Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.


Introduction
Let Ω ⊂ R n be a nonempty bounded open set and let f ∈ C(∂Ω). The Perron method (introduced on R in 1923 by Perron [47] and independently by Remak [48]) provides a unique function Pf that is harmonic in Ω and takes the boundary values f in a weak sense, i.e., Pf is a solution of the Dirichlet problem for the Laplace equation. A point x ∈ ∂Ω is regular if lim Ω y→x Pf (y) = f (x ) for all f ∈ C(∂Ω). Regular boundary points were characterized in 1924 by the so-called Wiener criterion and in terms of barriers, by Wiener [51] and Lebesgue [42], respectively.
A nonlinear analogue is to consider the Dirichlet problem for p-harmonic functions, which are solutions of the p-Laplace equation ∆p u := div(|∇u| p− ∇u) = , < p < ∞. This leads to a nonlinear potential theory, which has been studied since the 1960s, initially for R n , and later generalized to weighted R n , Riemannian manifolds, and other settings. For an extensive treatment in weighted R n , the reader may consult the monograph Heinonen-Kilpeläinen-Martio [33].
More recently, nonlinear potential theory has been developed on complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, < p < ∞, see, e.g., the monograph Björn-Björn [11] and the references therein. The Perron method was extended to this setting by Björn-Björn-Shanmugalingam [17] for bounded open sets and Hansevi [30] for unbounded open sets. Note that when R n is equipped with a measure dµ = w dx, our assumptions on µ are equivalent to assuming that w is p-admissible as in [33], and our de nition of p-harmonic functions is equivalent to the one in [33], see Appendix A.2 in [11].
Boundary regularity for p-harmonic functions on metric spaces was rst studied by Björn [22] and Björn-MacManus-Shanmugalingam [26]. Björn-Björn-Shanmugalingam [16] obtained the Kellogg property saying that the set of irregular boundary points has capacity zero. Björn-Björn [9] obtained the barrier characterization, showed that regularity is a local property, and also studied boundary regularity for obstacle problems showing that they have essentially the same regular boundary points as the Dirichlet problem. These studies were pursued on bounded open sets.
In this paper we study boundary regularity for p-harmonic functions on unbounded sets Ω in metric spaces X (satisfying the assumptions above). The boundary ∂Ω is considered within the one-point compactication X * = X∪{∞} of X, and is in particular always compact. We also impose the condition that the capacity Cp(X \ Ω) > .
In this generality it is not known if continuous functions f are resolutive, i.e., whether the upper and lower Perron solutions P Ω f and P Ω f coincide. We therefore make the following de nition.

