6.3, we retrieve the corresponding estimates for \(p = \text {const}\), cf. Lemma 2.7 in [53].

**Theorem 6.3**

*Remark 6.4*

The assumption that the complement of \(\Omega \) is uniformly \({p(\cdot )}\)-fat can be replaced by a growth estimate on the solution \(u\) near \(\partial \Omega \). In particular, we use the uniform \({p(\cdot )}\)-fatness only to be able to apply the Hölder continuity result (Lemma 3.4) giving (6.6).

*Proof*

The proof relies on ideas of the constant \(p\) case, see Eremenko and Lewis [26, Lemma 1] and Kilpeläinen and Zhong [43, Lemma 3.1]. However, the setting of variable exponent PDEs is causing difficulties in a straightforward extension of \(p=\text {const}\) arguments, namely the lack of homogeneity of \({p(\cdot )}\)-harmonic equation and the fact that the homogeneous Sobolev–Poincaré inequality (2.6) holds for norms but not for modular functions require more caution and delicate approach.

Using Theorems 3.7 and 6.3, we obtain the following weak doubling property of the \({p(\cdot )}\)-harmonic measure.

**Corollary 6.5**

*Proof*

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