The Perron method for $\varvec{p}$-harmonic functions in unbounded sets in $\mathbf {R}^n$ and metric spaces

Abstract

The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The upper and lower ($p$-harmonic) Perron solutions are studied for open sets, which are assumed to be $p$-parabolic if unbounded. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide), that the Perron solution agrees with the $p$-harmonic extension, and that Perron solutions are invariant under perturbation of the function on a set of capacity zero.

1 Introduction

The Dirichlet (boundary value) problem for $p$-harmonic functions, $1<p<\infty$, which is a nonlinear generalization of the classical Dirichlet problem, considers the $p$-Laplace equation,

$$\begin{aligned} \Delta _p u := {{\mathrm{div}}}(|\nabla u|^{p-2}\nabla u) = 0, \end{aligned}$$

(1.1)

with prescribed boundary values $u=f$ on the boundary $\partial \Omega$. A continuous weak solution of (1.1) is said to be $p$ -harmonic.

The nonlinear potential theory of $p$-harmonic functions has been developed since the 1960s; not only in $\mathbf {R}^n$, but also in weighted $\mathbf {R}^n$, Riemannian manifolds, and other settings. The books Malý–Ziemer [28] and Heinonen–Kilpeläinen–Martio [18] are two thorough treatments in $\mathbf {R}^n$ and weighted $\mathbf {R}^n$, respectively.

More recently, $p$-harmonic functions have been studied in complete metric spaces equipped with a doubling measure supporting a $p$-Poincaré inequality. It is not clear how to employ partial differential equations in such a general setting as a metric measure space. However, the equivalent variational problem of locally minimizing the $p$-energy integral,

$$\begin{aligned} \int |\nabla u|^p\,dx, \end{aligned}$$

(1.2)

among all admissible functions, becomes available when considering the notion of minimal $p$-weak upper gradient as a substitute for the modulus of the usual gradient. A continuous minimizer of (1.2) is $p$-harmonic. The reader might want to consult Björn–Björn [3] for the theory of $p$-harmonic functions and first-order analysis on metric spaces.

If the boundary value function f is not continuous, then it is not feasible to require that the solution u attains the boundary values as limits, i.e., to require that $u(y)\rightarrow f(x)$ as $y\rightarrow x$ ($y\in \Omega$) for all $x\in \partial \Omega$. This is actually often not possible even if f is continuous (see, e.g., Examples 13.3 and 13.4 in Björn–Björn [3]). It is therefore more reasonable to consider boundary data in a weaker (Sobolev) sense. Shanmugalingam [33] solved the Dirichlet problem for $p$-harmonic functions in bounded domains with Newtonian boundary data taken in Sobolev sense. This result was generalized by Hansevi [16] to unbounded domains with Dirichlet boundary data. For continuous boundary values, the problem was solved in bounded domains using uniform approximation by Björn–Björn–Shanmugalingam [6].

The Perron method for solving the Dirichlet problem for harmonic functions (on $\mathbf {R}^2$) was introduced in 1923 by Perron [29] (and independently by Remak [30]). The advantage of the method is that one can construct reasonable solutions for arbitrary boundary data. It provides an upper and a lower solution, and the major question is to determine when these solutions coincide, i.e., to determine when the boundary data is resolutive. The Perron method in connection with the usual Laplace operator has been studied extensively in Euclidean domains (see, e.g., Brelot [11] for the complete characterization of the resolutive functions) and has been extended to degenerate elliptic operators (see, e.g., Granlund–Lindqvist–Martio [14], Kilpeläinen [23], and Heinonen–Kilpeläinen–Martio [18]).

Björn–Björn–Shanmugalingam [7] extended the Perron method for $p$-harmonic functions to the setting of a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, and proved that Perron solutions are $p$-harmonic and agree with the previously obtained solutions for Newtonian boundary data in Shanmugalingam [33]. More recently, Björn–Björn–Shanmugalingam [9] have developed the Perron method for $p$-harmonic functions with respect to the Mazurkiewicz boundary. See also Estep–Shanmugalingam [12], A. Björn [2], and Björn–Björn–Sjödin [10].

The purpose of this paper is to extend the Perron method for solving the Dirichlet problem for $p$-harmonic functions to unbounded open sets in the setting of a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality. In particular, we show that quasicontinuous functions with finite Dirichlet energy, as well as continuous functions, are resolutive with respect to open sets, which are assumed to be $p$-parabolic if unbounded, and that the Perron solution is the unique $p$-harmonic solution that takes the required boundary data outside sets of capacity zero. We also show that Perron solutions are invariant under perturbations on sets of capacity zero.

The paper is organized as follows: In the next section, we establish notation, review some basic definitions relating to Sobolev-type spaces on metric spaces, and obtain a new convergence lemma. In Sect. 3, we review the obstacle problem associated with $p$-harmonic functions in unbounded sets and obtain a convergence theorem that will be important in the proof of Theorem 7.5 (the main result of this paper). Section 4 is devoted to $p$-parabolic sets. The necessary background on $p$-harmonic and superharmonic functions is given in Sect. 5, making it possible to define Perron solutions in Sect. 6, where we also extend the comparison principle for superharmonic functions to unbounded sets. In Sect. 7, we introduce a smaller capacity (and its related quasicontinuity property) before we obtain our main result (Theorem 7.5) on resolutivity (of quasicontinuous functions) along with some consequences.

2 Notation and preliminaries

We assume throughout the paper that $(X,\mathscr {M},\mu ,d)$ is a metric measure space (which we refer to as X) equipped with a metric d and a positive complete Borel measure $\mu$ such that $0<\mu (B)<\infty$ for all balls $B\subset X$. We use the following notation for balls,

$$\begin{aligned} B(x_0,r) := \{x\in X:d(x,x_0)<r\}, \end{aligned}$$

and for $B=B(x_0,r)$ and $\lambda >0$, we let $\lambda B=B(x_0,\lambda r)$. The $\sigma$-algebra $\mathscr {M}$ (on which $\mu$ is defined) is the completion of the Borel $\sigma$-algebra. Later we will impose additional requirements on the space and on the measure. We assume further that $1<p<\infty$ and that $\Omega$ is a nonempty (possibly unbounded) open subset of X.

The measure $\mu$ is said to be doubling if there exists a constant $C\ge 1$ such that

$$\begin{aligned} 0< \mu (2B) \le C \mu (B) < \infty \end{aligned}$$

for all balls $B\subset X$. Recall that a metric space is said to be proper if all bounded closed subsets are compact. In particular, this is true if the metric space is complete and the measure is doubling.

The characteristic function of a set E is denoted by $\chi _E$, and we let $\sup \varnothing =-\infty$ and $\inf \varnothing =\infty$. We say that the set E is compactly contained in A if (the closure of E) is a compact subset of A and denote this by $E\Subset A$. The extended real number system is denoted by . We use the notation $f_{+}=\max \{f, 0\}$ and $f_{-}=\max \{-f,0\}$. Continuous functions will be assumed to be real-valued. By a curve in X we mean a rectifiable nonconstant continuous mapping from a compact interval into X. A curve can thus be parametrized by its arc length $ds$.

Definition 2.1

A Borel function $g:X\rightarrow [0,\infty ]$ is said to be an upper gradient of a function whenever

$$\begin{aligned} |f(x)-f(y)| \le \int _\gamma g\,ds\end{aligned}$$

(2.1)

holds for each pair of points $x,y\in X$ and every curve $\gamma$ in X joining x and y. We make the convention that the left-hand side is infinite when at least one of the terms in the left-hand side is infinite.

A drawback of the upper gradients, introduced in Heinonen–Koskela [19, 20] is that they are not preserved by $L^{p}$-convergence. It is, however, possible to overcome this problem by relaxing the condition a bit (Koskela–MacManus [27]).

Definition 2.2

A measurable function $g:X\rightarrow [0,\infty ]$ is said to be a $p$ -weak upper gradient of a function whenever (2.1) holds for each pair of points $x,y\in X$ and $p$-almost every curve (see below) $\gamma$ in X joining x and y.

Note that a $p$-weak upper gradient is not required to be a Borel function (see the discussion in the notes to Chapter 1 in Björn–Björn [3]).

We say that a property holds for $p$ -almost every curve if it fails only for a curve family $\Gamma$ with zero $p$-modulus, i.e., if there exists a nonnegative $\rho \in L^{p}(X)$ such that $\int _\gamma \rho \,ds=\infty$ for every curve $\gamma \in \Gamma$.

A countable union of curve families, each with zero $p$-modulus, also has zero $p$-modulus. For proofs of this and other results in this section, we refer to Björn–Björn [3] or Heinonen–Koskela–Shanmugalingam–Tyson [21].

Shanmugalingam [32] used upper gradients to define so-called Newtonian spaces.