De nition 1.1. A boundary point
With a few exceptions, we limit ourselves to studying regularity at nite boundary points.
Our main results can be summarized as follows. Once the barrier characterization (b) has been shown, the locality (c) follows easily. Our proofs of these facts are however intertwined, and even though we use that these facts are already known to hold for bounded open sets, our proof is signi cantly longer than the proof in Björn-Björn [9] (or [11]). On the other hand, once (c) has been deduced, (a) follows from its version for bounded domains. Several other characterizations of regularity are also given, see Sections 5 and 9. We also study the associated (one-sided) obstacle problem with prescribed boundary values f and an obstacle ψ, where the solution is required to be greater than or equal to ψ q.e. in Ω (i.e., up to a set of capacity zero). This problem obviously reduces to the Dirichlet problem for p-harmonic functions when ψ ≡ −∞. In Section 8, we show that if x ∈ ∂Ω \ {∞} is a regular boundary point and f is continuous at x , then the solution u of the obstacle problem attains the boundary value at x in the limit, i.e., lim Ω y→x if and only if Cp-ess lim sup Ω y→x ψ(y) ≤ f (x ). The results in Section 8 generalize the corresponding results in Björn-Björn [9] to unbounded sets, with some improvements also for bounded sets. These results are new even on unweighted R n .
Boundary regularity for p-harmonic functions on R n was rst studied by Maz ya [45] who obtained the su ciency part of the Wiener criterion in 1970. Later on the full Wiener criterion has been obtained in various situations including weighted R n and for Cheeger p-harmonic functions on metric spaces, see [37], [43], [46], and [23]. The full Wiener criterion for p-harmonic functions de ned using upper gradients remains open even for bounded open sets in metric spaces (satisfying the assumptions above), but the su ciency has been obtained, see [26] and [24], and a weaker necessity condition, see [25]. An important consequence of Theorem 1.2 (c) is that the su ciency part of the Wiener criterion holds for unbounded open sets. (Hence also the porosity-type conditions in Corollary 11.25 in [11] imply regularity for unbounded open sets.) In nonlinear potential theory, the Kellogg property was rst obtained by Hedberg [31] and Hedberg-Wol [32] on R n (see also Kilpeläinen [36]). It was extended to homogeneous spaces by Vodop yanov [50], to weighted R n by Heinonen-Kilpeläinen-Martio [33], to subelliptic equations by , and to bounded open sets in metric spaces by Björn-Björn-Shanmugalingam [16]. In some of these papers boundary regularity was de ned in a di erent way than through Perron solutions, but these de nitions are now known to be equivalent. See also [1] and [41] for the Kellogg property for p(·)-harmonic functions on R n .
Granlund-Lindqvist-Martio [28] were the rst to de ne boundary regularity using Perron solutions for p-harmonic functions, p ≠ . They studied the case p = n in R n and obtained the barrier characterization in this case for bounded open sets. Kilpeläinen [36] generalized the barrier characterization to p > and also deduced resolutivity for continuous functions. The results in [36] covered both bounded and unbounded open sets in unweighted R n , and were extended to weighted R n (with a p-admissible measure) in Heinonen-Kilpeläinen-Martio [33,Chapter 9].
Very recently, Björn-Björn-Li [14] studied Perron solutions and boundary regular for p-harmonic functions on unbounded open sets in Ahlfors regular metric spaces. There is some overlap with the results in this paper, but it is not substantial and here we consider more general metric spaces than in [14].

Notation and preliminaries
We assume that (X, d, µ) is a metric measure space (which we simply refer to as X) equipped with a metric d and a positive complete Borel measure µ such that < µ(B) < ∞ for every ball B ⊂ X. It follows that X is separable, second countable, and Lindelöf (these properties are equivalent for metric spaces). For balls The σ-algebra on which µ is de ned is the completion of the Borel σ-algebra. We also assume that < p < ∞. Later we will impose further requirements on the space and on the measure. We will keep the discussion short, see the monographs Björn-Björn [11] and Heinonen-Koskela-Shanmugalingam-Tyson [35] for proofs, further discussion, and references on the topics in this section.
The measure µ is doubling if there exists a constant C ≥ such that < µ( B) ≤ Cµ(B) < ∞ for every ball B ⊂ X. A metric space is proper if all bounded closed subsets are compact, and this is in particular true if the metric space is complete and the measure is doubling. We use the standard notation f+ = max{f , } and f− = max{−f , }, and let χ E denote the characteristic function of the set E. Semicontinuous functions are allowed to take values in R := [−∞, ∞], whereas continuous functions will be assumed to be real-valued unless otherwise stated. For us, a curve in X is a recti able nonconstant continuous mapping from a compact interval into X, and it can thus be parametrized by its arc length ds.
By saying that a property holds for p-almost every curve, we mean that it fails only for a curve family Γ with zero p-modulus, i.e., there exists a nonnegative ρ ∈ L p (X) such that γ ρ ds = ∞ for every curve γ ∈ Γ.
for p-almost every curve γ : [ , lγ] → X, where we use the convention that the left-hand side is ∞ when at least one of the terms on the left-hand side is in nite.
Shanmugalingam [49] used p-weak upper gradients to de ne so-called Newtonian spaces.

De nition 2.2.
The Newtonian space on X, denoted N ,p (X), is the space of all extended real-valued functions f ∈ L p (X) such that where the in mum is taken over all p-weak upper gradients g of f .
Shanmugalingam [49] proved that the associated quotient space In this paper we assume that functions in N ,p (X) are de ned everywhere (with values in R), not just up to an equivalence class. This is important, in particular for the de nition of p-weak upper gradients to make sense.

