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

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## 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.

## Keywords

Dirichlet problem Obstacle problem \(p\)-Harmonic function \(p\)-Parabolic set Perron method Quasicontinuity## Mathematics Subject Classification

Primary 31E05 Secondary 31C45## 1 Introduction

*-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.

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

*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,

*X*.

*doubling*if there exists a constant \(C\ge 1\) such that

*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 Open image in new window (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 Open image in new window . 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

*upper gradient*of a function Open image in new window whenever

*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 Open image in new window 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

*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

*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

*Sobolev*)

*capacity*(with respect to

*A*) of \(E\subset A\) is the number

*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

*E*and

*A*of

*X*, where

*A*is measurable, the

*Dirichlet space with zero boundary values in*\(A{\setminus }E\), is

*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

*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

*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

*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.

*X*with zero \(p\)-modulus, such that for every curve \(\gamma \notin \Gamma \), it follows that

*u*, and let

*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

*g*is a \(p\)-weak upper gradient of

*u*, and thus \(u\in D^p(X)\), since

### Definition 2.8

*X*supports a (

*q*,

*p*)-

*Poincaré inequality*if there exist constants, \(C>0\) and \(\lambda \ge 1\) (the dilation constant), such that

*u*on

*X*, and all upper gradients

*g*of

*u*.

In (2.3), we have used the convenient notation Open image in new window . 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

*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

*lsc-regularization*of

*u*is the (lower semicontinuous) function \(u^*\) defined by

### 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 \(B':=\tfrac{1}{2}B\).

*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

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

*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 \({C_p}(\Omega {\setminus }E)=0\).

*E*, we know that \(v<\psi _j\) in \(U_j\), and because \(g_{\psi -v}=0\) a.e. 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

*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

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

*-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

*-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.)

*R*sufficiently large so that \(K\subset B:=B(0,R)\). Let

*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

*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

## 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

*superminimizer*in \(\Omega \) if

*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

*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 \(\partial V'\).

*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

*u*in \(\Omega \) that are bounded below and such that

*upper Perron solution*of

*f*is defined bySimilarly, we let \(\mathscr {L}_f(\Omega )\) be the set of all subharmonic functions

*v*in \(\Omega \) that are bounded above and such that

*lower Perron solution*of

*f*byIf Open image in new window , then we let Open image in new window . 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 Open image in new window and that Open image in new window if \(f\le h\). It also follows that Open image in new window .

In each component of \(\Omega \), Open image in new window 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

*u*is superharmonic and

*v*is subharmonic in \(\Omega \), then \(v\le u\) in \(\Omega \) whenever

### Corollary 6.3

If Open image in new window , then Open image in new window .

### Proof of Theorem 6.2

*N*such that

*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, Open image in new window , adapted to the topology that connects the domain to its Mazurkiewicz boundary. They also used the new capacity to define Open image in new window -quasicontinuous functions. By using Open image in new window , 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

*capacity*of a set \(E\subset \overline{\Omega }\) is the numberwhere \(\mathscr {V}_E\) 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

*-quasieverywhere*(or Open image in new window -

*q.e.*for short).

If \(E\subset \Omega \), then condition (7.1) becomes empty and Open image in new window .

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 Open image in new window . 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

### Definition 7.3

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

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

### Proposition 7.4

If Open image in new window 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 Open image in new window -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 Open image in new window -quasicontinuous on \(\overline{\Omega }{{\setminus }}\{\infty \}\).\(\square \)

The following is the main result of this paper.

### Theorem 7.5

Assume that Open image in new window is Open image in new window -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 \(Pf=Hf\).

*B*is the open unit ball centered at the origin. Then \(\Omega \) is \(p\)-hyperbolic. Furthermore, let

*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 Open image in new window .

### Proof of Theorem 7.5

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 Open image in new window -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 Open image in new window -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 Open image in new window and such that the restriction of \(Hf\) to \((\overline{\Omega }{{\setminus }}\{\infty \}){\setminus }U_k\) is continuous.

Now we derive that Open image in new window 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 \(\Omega \).

*m*and let \(\varepsilon =1/m\). By Lemma 7.2,

*x*such that

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 Open image in new window . 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 Open image in new window q.e. in \(\Omega \) (provided that *f* is bounded from below).

*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

*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 Open image in new window is Open image in new window -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 Open image in new window is zero Open image in new window -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 Open image in new window is an outer capacity, it follows that given \(\varepsilon >0\), we can find a relatively open subset *U* of \(\overline{\Omega }{{\setminus }}\{\infty \}\) with Open image in new window and such that \(E\subset U\), and hence *h* is Open image in new window -quasicontinuous on \(\overline{\Omega }{{\setminus }}\{\infty \}\). The subadditivity of the Open image in new window -capacity implies that this is true also for \(f+h\).

*f*and \(f+h\), which shows that \(f+h\) is resolutive and that

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

### Corollary 7.7

*u*is bounded and \(p\)-harmonic in \(\Omega \). Assume also that Open image in new window is Open image in new window -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 Open image in new window such that

### Proof

*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 Open image in new window is zero Open image in new window -q.e. on \(\partial \Omega {\setminus }\{\infty \}\), then *f* and \(f+h\) are resolutive with respect to \(\Omega \) and \(P(f+h)=Pf\).

### Proof

*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

*X*and \(f_j=\alpha \) outside \(K'_j\), it follows that \(f_j\in D^p(X)\).

*f*is resolutive. Similarly, we can see that also \(f+h\) is resolutive.

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

*u*is bounded and \(p\)-harmonic in \(\Omega \). If \(f\in C(\partial \Omega )\) and there is a set \(E\subset \partial \Omega \) with Open image in new window such that

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