Definition 2.3

The Newtonian space on X, denoted by $N^{1,p}(X)$, is the space of all everywhere defined, extended real-valued functions $u\in L^{p}(X)$ such that

$$\begin{aligned} \Vert u\Vert _{N^{1,p}(X)} := \biggl (\int _X|u|^p\,d\mu + \inf _g\int _X g^p\,d\mu \biggr )^{1/p}<\infty , \end{aligned}$$

where the infimum is taken over all upper gradients g of u.

Definition 2.4

An everywhere defined, measurable, extended real-valued function on X belongs to the Dirichlet space $D^p(X)$ if it has an upper gradient in $L^{p}(X)$.

It follows from Lemma 2.4 in Koskela–MacManus [27] that a measurable function belongs to $D^p(X)$ whenever it (merely) has a $p$-weak upper gradient in $L^{p}(X)$.

We emphasize that Newtonian and Dirichlet functions are defined everywhere (not just up to an equivalence class in the corresponding function space), which is essential for the notion of upper gradient to make sense. Shanmugalingam [32] proved that the associated normed (quotient) space defined by $N^{1,p}(X)/\sim$, where $u\sim v$ if and only if $\Vert u-v\Vert _{N^{1,p}(X)}=0$, is a Banach space.

A measurable set $A\subset X$ can be considered to be a metric space in its own right (with the restriction of d and $\mu$ to A). Thus the Newtonian space $N^{1,p}(A)$ and the Dirichlet space $D^p(A)$ are also given by Definitions 2.3 and 2.4, respectively. If X is proper, then $f\in L^{p}_\mathrm{loc}(\Omega )$, $f\in N^{1,p}_\mathrm{loc}(\Omega )$, and $f\in D^{p}_\mathrm{loc}(\Omega )$ if and only if $f\in L^{p}(\Omega ')$, $f\in N^{1,p}(\Omega ')$, and $f\in D^p(\Omega ')$, respectively, for all open $\Omega '\Subset \Omega$.

If $u\in D^p(X)$, then u has a minimal $p$ -weak upper gradient, denoted by $g_u$, which is minimal in the sense that $g_u\le g$ a.e. for all $p$-weak upper gradients g of u; see Shanmugalingam [33]. Minimal $p$-weak upper gradients $g_u$ are true substitutes for $|\nabla u|$ in metric spaces. One of the important properties of minimal $p$-weak upper gradients is that they are local in the sense that if two functions $u,v\in D^p(X)$ coincide on a set E, then $g_u=g_v$ a.e. on E. Furthermore, if $U=\{x\in X:u(x)>v(x)\}$, then $g_u\chi _U+g_v\chi _{X{\setminus }U}$ and $g_v\chi _U+g_u\chi _{X{\setminus }U}$ are minimal $p$-weak upper gradients of $\max \{u,v\}$ and $\min \{u,v\}$, respectively. The restriction of a minimal $p$-weak upper gradient to an open subset remains minimal with respect to that subset, and hence the results above about minimal $p$-weak upper gradients of functions in $D^p(X)$ extend to functions in $D^{p}_\mathrm{loc}(X)$ having minimal $p$-weak upper gradients in $L^{p}_\mathrm{loc}(X)$.

The notion of capacity of a set is important in potential theory, and various types and definitions can be found in the literature (see, e.g., Kinnunen–Martio [24] and Shanmugalingam [32]).

Definition 2.5

Let $A\subset X$ be measurable. The (Sobolev) capacity (with respect to A) of $E\subset A$ is the number

$$\begin{aligned} {C_p}(E;A) := \inf _u\Vert u\Vert _{N^{1,p}(A)}^p, \end{aligned}$$

where the infimum is taken over all $u\in N^{1,p}(A)$ such that $u\ge 1$ on E. When the capacity is taken with respect to X, we simplify the notation and write ${C_p}(E)$.

Whenever a property holds for all points except for those in a set of capacity zero, it is said to hold quasieverywhere (q.e.).

The capacity is countably subadditive, i.e., ${C_p}(\bigcup _{j=1}^\infty E_j)\le \sum _{j=1}^\infty {C_p}(E_j)$.

In order to be able to compare boundary values of Dirichlet and Newtonian functions, we introduce the following spaces.

Definition 2.6

For subsets E and A of X, where A is measurable, the Dirichlet space with zero boundary values in $A{\setminus }E$, is

$$\begin{aligned} D^p_0(E;A) := \{u|_{E\cap A}:u\in D^p(A)\text { and }u=0\text { in }A{\setminus }E\}. \end{aligned}$$

The Newtonian space with zero boundary values, $N^{1,p}_0(E;A)$, is defined analogously. We let $D^p_0(E)$ and $N^{1,p}_0(E)$ denote $D^p_0(E;X)$ and $N^{1,p}_0(E;X)$, respectively.

The condition “$u=0$ in $A{\setminus }E$” can actually be replaced by “$u=0$ q.e. in $A{\setminus }E$” without changing the obtained spaces.

If $E\subset X$ is measurable, $f\in D^p(E)$, $f_1,f_2\in D^p_0(E)$, and $f_1\le f\le f_2$ q.e. in E, then $f\in D^p_0(E)$ (this is Lemma 2.8 in Hansevi [16]).

The following convergence lemma will be used to prove Theorem 3.2, which in turn will be important when we prove Theorem 7.5.

Lemma 2.7

Let $G_1,G_2,\dots$ be open sets such that $G_1\subset G_2\subset \cdots \subset X=\bigcup _{k=1}^\infty G_k$ and let $\{u_j\}_{j=1}^\infty$ be a sequence of functions defined on X. Assume that $\{u_j\}_{j=1}^\infty$ is bounded in $L^{p}(G_k)$ for all $k=1,2,\dots$. Assume further that $\{g_j\}_{j=1}^\infty$ is bounded in $L^{p}(X)$, and that $g_j$ is a $p$-weak upper gradient of $u_j$ with respect to $G_j$ for each $j=1,2,\dots$. Then a function u belongs to $D^p(X)$ if $u_j\rightarrow u$ q.e. on X as $j\rightarrow \infty$.

Proof

Let k be a positive integer. Clearly, $g_j$ is a $p$-weak upper gradient of $u_j$ with respect to $G_k$ for every integer $j\ge k$. According to Lemma 3.2 in Björn–Björn–Parviainen [5], there are a $p$-weak upper gradient $\tilde{g}_k\in L^{p}(G_k)$ of u with respect to $G_k$ and a subsequence of $\{g_j\}_{j=1}^\infty$, denoted by $\{g_{k,j}\}_{j=1}^\infty$, such that $g_{k,j}\rightarrow \tilde{g}_k$ weakly in $L^{p}(G_k)$ as $j\rightarrow \infty$. Extend $\tilde{g}_k$ to X by letting $\tilde{g}_k=0$ on $X{\setminus }G_k$. Since $\{g_j\}_{j=1}^\infty$ is bounded in $L^{p}(X)$, there is an integer M such that $\Vert g_j\Vert _{L^{p}(X)}\le M$ for all $j=1,2,\dots$. The weak convergence implies that

$$\begin{aligned} \Vert \tilde{g}_k\Vert _{L^{p}(X)} = \Vert \tilde{g}_k\Vert _{L^{p}(G_k)} \le \liminf _{j\rightarrow \infty }\Vert g_{k,j}\Vert _{L^{p}(G_k)} \le \liminf _{j\rightarrow \infty }\Vert g_{k,j}\Vert _{L^{p}(X)} \le M, \end{aligned}$$

and hence the sequence $\{\tilde{g}_k\}_{k=1}^\infty$ is bounded in $L^{p}(X)$.

Since $L^{p}(X)$ is reflexive, it follows from Banach–Alaoglu’s theorem that there is a subsequence, also denoted by $\{\tilde{g}_k\}_{k=1}^\infty$, that converges weakly in $L^{p}(X)$ to a function g. By applying Mazur’s lemma (see, e.g., Theorem 3.12 in Rudin [31]) repeatedly to the sequences $\{\tilde{g}_k\}_{k=j}^\infty$, $j=1,2,\dots$, we can find convex combinations

$$\begin{aligned} g'_j = \sum _{k=j}^{N_j} a_{j,k}\tilde{g}_k \end{aligned}$$

such that $\Vert g'_j-g\Vert _{L^{p}(X)}<1/j$, and hence we obtain a sequence $\{g'_j\}_{j=1}^\infty$ that converges to g in $L^{p}(X)$. Note that $g\in L^{p}(X)$, and that for every $n=1,2,\dots$, the sequence $\{g'_j\}_{j=n}^\infty$ consists of $p$-weak upper gradients of u with respect to $G_n$. It suffices to show that g is a $p$-weak upper gradient of u to complete the proof.