De nition 2.
3. An everywhere de ned, measurable, extended real-valued function on X belongs to the Dirichlet space D p (X) if it has a p-weak upper gradient in L p (X).
A measurable set A ⊂ X can be considered to be a metric space in its own right (with the restriction of d and µ to A). Thus the Newtonian space N ,p (A) and the Dirichlet space D p (A) are also given by De nitions 2.2 and 2.3, respectively. If X is proper and Ω ⊂ X is open, then f ∈ N ,p loc (Ω) if and only if f ∈ N ,p (V) for every open V such that V is a compact subset of Ω, and similarly for D p loc (Ω). If f ∈ D p loc (X), then f has a minimal p-weak upper gradient g f ∈ L p loc (X) in the sense that g f ≤ g a.e. for all p-weak upper gradients g ∈ L p loc (X) of f . De nition 2.4. The (Sobolev) capacity of a set E ⊂ X is the number where the in mum is taken over all f ∈ N ,p (X) such that f ≥ on 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, and it is the correct gauge for distinguishing between two Newtonian functions: If f ∈ N ,p (X), then f ∼ h if and only if f = h q.e. Moreover, if f , h ∈ N ,p loc (X) and f = h a.e., then There is a subtle, but important, di erence to the standard theory on R n where the equivalence classes in the Sobolev space are (usually) up to sets of measure zero, while here the equivalence classes in N ,p (X) are up to sets of capacity zero. Moreover, under the assumptions from the beginning of Section 3, the functions in N ,p loc (X) and N ,p loc (Ω) are quasicontinuous. On weighted R n , the Newtonian space N ,p (X) therefore corresponds to the re ned Sobolev space mentioned on p. 96 in Heinonen-Kilpeläinen-Martio [33].
In order to be able to compare boundary values of Dirichlet and Newtonian functions, we need the following spaces.

De nition 2.5. For subsets E and
The Newtonian space with zero boundary values N ,p (E; A) is de ned analogously. We let D p (E) and N ,p (E) denote D p (E; X) and N ,p (E; X), respectively.
The condition "f = in A \ E" can in fact be replaced by "f = q.e. in A \ E" without changing the obtained spaces.
De nition 2.6. We say that X supports a p-Poincaré inequality if there exist constants, C > and λ ≥ (the dilation constant), such that for all balls B ⊂ X, all integrable functions f on X, and all p-weak upper gradients g of f .
In (2.1), we have used the convenient notation f B := B f dµ := µ(B) B f dµ. Requiring a Poincaré inequality to hold is one way of making it possible to control functions by their p-weak upper gradients.

The obstacle problem and p-harmonic functions
We assume from now on that < p < ∞, that X is a complete metric measure space supporting a p-Poincaré inequality, that µ is doubling, and that Ω ⊂ X is a nonempty (possibly unbounded) open subset such that Cp(X \ Ω) > .
One of our fundamental tools is the following obstacle problem, which in this generality was rst considered by Hansevi [29].
It was proved in Hansevi [29,Theorem 3.4] that the K ψ,f -obstacle problem has a unique (up to sets of capacity zero) solution whenever K ψ,f is nonempty. Furthermore, in this case, there is a unique lsc-regularized solution of the K ψ,f -obstacle problem, by Theorem 4.1 in [29].

De nition 3.2. A function
If (3.1) is only required to hold for all nonnegative φ ∈ N ,p (Ω), then u is a superminimizer. Moreover, a function is p-harmonic if it is a continuous minimizer.  [11]. Note that N ,p loc (Ω) = D p loc (Ω) (under our assumptions), by Proposition 4.14 in [11].
is continuous as an extended real-valued function, and K ψ,f ≠ ∅, then the lscregularized solution of the K ψ,f -obstacle problem is continuous, by Theorem 4.4 in Hansevi [29]. Thus the following de nition makes sense. It was rst used in this generality by Hansevi [29,De nition 4.6].
When proving Theorem 9.2 we will need the following generalization of Proposition 7.15 in [11], which may be of independent interest. The boundedness assumption cannot be dropped. To see this, let < p < n and Proof. Corollary 9.10 in [11] implies that u is superharmonic in V, and hence it follows from Corollary 7.9 and Theorem 7.14 in Kinnunen-Martio [38] (or Corollary 9.6 and Theorem 9.12 in [11]) that u is an lsc-regularized , and since X supports a p-Friedrichs inequality (De nition 2.6 in Björn-Björn [12]) and V is bounded, we see that φ ∈ N ,p (V), by Proposition 2.7 in [12].
Hence u is the lsc-regularized solution of the Ku,u(V)-obstacle problem.