By Fuglede’s lemma (Lemma 3.4 in Shanmugalingam [32]), we can find a subsequence, also denoted by $\{g'_j\}_{j=1}^\infty$, and a collection of curves $\Gamma$ in X with zero $p$-modulus, such that for every curve $\gamma \notin \Gamma$, it follows that

$$\begin{aligned} \int _\gamma g'_j\,ds\rightarrow \int _\gamma g\,ds\quad \text {as }j\rightarrow \infty . \end{aligned}$$

(2.2)

For every $n=1,2,\dots$, let $\Gamma _{n,j}$, $j=n,n+1,\dots$, be the collection of curves in $G_n$ along which $g'_j$ is not an upper gradient of u, and let

$$\begin{aligned} \Gamma ' = \Gamma \cup \bigcup _{n=1}^\infty \bigcup _{j=n}^\infty \Gamma _{n,j}. \end{aligned}$$

Then $\Gamma '$ has zero $p$-modulus.

Let $\gamma \notin \Gamma '$ be an arbitrary curve in X with endpoints x and y. Since $\gamma$ is compact and $G_1,G_2,\dots$  are open sets that exhaust X, we can find an integer N such that $\gamma \subset G_N$ and

$$\begin{aligned} |u(x)-u(y)| \le \int _\gamma g'_j\,ds, \quad j=N,N+1,\dots . \end{aligned}$$

It follows that g is a $p$-weak upper gradient of u, and thus $u\in D^p(X)$, since

$$\begin{aligned} |u(x)-u(y)| \le \lim _{j\rightarrow \infty }\int _\gamma g'_j\,ds= \int _\gamma g\,ds. \end{aligned}$$

$\square$

Definition 2.8

Let $q\ge 1$. We say that X supports a (q, p)-Poincaré inequality if there exist constants, $C>0$ and $\lambda \ge 1$ (the dilation constant), such that

(2.3)

for all balls $B\subset X$, all integrable functions u on X, and all upper gradients g of u.

In (2.3), we have used the convenient notation . We usually write $p$-Poincaré inequality instead of (1, p)-Poincaré inequality.

Requiring a Poincaré inequality to hold is one way of making it possible to control functions by their upper gradients.

3 The obstacle problem

In this section, we also assume that X is proper and supports a (p, p)-Poincaré inequality, and that ${C_p}(X{\setminus }\Omega )>0$.

Inspired by Kinnunen–Martio [25], the following obstacle problem, which is a generalization that allows for unbounded sets, was defined in Hansevi [16].

Definition 3.1

Let $V\subset X$ be a nonempty open subset such that ${C_p}(X{\setminus }V)>0$. For and $f\in D^p(V)$, define

$$\begin{aligned} \mathscr {K}_{\psi ,f}(V) = \{v\in D^p(V):v-f\in D^p_0(V) \text { and }v\ge \psi \text { q.e. in }V\}. \end{aligned}$$

A function u is said to be a solution of the $\mathscr {K}_{\psi ,f}(V)$-obstacle problem (with obstacle $\psi$ and boundary values f ) whenever $u\in \mathscr {K}_{\psi ,f}(V)$ and

$$\begin{aligned} \int _V g_u^p\,d\mu \le \int _V g_v^p\,d\mu \quad \text {for all }v\in \mathscr {K}_{\psi ,f}(V). \end{aligned}$$

When $V=\Omega$, we usually denote $\mathscr {K}_{\psi ,f}(\Omega )$ by $\mathscr {K}_{\psi ,f}$ for short.

It was proved in Hansevi [16] that the $\mathscr {K}_{\psi ,f}$-obstacle problem has a unique (up to sets of capacity zero) solution under the natural condition of $\mathscr {K}_{\psi ,f}$ being nonempty. If the measure $\mu$ is doubling, then there is a unique lsc-regularized solution of the $\mathscr {K}_{\psi ,f}$-obstacle problem whenever $\mathscr {K}_{\psi ,f}$ is nonempty (Theorem 4.1 in Hansevi [16]). The lsc-regularization of u is the (lower semicontinuous) function $u^*$ defined by

$$\begin{aligned} u^*(x) = \mathop {{{\mathrm{ess\,lim\,inf}}}}\limits _{y\rightarrow x}u(y) := \lim _{r\rightarrow 0}\mathop {{{\mathrm{ess\,inf}}}}\limits _{B(x,r)} u. \end{aligned}$$

We conclude this section with a proof of a new convergence theorem that will be used in the proof of Theorem 7.5. It is a generalization of Proposition 10.18 in Björn–Björn [3] to unbounded sets and Dirichlet functions. The special case when $\psi _j=f_j\in N^{1,p}(\Omega )$ had previously been proved in Kinnunen–Shanmugalingam [26], and a similar result for the double obstacle problem was obtained in Farnana [13].

Theorem 3.2

Let $\{\psi _j\}_{j=1}^\infty$ and $\{f_j\}_{j=1}^\infty$ be sequences of functions in $D^p(\Omega )$ that are decreasing q.e. to functions $\psi$ and f in $D^p(\Omega )$, respectively, and are such that $\Vert g_{\psi _j-\psi }\Vert _{L^{p}(\Omega )}\rightarrow 0$ and $\Vert g_{f_j-f}\Vert _{L^{p}(\Omega )}\rightarrow 0$ as $j\rightarrow \infty$. If $u_j$ is a solution of the $\mathscr {K}_{\psi _j,f_j}$-obstacle problem for each $j=1,2,\dots$, then the sequence $\{u_j\}_{j=1}^\infty$ is decreasing q.e. in $\Omega$ to a function which is a solution of the $\mathscr {K}_{\psi ,f}$-obstacle problem.

Proof

The comparison principle (Lemma 3.6 in Hansevi [16]) asserts that $u_{j+1}\le u_j$ q.e. in $\Omega$ for each $j=1,2,\dots$, and hence by the subadditivity of the capacity there exists a function u such that $\{u_j\}_{j=1}^\infty$ is decreasing to u q.e. in $\Omega$. We will show that u is a solution of the $\mathscr {K}_{\psi ,f}$-obstacle problem.

Let $w_j=u_j-f_j$ and $w=u-f$, all functions extended by zero outside $\Omega$. Let $B\subset X$ be a ball such that $B\cap \Omega$ is nonempty and ${C_p}(B'{\setminus }\Omega )>0$ where $B':=\tfrac{1}{2}B$.

We claim that the sequences $\{g_{w_j}\}_{j=1}^\infty$ and $\{w_j\}_{j=1}^\infty$ are bounded in $L^{p}(X)$ and $L^{p}(kB)$, respectively, for every $k=1,2,\dots$. To show this, let k be a positive integer. Let $S=\bigcap _{j=1}^\infty S_j$, where $S_j:=\{x\in X:w_j(x)=0\}$. Proposition 4.14 in Björn–Björn [3] asserts that $w_j\in N^{1,p}_\mathrm{loc}(X)$, and since

$$\begin{aligned} {C_p}(kB'\cap S_j) \ge {C_p}(kB'\cap S) \ge {C_p}(kB'{\setminus }\Omega ) \ge {C_p}(B'{\setminus }\Omega ) > 0, \end{aligned}$$

Maz$'$ya’s inequality (Theorem 5.53 in Björn–Björn [3]) implies the existence of constants $C_{kB,\Omega }>0$ and $\lambda \ge 1$ such that

$$\begin{aligned} \int _{kB}|w_j|^p\,d\mu \le C_{kB,\Omega }\int _{\lambda kB}g_{w_j}^p\,d\mu . \end{aligned}$$

Let $h_j=\max \{f_j,\psi _j\}$. Then $0\le h_j-f_j=(\psi _j-f_j)_{+}\le (u_j-f_j)_{+}$ q.e. in $\Omega$, and hence Lemma 2.8 in Hansevi [16] asserts that $h_j-f_j\in D^p_0(\Omega )$. Clearly, $h_j\in \mathscr {K}_{\psi _j,f_j}$, and as $u_j$ is a solution of the $\mathscr {K}_{\psi _j,f_j}$-obstacle problem, it follows that $\Vert g_{u_j}\Vert _{L^{p}(\Omega )}\le \Vert g_{h_j}\Vert _{L^{p}(\Omega )}$. We also know that $g_{h_j}\le g_{\psi _j}+g_{f_j}$ a.e. in $\Omega$, and therefore the claim follows because

$$\begin{aligned} C_{kB,\Omega }^{\,-1/p}\Vert w_j\Vert _{L^{p}(kB)}&\le \Vert g_{w_j}\Vert _{L^{p}(X)} \nonumber \\&\le \Vert g_{u_j}\Vert _{L^{p}(\Omega )} + \Vert g_{f_j}\Vert _{L^{p}(\Omega )} \nonumber \\&\le \Vert g_{h_j}\Vert _{L^{p}(\Omega )} + \Vert g_{f_j}\Vert _{L^{p}(\Omega )} \nonumber \\&\le \Vert g_{\psi _j}\Vert _{L^{p}(\Omega )} + 2\Vert g_{f_j}\Vert _{L^{p}(\Omega )} \nonumber \\&\le \Vert g_{\psi _j-\psi }\Vert _{L^{p}(\Omega )} + \Vert g_\psi \Vert _{L^{p}(\Omega )} + 2\Vert g_{f_j-f}\Vert _{L^{p}(\Omega )} + 2\Vert g_{f}\Vert _{L^{p}(\Omega )} . \end{aligned}$$

(3.1)

Lemma 2.7 applies here and asserts that $w\in D^p(X)$, and hence $u-f\in D^p_0(\Omega )$. As $f\in D^p(\Omega )$, this also shows that $u\in D^p(\Omega )$. Since ${C_p}$ is countably subadditive, $u\ge \psi$ q.e. in $\Omega$, and hence $u\in \mathscr {K}_{\psi ,f}$.

Let v be an arbitrary function that belongs to $\mathscr {K}_{\psi ,f}$. We complete the proof by showing that

$$\begin{aligned} \int _\Omega g_u^p\,d\mu \le \int _\Omega g_v^p\,d\mu . \end{aligned}$$

(3.2)

Let $\varphi _j=\max \{v+f_j-f,\psi _j\}$. Clearly, $\varphi _j\ge \psi _j$ and $\varphi _j\in D^p(\Omega )$. Furthermore,

$$\begin{aligned} v-f \le \max \{v-f,\psi _j-f_j\} = \varphi _j-f_j \le \max \{v-f,(u_j-f_j)_{+}\} \quad \text {q.e. in }\Omega , \end{aligned}$$

and hence $\varphi _j-f_j\in D^p_0(\Omega )$ by Lemma 2.8 in Hansevi [16]. We conclude that $\varphi _j$ belongs to $\mathscr {K}_{\psi _j,f_j}$, and therefore

$$\begin{aligned} \int _\Omega g_{u_j}^p\,d\mu \le \int _\Omega g_{\varphi _j}^p\,d\mu . \end{aligned}$$

Let E be the set where $\{f_j\}_{j=1}^\infty$ decreases to f, $\{\psi _j\}_{j=1}^\infty$ decreases to $\psi$, and simultaneously $v\ge \psi$. Then ${C_p}(\Omega {\setminus }E)=0$.

Let $U_j=\{x\in E:(f_j-f)(x)<(\psi _j-v)(x)\}$. Clearly, $\varphi _j-v=\psi _j-v$ in $U_j$ and $\varphi _j-v=f_j-f$ in $E{\setminus }U_j$, and hence it follows that

$$\begin{aligned} \int _\Omega g_{\varphi _j-v}^p\,d\mu&\le \int _{U_j}(g_{\psi _j-\psi }+g_{\psi -v})^p\,d\mu + \int _{E{\setminus }U_j}g_{f_j-f}^p\,d\mu \nonumber \\&\le 2^p\int _{U_j}g_{\psi -v}^p\,d\mu + 2^p\int _\Omega g_{\psi _j-\psi }^p\,d\mu + \int _\Omega g_{f_j-f}^p\,d\mu , \end{aligned}$$

(3.3)

where the last two integrals tend to zero as $j\rightarrow \infty$.

Let $V_j=\{x\in E:\psi (x)<v(x)<\psi _j(x)\}$. Since $f_j-f\ge 0$ in E, we know that $v<\psi _j$ in $U_j$, and because $g_{\psi -v}=0$ a.e. in

$$\begin{aligned} \{x\in E:v(x)\le \psi (x)\}=\{x\in E:v(x)=\psi (x)\}, \end{aligned}$$

it follows that

$$\begin{aligned} \int _{U_j}g_{\psi -v}^p\,d\mu \le \int _{V_j}g_{\psi -v}^p\,d\mu . \end{aligned}$$

(3.4)

The fact that $\{\psi _j\}_{j=1}^\infty$ is decreasing to $\psi$ in E implies that $g_{\psi -v}\chi _{V_j}\rightarrow 0$ everywhere in E as $j\rightarrow \infty$, and since $|g_{\psi -v}\chi _{V_j}|\le g_{\psi -v}\le g_\psi +g_v$ a.e. in E and $g_\psi +g_v\in L^{p}(E)$, dominated convergence asserts that

$$\begin{aligned} \int _{V_j}g_{\psi -v}^p\,d\mu = \int _E g_{\psi -v}^p\chi _{V_j}\,d\mu \rightarrow 0 \quad \text {as }j\rightarrow \infty . \end{aligned}$$

(3.5)

It follows from (3.3), (3.4), and (3.5) that $g_{\varphi _j}\rightarrow g_v$ in $L^{p}(\Omega )$ as $j\rightarrow \infty$.

Let

$$\begin{aligned} \Omega _k = \{x\in kB\cap \Omega :{{\mathrm{dist}}}(x,\partial \Omega )>\delta /k\}, \quad k=1,2,\dots , \end{aligned}$$

where $\delta >0$ is sufficiently small so that $\Omega _1$ is nonempty. It is clear that

$$\begin{aligned} \Omega _1\Subset \Omega _2\Subset \cdots \Subset \Omega = \bigcup _{k=1}^\infty \Omega _k. \end{aligned}$$

Fix a positive integer k. Then $g_u$ and $g_{u_j}$ are minimal $p$-weak upper gradients of u and $u_j$, respectively, with respect to $\Omega _k$. By Proposition 4.14 in Björn–Björn [3], the functions f and $f_j$ belong to $L^{p}_\mathrm{loc}(\Omega )$, and hence f and $f_j$ are in $L^{p}(\Omega _k)$. Furthermore, $\{f_j\}_{j=1}^\infty$ is decreasing to f q.e. in $\Omega$, and therefore $|f_j-f|\le |f_1-f|$ q.e. in $\Omega$. By (3.1), we can see that $\{w_j\}_{j=1}^\infty$ is bounded in $L^{p}(kB)$, and also that $\{g_{u_j}\}_{j=1}^\infty$ is bounded in $L^{p}(\Omega )$. Since

$$\begin{aligned} \Vert u_j\Vert _{L^{p}(\Omega _k)} \le \Vert w_j\Vert _{L^{p}(kB)} + \Vert f_1-f\Vert _{L^{p}(\Omega _k)} + \Vert f\Vert _{L^{p}(\Omega _k)}, \end{aligned}$$

it follows that $\{u_j\}_{j=1}^\infty$ is bounded in $N^{1,p}(\Omega _k)$, and because $u_j\rightarrow u$ q.e. in $\Omega$ as $j\rightarrow \infty$, Corollary 3.3 in Björn–Björn–Parviainen [5] asserts that

$$\begin{aligned} \int _{\Omega _k}g_u^p\,d\mu \le \liminf _{j\rightarrow \infty }\int _{\Omega _k}g_{u_j}^p\,d\mu \le \liminf _{j\rightarrow \infty }\int _\Omega g_{u_j}^p\,d\mu \le \liminf _{j\rightarrow \infty }\int _\Omega g_{\varphi _j}^p\,d\mu = \int _\Omega g_v^p\,d\mu . \end{aligned}$$

Letting $k\rightarrow \infty$ yields (3.2) and the proof is complete.$\square$

If $\mu$ is doubling, then X is proper if and only if X is complete (see, e.g., Proposition 3.1 in Björn–Björn [3]). Hölder’s inequality implies that X supports a $p$-Poincaré inequality if X supports a (p, p)-Poincaré inequality. The converse is true when $\mu$ is doubling; see Theorem 5.1 in Hajłasz–Koskela [15]. Thus adding the assumption that $\mu$ is doubling leads to the rather standard assumptions stated below.

We assume from now on that $1<p<\infty$, that X is a complete metric measure space supporting a $p$ -Poincaré inequality, that $\mu$ is doubling, and that $\Omega \subset X$ is a nonempty (possibly unbounded) open subset with ${C_p}(X{\setminus }\Omega )>0$.

4 p-Parabolicity

Note the standing assumptions described at the end of the previous section.

In the proof of Theorem 7.5, we need $\Omega$ to be $p$-parabolic if it is unbounded.

Definition 4.1

If $\Omega$ is unbounded, then we say that $\Omega$ is $p$ -parabolic if for every compact $K\subset \Omega$, there exist functions $u_j\in N^{1,p}(\Omega )$ such that $u_j\ge 1$ on K for all $j=1,2,\dots$, and

$$\begin{aligned} \int _\Omega g_{u_j}^p\,d\mu \rightarrow 0 \quad \text {as }j\rightarrow \infty . \end{aligned}$$

(4.1)

Otherwise, $\Omega$ is said to be $p$ -hyperbolic.

In Definition 4.1, we may as well use $u_j\in D^p(\Omega )$ with bounded support such that $\chi _K\le u_j\le 1$, $j=1,2,\dots$ (see, e.g., the proof of Lemma 5.43 in Björn–Björn [3]).

Remark 4.2

If $\Omega _1\subset \Omega _2$, then $\Omega _1$ is $p$-parabolic whenever $\Omega _2$ is $p$-parabolic.