Perron solutions
In addition to the assumptions given at the beginning of Section 3, from now on we make the convention that if Ω is unbounded, then the point at in nity, ∞, belongs to the boundary ∂Ω. Topological notions should therefore be understood with respect to the one-point compacti cation X * := X ∪ {∞}.
Note that this convention does not a ect any of the de nitions in Sections 2 or 3, as ∞ is not added to X (it is added solely to ∂Ω).
Since continuous functions are assumed to be real-valued, every function in C(∂Ω) is bounded even if Ω is unbounded.

De nition 4.1.
Given a function f : ∂Ω → R, let U f (Ω) be the collection of all functions u that are superharmonic in Ω, bounded from below, and such that The upper Perron solution of f is de ned by Let L f (Ω) be the collection of all functions v that are subharmonic in Ω, bounded from above, and such that lim sup The lower Perron solution of f is de ned by If P Ω f = P Ω f , then we denote the common value by P Ω f . Moreover, if P Ω f is real-valued, then f is said to be resolutive (with respect to Ω). We often write Pf instead of P Ω f , and similarly for Pf and Pf .

De nition 4.2.
Assume that Ω is unbounded. Then Ω is p-parabolic if for every compact K ⊂ Ω, there exist functions u j ∈ N ,p (Ω) such that u j ≥ on K for all j = , , ... , and For examples of p-parabolic sets, see, e.g., Hansevi [30]. The main reason for introducing p-parabolic sets in [30] was to be able to obtain resolutivity results. We formulate this in a special case, which will be su cient for us. Recall from Section 2 that under our standing assumptions, the equivalence classes in D p (X) only contain quasicontinuous representatives. This fact is crucial for the validity of the second part of Theorem 4.3.

Boundary regularity
For unbounded p-hyperbolic sets resolutivity of continuous functions is not known, which will be an obstacle to overcome in some of our proofs below. This explains why regularity was de ned using upper Perron solutions in De nition 1.1. In our de nition it is not required that Ω is bounded, but if it is, then it follows from Theorem 4.3 that it coincides with the de nitions of regularity in Björn-Björn-Shanmugalingam [16], [17], and Björn-Björn [9], [11], where regularity is de ned using Pf or Hf . Thus we can use the boundary regularity results from these papers when considering bounded sets. Since for all f ∈ C(∂Ω).
The particular form of dx is not important. The same characterization holds for any nonnegative continuous function d : X * → [ , ∞) which is zero at and only at x . For the later applications in this paper it will also be important that d ∈ D p (X), which is true for dx .
Letting α → f (x ) yields the desired result.  (e) ⇒ (a) This is analogous to the proof of (c) ⇒ (d).
We will mainly concentrate on the regularity of nite points in the rest of the paper.

Barrier characterization of regular points
De nition 6.1. A function u is a barrier (with respect to Ω) at x ∈ ∂Ω if (i) u is superharmonic in Ω; (ii) lim Ω y→x u(y) = ; (iii) lim inf Ω y→x u(y) > for every x ∈ ∂Ω \ {x }.
Superharmonic functions satisfy the strong minimum principle, i.e., if u is superharmonic and attains its minimum in some component G of Ω, then u| G is constant (see Theorem 9.13 in [11]). This implies that a barrier is always nonnegative, and furthermore, that a barrier is positive if every component G ⊂ Ω has a boundary point in ∂G \ {x }.  d(x, x ) for all x ∈ Ω ∩ B.
We rst show that parts (c) to (f) are equivalent, and that (c) ⇒ (b) ⇒ (a). To conclude the proof we then show that (a) ⇒ (c), which is by far the most complicated part of the proof.
In the next section, we will use this characterization to obtain the Kellogg property for unbounded sets. In the proof below we will however need the Kellogg property for bounded sets, which for metric spaces is due to Björn-Björn-Shanmugalingam [16,Theorem 3.9]. (See alternatively [11,Theorem 10.5]. ) We do not know if the corresponding characterizations of regularity at ∞ holds, but the proof below shows that the existence of a barrier implies regularity also at ∞.
Proof. (c) ⇒ (e) Suppose that u is a positive barrier with respect to Ω at x . Then u is superharmonic in Ω ∩ B, by Corollary 9.10 in [11]. Clearly, u satis es condition (ii) in De nition 6.1 with respect to Ω ∩ B, and since u is positive and lower semicontinuous in Ω, u also satis es condition (iii) in De nition 6.1 with respect to Ω ∩ B. Thus u is a positive barrier with respect to Ω ∩ B at x .