Holopainen–Shanmugalingam [22] proposed a definition of $p$-harmonic Green functions (i.e., fundamental solutions of the $p$-Laplace operator) on metric spaces. The functions they defined did, however, not share all characteristics with Green functions, and therefore they gave them another name; they called them $p$ -singular functions. Theorem 3.14 in [22] asserts that if X is locally linearly locally connected (see Sect. 2 in [22] for the definition), then the space X is $p$-hyperbolic if and only if for every $y\in X$ there exists a $p$-singular function with singularity at y.

Example 4.3

The space $\mathbf {R}^n$, $n\ge 1$, is $p$-parabolic if and only if $p\ge n$. (It follows that all open subsets of $\mathbf {R}^n$ are $p$-parabolic for all $p\ge n$; see Remark 4.2.)

To see this, assume that $p\ge n$ and let $K\subset \mathbf {R}^n$ be compact. Choose R sufficiently large so that $K\subset B:=B(0,R)$. Let

$$\begin{aligned} u_j(x) = \min \biggl \{1,\biggl (1-\frac{\log |x/R|}{j}\biggr )_+\,\biggr \}, \quad j=1,2,\dots . \end{aligned}$$

(4.2)

Then $\{u_j\}_{j=1}^\infty$ is a sequence of admissible functions for (4.1), and

$$\begin{aligned} g_{u_j} = (j\,|x|)^{-1}\chi _{B_j{\setminus }B}, \quad j=1,2,\dots , \end{aligned}$$

where $B_j:=B(0,Re^j)$. It follows that

$$\begin{aligned} \int _{\mathbf {R}^n}g_{u_j}^p\,dx= C_n\int _R^{Re^j}\frac{r^{n-1}}{(jr)^p}\,dr= C_n{\left\{ \begin{array}{ll} \dfrac{R^{n-p}(1-e^{-j(p-n)})}{(p-n)j^p} &{} \text {if }p>n, \\ \,j^{1-p} &{} \text {if }p=n, \end{array}\right. } \end{aligned}$$

and hence $\int _{\mathbf {R}^n}g_{u_j}^p\,dx\rightarrow 0$ as $j\rightarrow \infty$.

The necessity follows from Theorem 3.14 in Holopainen–Shanmugalingam [22], because if we assume that $p<n$ and let $y\in \mathbf {R}^n$, then

$$\begin{aligned} f(x)=|x-y|^{\tfrac{p-n}{p-1}}, \quad x\in \mathbf {R}^n, \end{aligned}$$

is a Green function with singularity at y that is $p$-harmonic in $\mathbf {R}^n{\setminus }\{y\}$.

A set can be $p$-parabolic if it does not “grow too much” towards infinity, even though the surrounding space is not $p$-parabolic.

Example 4.4

Let $n\ge 2$ and assume that $1<p<n$. Let

$$\begin{aligned} \Omega _f = \{x=(x',\tilde{x})\in \mathbf {R}\times \mathbf {R}^{n-1}:0<x'<f(|\tilde{x}|)\}, \end{aligned}$$

where

$$\begin{aligned} f(r) \le {\left\{ \begin{array}{ll} C &{} \text {if } r < 1, \\ Cr^q &{} \text {if } r\ge 1, \end{array}\right. } \end{aligned}$$

and $q\le p-n+1$ (note that $q<1$ since $p<n$).

Let $K\subset \Omega _f$ be compact. Choose R sufficiently large so that $K\subset B:=B(0,R)$. It can be chosen large enough so that $|\tilde{x}|\ge R/2\ge 1$ for all $(x',\tilde{x})\in \Omega _f{\setminus }B$. This is possible since $q<1$ and $f(r)<Cr^q$. Define the sequence of admissible functions $\{u_j\}_{j=1}^\infty$ as in (4.2). Then

$$\begin{aligned} \int _{\Omega _f}g_{u_j}^p\,dx&= \int _{\mathbf {R}^{n-1}}\int _0^{f(|\tilde{x}|)} \frac{\chi _{B_j{\setminus }B}}{(j|x|)^p}\,dx'\,d\tilde{x} \\&\le \frac{C_{n-1}}{j^p}\int _{R/2}^{Re^j}\frac{f(r)}{r^p}\,r^{n-2}\,dr= \frac{C'_{n-1}}{j^p}\int _{R/2}^{Re^j}r^{q-p+n-2}\,dr=: I_j. \end{aligned}$$

Since

$$\begin{aligned} \int _{R/2}^{Re^j} r^{q-p+n-2}\,dr= {\left\{ \begin{array}{ll} j+\log {2} &{} \text {if }q=p-n+1, \\ \dfrac{(e^{j(q-p+n-1)}-2^{-(q-p+n-1)})R^{q-p+n-1}}{q-p+n-1} &{} \text {if }q<p-n+1, \end{array}\right. } \end{aligned}$$

it follows that $\int _{\Omega _f}g_{u_j}^p\,dx\le I_j\rightarrow 0$ as $j\rightarrow \infty$. Thus $\Omega _f$ is $p$-parabolic (while $\mathbf {R}^n$ is not $p$-parabolic since $p<n$ in this case).

5 p-Harmonic and superharmonic functions

The standing assumptions are described at the end of Sect. 3.

There are many equivalent definitions of (super)minimizers (or, more accurately, $p$-(super)minimizers) in the literature (see, e.g., Proposition 3.2 in A. Björn [1]).

Definition 5.1

We say that a function $u\in N^{1,p}_\mathrm{loc}(\Omega )$ is a superminimizer in $\Omega$ if

$$\begin{aligned} \int _{\varphi \ne 0}g_u^p\,d\mu \le \int _{\varphi \ne 0}g_{u+\varphi }^p\,d\mu \end{aligned}$$

(5.1)

holds for all nonnegative $\varphi \in N^{1,p}_0(\Omega )$, and a minimizer in $\Omega$ if (5.1) holds for all $\varphi \in N^{1,p}_0(\Omega )$. Moreover, a function is $p$ -harmonic if it is a continuous minimizer.

According to Proposition 3.2 in A. Björn [1], it is in fact only necessary to test (5.1) with (all nonnegative and all, respectively) $\varphi \in {{{\mathrm{Lip}}}_c}(\Omega )$.

Proposition 3.9 in Hansevi [16] asserts that a function u is a superminimizer in $\Omega$ if u is a solution of the $\mathscr {K}_{\psi ,f}$-obstacle problem.

The following definition makes sense due to Theorem 4.4 in Hansevi [16]. Because Proposition 2.7 in Björn–Björn [4] asserts that $D^p_0(\Omega )=N^{1,p}_0(\Omega )$ if $\Omega$ is bounded, it is a generalization of Definition 8.31 in Björn–Björn [3] to Dirichlet functions and to unbounded sets.

Definition 5.2

Let $V\subset X$ be a nonempty open subset with ${C_p}(X{\setminus }V)>0$. The $p$ -harmonic extension $H_V f$ of $f\in D^p(V)$ to V is the continuous solution of the $\mathscr {K}_{-\infty ,f}(V)$-obstacle problem. When $V=\Omega$ we usually write $Hf$ instead of $H_\Omega f$.

If f is defined outside V, then we sometimes consider $H_V f$ to be equal to f in some set outside V where f is defined.

A Lipschitz function f on $\partial V$ can be extended to a Lipschitz function $\bar{f}$ on $\overline{V}$ (see, e.g., Theorem 6.2 in Heinonen [17]), and $\bar{f}\in N^{1,p}(\overline{V})$ if V is bounded. The comparison principle (Lemma 4.7 in Hansevi [16]) implies that $H_V \bar{f}$ does not depend on the particular choice of extension $\bar{f}$. We can therefore define the $p$-harmonic extension for Lipschitz functions on the boundary by $H_V f:=H_V\bar{f}$ if V is bounded.

Proposition 5.3

If $\{f_j\}_{j=1}^\infty$ is a sequence of functions in $D^p(\Omega )$ that is decreasing q.e. in $\Omega$ to $f\in D^p(\Omega )$ and $\Vert g_{f_j-f}\Vert _{L^{p}(\Omega )}\rightarrow 0$ as $j\rightarrow \infty$, then $Hf_j$ decreases to $Hf$ locally uniformly in $\Omega$.

Proof

By the comparison principle (Lemma 4.7 in Hansevi [16]), it follows that $Hf_j\ge Hf_{j+1}\ge Hf$ in $\Omega$ for all $j=1,2,\dots$. Since $Hf_j$ and $Hf$ are the continuous solutions of the $\mathscr {K}_{f_j,Hf}$- and $\mathscr {K}_{f,Hf}$-obstacle problems, respectively, it follows from Theorem 3.2 that $Hf_j$ decreases to $Hf$ q.e. in $\Omega$ as $j\rightarrow \infty$.

Because $Hf$ is continuous, and therefore locally bounded, Proposition 5.1 in Shanmugalingam [34] implies that $Hf_j\rightarrow Hf$ locally uniformly in $\Omega$ as $j\rightarrow \infty$.$\square$

In order to define Perron solutions, we need superharmonic functions. We follow Kinnunen–Martio [25], however, we use a slightly different, nevertheless equivalent, definition (see, e.g., Proposition 9.26 in Björn–Björn [3]).