(f) ⇒ (c) Suppose that u is a continuous barrier with respect to
Then v is continuous, and hence superharmonic in Ω by Lemma 9. Because u is lower semicontinuous and satis es condition (iii) in De nition 6.1, we have δ := inf ∂Ω\U u > .
Since h is bounded from below and superharmonic, we see that h ∈ U f , and hence Pf ≤ h in Ω. As u is a barrier, it follows that lim sup Letting α → f (x ), and appealing to Theorem 5.1 shows that x is regular.
(a) ⇒ (c) Assume that x is regular. We begin with the case when Cp({x }) > . Let dx ∈ D p (X) be given by (5.1). We let u be the continuous solution of the K dx ,dx -obstacle problem, which is superharmonic (see Section 3) and hence satis es condition (i) in De nition 6.1. We also extend u to X by letting u = dx outside Ω so that u ∈ D p (X). Then ≤ u ≤ (as ≤ dx ≤ ), and thus U := {x ∈ Ω : u(x) > dx (x)} ⊂ B(x , ). Since u and dx are continuous, we see that U is open and u = dx on ∂U.
Suppose that x ∈ ∂U. Proposition 3.7 in Hansevi [29] implies that u is the continuous solution of the K dx ,dx (U)-obstacle problem. Since u > dx in U, we see that u| U = H U dx , and hence, by Theorem 4.3, The Kellogg property for bounded sets (Theorem 3.9 in Björn-Björn-Shanmugalingam [16] or Theorem 10.5 in [11]) implies that x is regular with respect to U as Cp({x }) > . It thus follows that On the other hand, if x ∈ ∂(Ω \ U), then Then u is p-harmonic, see Section 4, and in particular continuous. Thus u satis es condition (i) in De nition 6.1. Because x is regular and w is continuous at x and bounded, it follows from Theorem 5.1 that u satis es condition (ii) in De nition 6.1, as and hence Lemma 4.3 in Björn-Björn [9] (or Lemma 4.5 in [11]) implies that Cp(∂G) > . Let B be a su ciently large ball so that Cp(B ∩ ∂G) > . Since Cp({x }) = , it follows from the Kellogg property for bounded sets (Theorem 3.9 in Björn-Björn-Shanmugalingam [16] or Theorem 10.5 in [11]) that there is a point x ∈ (B ∩ ∂G) \ {x } that is regular with respect to G := G ∩ B . As in (6.1) for U, we see that v| G = P G v, and it follows that lim Thus v ≢ in G. As v ≤ is p-harmonic in G (by Theorem 4.4 in Hansevi [29]), it follows from the strong maximum principle (see Corollary 6.4 in Kinnunen-Shanmugalingam [39] or [11,Theorem 8.13]), that v < in G (and thus also in V). We conclude that v < in (Ω ∪ B) \ {x }.
Let m = sup ∂B v. By compactness, we get that −r ≤ m < . Since v| (Ω∪ B)\B is the continuous solution of the K h,v ((Ω ∪ B) \ B)-obstacle problem (by Proposition 3.7 in [29]) and h = −r in (Ω ∪ B) \ B, we see that Moreover, as v is continuous in B, it follows that lim sup and hence lim sup Since v is bounded and superharmonic in Ω, de ning w in the particular way on ∂Ω as we did in (6.2) makes sure that w ∈ Lw, and hence u ≥ w in Ω. It follows that u is positive and satis es condition (iii) in De nition 6.1, as Thus u is a positive continuous barrier at x . Proof. Cover ∂Ω \ {∞} by a countable set of balls {B j } ∞ j= and let I j = I ∩ B j . Furthermore, let I j be the set of irregular boundary points of Ω ∩ B j , j = , , ... . Theorem 6.2 (using that ¬(a) ⇒ ¬(d)) implies that I j ⊂ I j . Moreover, Cp(I j ) = , by the Kellogg property for bounded sets (Theorem 3.9 in Björn-Björn-Shanmugalingam [16] or Theorem 10.5 in [11]). Hence Cp(I j ) = for all j, and thus by the subadditivity of the capacity, Cp(I) = .