Definition 5.4

We say that a function $u:\Omega \rightarrow (-\infty ,\infty ]$ is superharmonic in $\Omega$ if

• (a) u is lower semicontinuous;

• (b) u is not identically $\infty$ in any component of $\Omega$;

• (c) for every nonempty open set $V'\Subset \Omega$ and all $v\in {{\mathrm{Lip}}}(\partial V')$, we have $H_{V'}v\le u$ in $V'$ whenever $v\le u$ on $\partial V'$.

A function $u:\Omega \rightarrow [-\infty ,\infty )$ is subharmonic in $\Omega$ if the function $-u$ is superharmonic.

6 Perron solutions

The standing assumptions are described at the end of Sect. 3. We make the convention from now on that the point at infinity, $\infty$, belongs to the boundary $\partial \Omega$ if $\Omega$ is unbounded. Topological notions should therefore be understood with respect to the one-point compactification $X^*:=X\cup \{\infty \}$.

Definition 6.1

Given a function , we let $\mathscr {U}_f(\Omega )$ be the set of all superharmonic functions u in $\Omega$ that are bounded below and such that

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}u(y) \ge f(x) \quad \text {for all }x\in \partial \Omega . \end{aligned}$$

Then the upper Perron solution of f is defined by

Similarly, we let $\mathscr {L}_f(\Omega )$ be the set of all subharmonic functions v in $\Omega$ that are bounded above and such that

$$\begin{aligned} \limsup _{\Omega \ni y\rightarrow x}v(y) \le f(x) \quad \text {for all }x\in \partial \Omega , \end{aligned}$$

and define the lower Perron solution of f by

If , then we let . Moreover, if $P_\Omega f$ is real-valued, then f is said to be resolutive (with respect to $\Omega$). We often write $Pf$ instead of $P_\Omega f$.

Immediate consequences of the above definition are that and that if $f\le h$. It also follows that .

In each component of $\Omega$, is either $p$-harmonic or identically $\pm \infty$, see, e.g., Björn–Björn [3] (their proof applies also to unbounded $\Omega$). Thus Perron solutions are reasonable candidates for solutions of the Dirichlet problem.

The following theorem extends the comparison principle, which is fundamental for the nonlinear potential theory of superharmonic functions, and also plays an important role for the Perron method.

Theorem 6.2

If u is superharmonic and v is subharmonic in $\Omega$, then $v\le u$ in $\Omega$ whenever

$$\begin{aligned} \infty \ne \limsup _{\Omega \ni y\rightarrow x}v(y) \le \liminf _{\Omega \ni y\rightarrow x}u(y) \ne -\infty \end{aligned}$$

(6.1)

for all $x\in \partial \Omega$ (i.e., also for $x=\infty$ if $\Omega$ is unbounded ).

Corollary 6.3

If , then .

Proof of Theorem 6.2

Fix $\varepsilon >0$. For each $x\in \partial \Omega$, it follows from (6.1) that

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}(u(y)-v(y)) \ge \liminf _{\Omega \ni y\rightarrow x}u(y) - \limsup _{\Omega \ni y\rightarrow x}v(y) \ge 0, \end{aligned}$$

and hence there is an open set $U_x\subset X^*$ such that $x\in U_x$ and

$$\begin{aligned} u-v \ge -\varepsilon \quad \text {in }U_x\cap \Omega . \end{aligned}$$

Let $\Omega _1,\Omega _2,\dots$ be open sets such that $\Omega _1\Subset \Omega _2\Subset \cdots \Subset \Omega =\bigcup _{k=1}^\infty \Omega _k$. Then

$$\begin{aligned} \overline{\Omega }\subset \bigcup _{k=1}^\infty \Omega _k \;\cup \bigcup _{x\in \partial \Omega }U_x. \end{aligned}$$

Since $\overline{\Omega }$ is compact (with respect to the topology of $X^*$), there exist integers $k>1/\varepsilon$ and N such that

$$\begin{aligned} \overline{\Omega }\subset \Omega _k\cup U_{x_1}\cup \cdots \cup U_{x_N}. \end{aligned}$$

It follows that $v\le u+\varepsilon$ on $\partial \Omega _k$. Since v is upper semicontinuous (and does not take the value $\infty$), it follows that there is a decreasing sequence $\{\varphi _j\}_{j=1}^\infty \subset {{\mathrm{Lip}}}(\overline{\Omega }_k)$ such that $\varphi _j\rightarrow v$ on $\overline{\Omega }_k$ as $j\rightarrow \infty$ (see, e.g., Proposition 1.12 in Björn–Björn [3]).

Since $u+\varepsilon$ is lower semicontinuous, the compactness of $\partial \Omega _k$ shows that there exists an integer M such that $\varphi _M\le u+\varepsilon$ on $\partial \Omega _k$, and, by (c) in Definition 5.4, also that $H_{\Omega _k}\varphi _M\le u+\varepsilon$ in $\Omega _k$. Similarly, $v\le H_{\Omega _k}\varphi _M$, and thus $v\le u+\varepsilon$ in $\Omega _k$. Letting $\varepsilon \rightarrow 0$ (and hence letting $k\rightarrow \infty$) implies that $v\le u$ in $\Omega$.$\square$

7 Resolutivity of functions on $\partial \Omega$

In addition to the standing assumptions described at the end of Sect. 3, we assume that $\Omega$ is $p$ -parabolic if $\Omega$ is unbounded (see Definition 4.1). For the convention about the point at infinity, see the beginning of Sect. 6.

When Björn–Björn–Shanmugalingam [9] extended the Perron method to the Mazurkiewicz boundary of bounded domains that are finitely connected at the boundary, they introduced a new capacity, , adapted to the topology that connects the domain to its Mazurkiewicz boundary. They also used the new capacity to define -quasicontinuous functions. By using , which is smaller than the usual Sobolev capacity (see the appendix of [9]), we allow for perturbations on larger sets and we obtain resolutivity for more functions.

Definition 7.1

The -capacity of a set $E\subset \overline{\Omega }$ is the number

where $\mathscr {V}_E$ is the family of all functions $u\in N^{1,p}(\Omega )$ that satisfy both $u(x)\ge 1$ for all $x\in E\cap \Omega$ and

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}u(y) \ge 1 \quad \text {for all }x\in E\cap \partial \Omega . \end{aligned}$$

(7.1)

When a property holds for all points except for points in a set of -capacity zero, it is said to hold -quasieverywhere (or -q.e. for short).

If $E\subset \Omega$, then condition (7.1) becomes empty and .

The capacity shares several properties with the Sobolev capacity, e.g., monotonicity and countable subadditivity. Moreover, is an outer capacity, i.e., if $E\subset \overline{\Omega }$, then

These results are proved in Björn–Björn–Shanmugalingam [9] (a slightly modified version of their proof that is outer is valid in our setting as well).

To prove Theorem 7.5, we need the following version of Lemma 5.3 in Björn–Björn–Shanmugalingam [7].

Lemma 7.2

Assume that $\{U_k\}_{k=1}^\infty$ is a decreasing sequence of relatively open subsets of $\overline{\Omega }$ with . Then there exists a sequence of nonnegative functions $\{\psi _j\}_{j=1}^\infty$ that decreases to zero q.e. in $\Omega$, such that $\Vert \psi _j\Vert _{N^{1,p}(\Omega )}<2^{-j}$ and $\psi _j\ge k-j$ in $U_k\cap \Omega$.

Proof

For each $k=1,2,\dots$, there exists a nonnegative function $u_k$ such that $u_k=1$ in $U_k\cap \Omega$ and $\Vert u_k\Vert _{N^{1,p}(\Omega )}<2^{-k}$ because . Letting

$$\begin{aligned} \psi _j = \sum _{k=j+1}^\infty u_k, \quad j=1,2,\dots , \end{aligned}$$

yields a decreasing sequence of nonnegative functions such that $\Vert \psi _j\Vert _{N^{1,p}(\Omega )}<2^{-j}$ and $\psi _j\ge k-j$ in $U_k\cap \Omega$. Corollary 3.9 in Shanmugalingam [32] implies the existence of a subsequence of $\{\psi _j\}_{j=1}^\infty$ that converges to zero q.e. in $\Omega$, and since $\{\psi _j\}_{j=1}^\infty$ is nonnegative and decreasing, this shows that $\{\psi _j\}_{j=1}^\infty$ decreases to zero q.e. in $\Omega$.$\square$

Definition 7.3

Let f be an extended real-valued function defined on $\overline{\Omega }{{\setminus }}\{\infty \}$. We say that f is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ if for every $\varepsilon >0$ there is a relatively open subset U of $\overline{\Omega }{{\setminus }}\{\infty \}$ with such that the restriction of f to $(\overline{\Omega }{{\setminus }}\{\infty \}){\setminus }U$ is continuous and real-valued.