The Kellogg property
As a consequence of Theorem 7.1 we obtain the following result, which in the bounded case is a direct consequence of the results in Björn-Björn-Shanmugalingam [16], [17].

Theorem 7.2. If f ∈ C(∂Ω), then there exists a bounded p-harmonic function u on Ω such that there is a set
If moreover, Ω is bounded or p-parabolic, then u is unique and u = Pf .
Existence holds also for p-hyperbolic sets, which follows from the proof below, but uniqueness can fail. To see this, let < p < n and Ω = R n \ B( , ) in unweighted R n . Then both u(x) = |x| (p−n)/(p− ) and v ≡ are functions that are p-harmonic in Ω and satisfy (7.1) when f ≡ , with E = ∅.
Proof. Let u = Pf and let E be the set of irregular boundary points in ∂Ω \ {∞}. Then Cp(E) = by the Kellogg property (Theorem 7.1), and u is bounded, p-harmonic, and satis es (7.1), which shows the existence. For uniqueness, suppose that Ω is bounded or p-parabolic, and that u is a bounded p-harmonic function that satis es (7.1). By Lemma 5.2 in Björn-Björn-Shanmugalingam [19], Cp(E, Ω) ≤ Cp(E) (the proof is valid also if Ω is unbounded), and hence Corollary 7.9 in Hansevi [30] implies that u = Pf .
Another consequence of the barrier characterization is the following restriction result.

Proposition 7.3. Let x ∈ ∂Ω \ {∞} be regular, and let V ⊂ Ω be open and such that x ∈ ∂V. Then x is regular also with respect to V.
Proof. Using the barrier characterization the proof of this fact is almost identical to the proof of the implication (c) ⇒ (e) in Theorem 6.2. We leave the details to the reader. Roughly speaking, m is the lim inf of f at x in the Sobolev sense and M is the corresponding lim sup. Observe that it is not possible to replace M by M , as it can happen that Cp-ess lim sup Ω y→x ψ(y) > M , see Example 5.7 in Björn-Björn [9] (or Example 11.10 in [11]).

Boundary regularity for obstacle problems
In the case when Ω is bounded, this improves upon Theorem 5.6 in [9] (and Theorem 11.6 in [11]) in two ways: By allowing for f ∈ D p (Ω) and by having (two) equalities in (8.1), instead of inequalities. Proof. Let h ∈ D p (Ω; B) for some ball B. Extend h to B by letting h be equal to zero in B \ Ω so that h ∈ D p (B). Theorem 4.14 in [11] implies that h ∈ N ,p loc (B), and hence h ∈ N ,p (τB). As h = in τB \ Ω, it follows that h ∈ N ,p (Ω; τB).  Lemma 5.3 in Björn-Björn [9] (or Lemma 2.37 in [11]) shows that (u − k)+ ∈ N ,p (Ω; B). Because max{u, k} = k + (u − k)+, we see that (v − k)+ ∈ N ,p (V; B) and v ∈ N ,p (B). Let U = Ω ∩ B. The boundary weak Harnack inequality (Lemma 5.5 in [9] or Lemma 11.4 in [11]) implies that H V v is bounded from above on U.

It follows from
By Lemma 4.7 in Hansevi [29], it follows that and hence H V v is a solution of the K ψ,v (V)-obstacle problem. Furthermore, Proposition 3.7 in [29] shows that u is a solution of the K ψ,u (V)-obstacle problem, and thus u ≤ H V v in V, by Lemma 4.2 in [29]. Hence u is bounded from above on U, and thus v is bounded on U.
Taking in mum over all k > M shows that Now let k > lim sup Ω y→x u(y) be real. Then there is a ball B x such that u ≤ k in B ∩ Ω, and hence (u − k)+ ≡ in B ∩ Ω. It follows that and thus ( f − k)+ ∈ D p (Ω; B), by Lemma 2.8 in Hansevi [29]. This implies that k ≥ M , and hence taking in mum over all k > lim sup Ω y→x u(y) shows that We also know that u ≥ ψ q.e., so that lim sup To prove the other equality, let k < lim inf Ω y→x u(y). Then there is a ball B x such that k ≤ u in B ∩ Ω, and hence (k − u)+ ≡ in B ∩ Ω. Lemma 2.8 in Hansevi [29] implies that ( f − k)− ∈ D p (Ω; B), since Thus k ≤ m, and hence taking supremum over all k < lim inf Ω y→x u(y) shows that lim inf Ω y→x u(y) ≤ m. In both cases we allow f (x ) to be ±∞.
Note that it is possible to have f (x ) < Cp-ess lim sup Ω y→x ψ(y) and still have a solvable obstacle problem, see Example 5.7 in Björn-Björn [9] (or Example 11.10 in [11]). The proof of Theorem 8.3 is similar to the proof of Theorem 5.1 in Björn-Björn [9] (or Theorem 11.8 in [11]), but appealing to Theorem 8.1 above instead of Theorem 5.6 in [9] (or Theorem 11.6 in [11]). That one can allow for f (x ) = ±∞ seems not to have been noticed before.
Proof. Let m, M, and M be de ned as in Theorem 8.1. We rst show that M ≤ f (x ). If f (x ) = ∞ there is nothing to prove, so assume that f (x ) ∈ [−∞, ∞) and let α > f (x ) be real. Also let B = B(x , r) be chosen so that As u is superharmonic (see Section 3), it is a positive continuous barrier at x .
(d) ⇒ (a) Since u is a barrier at x , Theorem 6.2 implies that x is regular.
for all f ∈ D p (Ω ∪ (B ∩ Ω)) that are superharmonic in Ω and such that f | ∂Ω is lower semicontinuous at x . (a) ⇒ (d) Theorem 6.2 asserts that the point x is regular with respect to V := Ω ∩ B. If f (x ) = −∞ there is nothing to prove, so assume that f (x ) ∈ (−∞, ∞] and let α < f (x ) be real.
As f | ∂Ω is lower semicontinuous at x , there is r such that < r < dist(x , ∂B) and f ≥ α in B(x , r) ∩ ∂V. Let h = min{f , α}, which is also superharmonic in Ω, by Lemma 9.3 in [11]. It thus follows from Lemma 3.5 that h is the lsc-regularized solution of the K h,h (V)-obstacle problem. Since h − α = in B(x , r) ∩ ∂V, we get that h − α ∈ D p (V; B(x , r)). Letting α → f (x ) yields the desired result.
We now assume that Ω is bounded or p-parabolic.  which shows that lim Ω y→x Pdx (y) = . Thus x is regular by Theorem 5.1.
The following two results remove the assumption of bounded sets from the p-harmonic versions of Lemma 7.4 and Theorem 7.5 in Björn [6] (or Theorem 11.27 and Lemma 11.32 in [11]). The lemma follows directly from the su ciency part of the Wiener criterion, see [6] or [11]. With straightforward modi cations of the proof of Theorem 7.5 in [6] (or Theorem 11.27 in [11]), we obtain a proof for Theorem 9.3. For the reader's convenience, we give the proof here. and thus x is irregular both with respect to Ω := ∞ k= G k and with respect to Ω := ∞ k= G k+ , by Theorem 5.1. Since Ω and Ω are disjoint, this contradicts Lemma 9.4. We conclude that there are only nitely many components of Ω containing points from the sequence {y j } ∞ j= . Thus there is a component G that contains in nitely many of the points from the sequence {y j } ∞ j= . So there is a subsequence {y j k } ∞ k= such that y j k ∈ G for every k = , , ... . It follows that x ∈ ∂G and as lim k→∞ Pdx (y j k ) > , x must be irregular with respect to G. Finally, if G is any other component of Ω with x ∈ ∂G , then, by Lemma 9.4, x is regular with respect to G .