Since the -capacity is smaller than the Sobolev capacity (which is used to define quasicontinuity), it follows that quasicontinuous functions are also -quasicontinuous.

Proposition 7.4

If is a function such that $f=0$ q.e. on $\partial \Omega {\setminus }\{\infty \}$ and $f|_\Omega \in D^p_0(\Omega )$, then f is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$.

Proof

Extend f to X by letting f be equal to zero outside $\overline{\Omega }$ so that $f\in D^p(X)$. Then $f\in N^{1,p}_\mathrm{loc}(X)$ by Proposition 4.14 in Björn–Björn [3], and hence Theorem 1.1 in Björn–Björn–Shanmugalingam [8] asserts that f is quasicontinuous on X, and therefore -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$.$\square$

The following is the main result of this paper.

Theorem 7.5

Assume that is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ and such that $f|_\Omega \in D^p(\Omega )$, which in particular hold if $f\in D^p(X)$. Then f is resolutive with respect to $\Omega$ and $Pf=Hf$.

To see that $p$-parabolicity is needed in Theorem 7.5 if $\Omega$ is unbounded, let $n>p$ and let , where B is the open unit ball centered at the origin. Then $\Omega$ is $p$-hyperbolic. Furthermore, let

$$\begin{aligned} f(x)=|x|^{\tfrac{p-n}{p-1}}, \quad x\in \overline{\Omega }. \end{aligned}$$

Then f satisfies the hypothesis of Theorem 7.5. Because $f\equiv 1$ on $\partial B$ and the $p$-harmonic extension does not consider the point at infinity, it is clear that $Hf\equiv 1$. However, $Pf\equiv f$, since f is in fact $p$-harmonic (it is easy to verify that f is a solution of the $p$-Laplace Eq. (1.1)) and continuous on $\overline{\Omega }$, and hence $f\in \mathscr {U}_f(\Omega )$ and $f\in \mathscr {L}_f(\Omega )$, which implies that .

Proof of Theorem 7.5

Suppose that $\Omega$ is unbounded and $p$-parabolic. Let $\{K_j\}_{j=1}^\infty$ be an increasing sequence of compact sets such that

$$\begin{aligned} K_1\Subset K_2\Subset \cdots \Subset \Omega = \bigcup _{j=1}^\infty K_j \end{aligned}$$

and let $x_0\in X$. For each $j=1,2,\dots$, we can find a function $u_j\in D^p(\Omega )$ such that $\chi _{K_j}\le u_j\le 1$, $u_j=0$ in $\Omega {\setminus }B_j$ for some ball $B_j\supset K_j$ centered at $x_0$, and

$$\begin{aligned} \Vert g_{u_j}\Vert _{L^{p}(\Omega )}<2^{-j}. \end{aligned}$$

(7.2)

Let

$$\begin{aligned} \xi _j = \sum _{k=j}^\infty (1-u_k), \quad j=1,2,\dots . \end{aligned}$$

(7.3)

Then $\xi _j\ge 0$ and

$$\begin{aligned} \Vert g_{\xi _j}\Vert _{L^{p}(\Omega )} \le \sum _{k=j}^\infty \Vert g_{u_k}\Vert _{L^{p}(\Omega )} < \sum _{k=j}^\infty 2^{-k} = 2^{1-j}. \end{aligned}$$

(7.4)

Let $\Omega _j=\bigcup _{n=1}^{\,j}B_n\cap \Omega$, $j=1,2,\dots$. Then $\Omega _1\subset \Omega _2\subset \cdots \subset \Omega =\bigcup _{j=1}^\infty \Omega _j$. Since $u_j=0$ in $\Omega {\setminus }\Omega _j$, it is easy to see that

$$\begin{aligned} \lim _{\Omega \ni y\rightarrow \infty }\xi _j(y) = \infty \quad \text {for all }j=1,2,\dots . \end{aligned}$$

(7.5)

Furthermore, since $\{\xi _j\}_{j=1}^\infty$ is decreasing and $\xi _j=0$ on $K_j$ for each $j=1,2,\dots$, it follows that $\{\xi _j\}_{j=1}^\infty$ decreases to zero in $\Omega$.

On the other hand, if $\Omega$ is bounded, then we let $\xi _j\equiv 0$ in $\Omega$, $j=1,2,\dots$.

The $p$-harmonic extension $Hf$ is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ (when we consider $Hf$ to be equal to f on $\partial \Omega$), since Proposition 7.4 asserts that $Hf-f$ is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ as $(Hf-f)|_\Omega \in D^p_0(\Omega )$. We can therefore find a decreasing sequence $\{U_k\}_{k=1}^\infty$ of relatively open subsets of $\overline{\Omega }{{\setminus }}\{\infty \}$ with and such that the restriction of $Hf$ to $(\overline{\Omega }{{\setminus }}\{\infty \}){\setminus }U_k$ is continuous.

Now we derive that q.e. in $\Omega$ if f is bounded from below. Without loss of generality, we may as well assume that $f\ge 0$. Then the comparison principle (Lemma 4.7 in Hansevi [16]) implies that $Hf\ge 0$ in $\Omega$.

Consider the sequence of nonnegative functions $\{\psi _j\}_{j=1}^\infty$ given by Lemma 7.2, and define $h_j:\Omega \rightarrow [0,\infty ]$ by letting

$$\begin{aligned} h_j = Hf + \xi _j + \psi _j, \quad j=1,2,\dots . \end{aligned}$$

Then $h_j\in D^p(\Omega )$ and $\{h_j\}_{j=1}^\infty$ decreases to $Hf$ q.e. in $\Omega$.

Let $\varphi _j$ be the lsc-regularized solution of the $\mathscr {K}_{h_j,h_j}$-obstacle problem, $j=1,2,\dots$. By (7.4) and Lemma 7.2,

$$\begin{aligned} \Vert g_{h_j-Hf}\Vert _{L^{p}(\Omega )} \le \Vert g_{\xi _j}\Vert _{L^{p}(\Omega )} + \Vert g_{\psi _j}\Vert _{L^{p}(\Omega )} < 2^{1-j} + 2^{-j} \rightarrow 0 \quad \text {as }j\rightarrow \infty , \end{aligned}$$

and as $Hf$ is a solution of the $\mathscr {K}_{Hf,Hf}$-obstacle problem, it follows from Theorem 3.2 that $\{\varphi _{j}\}_{j=1}^\infty$ decreases to $Hf$ q.e. in $\Omega$. This will be used later in the proof.

Next we show that

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}\varphi _j(y) \ge f(x) \quad \text {for all }x\in \partial \Omega . \end{aligned}$$

(7.6)

Fix a positive integer m and let $\varepsilon =1/m$. By Lemma 7.2,

$$\begin{aligned} h_j(y) \ge \psi _j(y) \ge m \quad \text {for all }y\in U_{m+j}\cap \Omega . \end{aligned}$$

(7.7)

Let $x\in \partial \Omega {\setminus }\{\infty \}$. If $x\notin U_{m+j}$, then as the restriction of $Hf$ to $(\overline{\Omega }{{\setminus }}\{\infty \}){\setminus }U_{m+j}$ is continuous, there is a relative neighborhood $V_x\subset \overline{\Omega }{{\setminus }}\{\infty \}$ of x such that

$$\begin{aligned} h_j(y) \ge Hf(y) \ge Hf(x)-\varepsilon = f(x)-\varepsilon \quad \text {for all }y\in (V_x\cap \Omega ){\setminus }U_{m+j}. \end{aligned}$$

(7.8)

By combining (7.7) and (7.8), we see that for $x\in (\partial \Omega {\setminus }\{\infty \}){\setminus }U_{m+j}$,

$$\begin{aligned} h_j(y) \ge \min \{f(x)-\varepsilon ,m\} \quad \text {for all }y\in V_x\cap \Omega . \end{aligned}$$

(7.9)

On the other hand, if $x\in U_{m+j}$, then we let $V_x=U_{m+j}$, and see that (7.9) holds also in this case due to (7.7). Because $\varphi _j\ge h_j$ q.e. in $\Omega$ and $\varphi _j$ is lsc-regularized, it follows that

$$\begin{aligned} \varphi _j(y) \ge \min \{f(x)-\varepsilon ,m\} \quad \text {for all }y\in V_x\cap \Omega , \end{aligned}$$

and hence

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}\varphi _j(y) \ge \min \{f(x)-\varepsilon ,m\}. \end{aligned}$$

Letting $m\rightarrow \infty$ (and thus letting $\varepsilon \rightarrow 0$) establishes that

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow x}\varphi _j(y) \ge f(x) \quad \text {for all }x\in \partial \Omega {\setminus }\{\infty \}. \end{aligned}$$

Finally, if $\Omega$ is unbounded, then $\varphi _j\ge h_j$ q.e. in $\Omega$ and $h_j\ge \xi _j$ everywhere in $\Omega$. From the lsc-regularity of $\varphi _j$ and (7.5), it follows that

$$\begin{aligned} \liminf _{\Omega \ni y\rightarrow \infty }\varphi _j(y) \ge \lim _{\Omega \ni y\rightarrow \infty }\xi _j(y) = \infty , \end{aligned}$$

and hence we have shown that (7.6) holds.

Since $\varphi _j$ is an lsc-regularized superminimizer, Proposition 7.4 in Kinnunen–Martio [25] asserts that $\varphi _j$ is superharmonic. As $\varphi _j$ is bounded from below and (7.6) holds, it follows that $\varphi _j\in \mathscr {U}_f(\Omega )$, and hence we know that . Because $h_j\in D^p(\Omega )$ and $\{h_j\}_{j=1}^\infty$ decreases to $Hf$ q.e. in $\Omega$, $\Vert g_{h_j-Hf}\Vert _{L^{p}(\Omega )}\rightarrow 0$ as $j\rightarrow \infty$, and $Hf$ is a solution of the $\mathscr {K}_{Hf,Hf}$-obstacle problem, it follows from Theorem 3.2 that $\{\varphi _{j}\}_{j=1}^\infty$ decreases to $Hf$ q.e. in $\Omega$. We therefore conclude that q.e. in $\Omega$ (provided that f is bounded from below).

Now we remove the extra assumption of f being bounded from below, and let $f_k=\max \{f,-k\}$, $k=1,2,\dots$. Then $\{f_k\}_{k=1}^\infty$ is decreasing to f. Proposition 4.14 in Björn–Björn [3] implies that $f\in L^{p}_\mathrm{loc}(\Omega )$. Hence $\mu (\{x\in \Omega :|f(x)|=\infty \})=0$, and therefore $\chi _{\{x\in \Omega \,:\,f(x)<-k\}}\rightarrow 0$ a.e. in $\Omega$ as $k\rightarrow \infty$. Since

$$\begin{aligned} g_{f_k-f} = g_{\max \{0,-f-k\}} = g_f\chi _{\{x\in \Omega \,:\,f(x)<-k\}} \quad \text {a.e. in }\Omega , \end{aligned}$$

implies that $g_{f_k-f}\rightarrow 0$ a.e. in $\Omega$ as $k\rightarrow \infty$, and because $g_f\in L^{p}(\Omega )$ and

$$\begin{aligned} g_{f_k-f} \le g_{f_k}+g_f \le 2g_f \quad \text {a.e. in }\Omega , \end{aligned}$$

it follows by dominated convergence that $g_{f_k-f}\rightarrow 0$ in $L^{p}(\Omega )$ as $k\rightarrow \infty$. Thus Proposition 5.3 asserts that

$$\begin{aligned} Hf_k\rightarrow Hf \quad \text {in }\Omega \text { as }k\rightarrow \infty . \end{aligned}$$

Since $f_k$ is bounded from below, it follows that

As both and $Hf$ are continuous, we conclude that everywhere in $\Omega$. By Corollary 6.3, it follows that

which implies that f is resolutive and that $Pf=Hf$.$\square$

Perron solutions are invariant under perturbation of the function on a set of capacity zero.

Theorem 7.6

Assume that is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ and such that $f|_\Omega \in D^p(\Omega )$, which in particular hold if $f\in D^p(X)$. Assume also that is zero -q.e. on $\partial \Omega {\setminus }\{\infty \}$. Then $f+h$ is resolutive with respect to $\Omega$ and $P(f+h)=Pf$.

Proof

Extend h by zero in $\Omega$ and let $E=\{x\in \overline{\Omega }:h(x)\ne 0\}$. Since is an outer capacity, it follows that given $\varepsilon >0$, we can find a relatively open subset U of $\overline{\Omega }{{\setminus }}\{\infty \}$ with and such that $E\subset U$, and hence h is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$. The subadditivity of the -capacity implies that this is true also for $f+h$.

Since $f+h=f$ in $\Omega$ and $f|_\Omega \in D^p(\Omega )$, we know that $H(f+h)=Hf$. We complete the proof by applying Theorem 7.5 to both f and $f+h$, which shows that $f+h$ is resolutive and that

$$\begin{aligned} P(f+h) = H(f+h) = Hf = Pf. \end{aligned}$$

$\square$

The following uniqueness result is a direct consequence of Theorem 7.6.

Corollary 7.7

Assume that u is bounded and $p$-harmonic in $\Omega$. Assume also that is -quasicontinuous on $\overline{\Omega }{{\setminus }}\{\infty \}$ and such that $f|_\Omega \in D^p(\Omega )$. Then $u=Pf$ in $\Omega$ whenever there exists a set $E\subset \partial \Omega$ with such that

$$\begin{aligned} \lim _{\Omega \ni y\rightarrow x}u(y)=f(x) \quad \text {for all }x\in \partial \Omega {\setminus }E. \end{aligned}$$

Proof

Since , Theorem 7.6 applies to f and $h:=\infty \chi _E$ (and clearly also to f and $-h$), and because $u\in \mathscr {U}_{f-h}(\Omega )$ and $u\in \mathscr {L}_{f+h}(\Omega )$ (since u is bounded), it follows that

$\square$

The obtained resolutivity results can now be extended to continuous functions. Björn–Björn–Shanmugalingam [7],[9] proved the following result for bounded domains.

Theorem 7.8

If $f\in C(\partial \Omega )$ and is zero -q.e. on $\partial \Omega {\setminus }\{\infty \}$, then f and $f+h$ are resolutive with respect to $\Omega$ and $P(f+h)=Pf$.

Proof

We start by choosing a point $x_0\in \partial \Omega$. If $\Omega$ is unbounded, then we let $x_0=\infty$. Let $\alpha =f(x_0)\in \mathbf {R}$ and let j be a positive integer. Since $f\in C(\partial \Omega )$, there exists a compact set $K_j\subset X$ such that $|f(x)-\alpha |<1/3j$ for all $x\in \partial \Omega {\setminus }K_j$. Let

$$\begin{aligned} K'_j=\{x\in X:{{\mathrm{dist}}}(x,K_j)\le 1\}. \end{aligned}$$

We can find functions $\varphi _j\in {{{\mathrm{Lip}}}_c}(X)$ such that $|\varphi _j-f|\le 1/3j$ on $\partial \Omega \cap K'_j$. Let $f_j=(\varphi _j-\alpha )\eta _j+\alpha$, where

$$\begin{aligned} \eta _j(x) := {\left\{ \begin{array}{ll} 1, &{} x\in K_j, \\ 1-{{\mathrm{dist}}}(x,K_j), &{} x\in K'_j{\setminus }K_j, \\ 0, &{} x\in X{\setminus }K'_j. \end{array}\right. } \end{aligned}$$

Since $f_j$ is Lipschitz on X and $f_j=\alpha$ outside $K'_j$, it follows that $f_j\in D^p(X)$.

Let $x\in \partial \Omega$. Then $|f_j(x)-f(x)|\le 1/3j$ whenever $x\notin K'_j{\setminus }K_j$. Otherwise it follows that

$$\begin{aligned} |f_j(x)-f(x)|&= |(\varphi _j(x)-\alpha )\eta _j(x)+\alpha -f(x)| \le |\varphi _j(x)-\alpha )|+|\alpha -f(x)| \\&\le |\varphi _j(x)-f(x)|+2|f(x)-\alpha | < \frac{1}{j}, \end{aligned}$$

and thus we know that $f-1/j \le f_j\le f+1/j$ on $\partial \Omega$. It follows directly from Definition 6.1 that , and we also get corresponding inequalities for , , and .

Theorem 7.6 asserts that $f_j$ and $f_j+h$ are resolutive and that $P(f_j+h)=Pf_j$. It follows that

(7.10)

Applying Corollary 6.3 to (7.10) yields . Letting $j\rightarrow \infty$ shows that f is resolutive. Similarly, we can see that also $f+h$ is resolutive.

Finally, we have

(7.11)

Interchanging and with and , respectively, in (7.11) yields $P(f+h)-Pf\ge -2/j$, and hence $|P(f+h)-Pf|<2/j$. Letting $j\rightarrow \infty$ shows that $P(f+h)=Pf$.$\square$

We conclude this paper with the following uniqueness result, corresponding to Corollary 7.7, that follows directly from Theorem 7.8. The proof is identical to the proof of Corollary 7.7, except for applying Theorem 7.8 (instead of Theorem 7.6).

Corollary 7.9

Assume that u is bounded and $p$-harmonic in $\Omega$. If $f\in C(\partial \Omega )$ and there is a set $E\subset \partial \Omega$ with such that

$$\begin{aligned} \lim _{\Omega \ni y\rightarrow x}u(y)=f(x) \quad \text {for all }x\in \partial \Omega {\setminus }E, \end{aligned}$$

then $u=Pf$ in $\Omega$.