The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces

We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1 < p < \infty $\end{document}. After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted Rn. We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.


Introduction
Superharmonic functions play a fundamental role in the classical potential theory. Unlike harmonic functions (i.e. solutions of the Laplace equation ∆u = 0), they need not be continuous but are finely continuous. In fact, the fine topology is the coarsest topology that makes all superharmonic functions continuous, see Cartan [19]. The fine topology is closely related to the Dirichlet boundary value problem for the Laplace equation on open sets. It follows from the famous Wiener criterion [50] that a boundary point x 0 ∈ ∂Ω of a Euclidean domain Ω is irregular for ∆u = 0 if and only if Ω ∪ {x 0 } is finely open, i.e. if the complement R n \ Ω is thin at x 0 in a capacity density sense. In this case, the complement and the boundary are simply too small in the potential theoretical sense to ensure that continuous boundary data enforce continuity of the corresponding solution at x 0 . These facts have lead to the development of fine potential theory and finely (super)harmonic functions associated with ∆u = 0 on finely open sets, see the monograph [21] of Fuglede, the papers [22]- [26], [42], [44], [45], and the book [43] by Lukeš-Malý-Zajíček, which contain additional results and references.
In the nonlinear case, for equations associated with the p-Laplacian ∆ p and 1 < p = 2, the first similar study was conducted by Kilpeläinen-Malý [30], who studied p-fine (super)solutions for such equations on quasiopen subsets of unweighted R n . That theory was further extended by Latvala [39], [40], in particular for p = n. Eigenvalue problems for the p-Laplacian in quasiopen subsets of R n were considered in Fusco-Mukherjee-Zhang [27]. We are not aware of any other papers dealing with p-fine (super)solutions, and in particular none beyond unweighted R n .
The Wiener criterion was extended to the nonlinear theory associated with pharmonic functions on subsets of (unweighted and weighted) R n in [28], [31], [41], [47] and [48], and partially also to metric spaces, see [13]- [15]. It has also been related to fine continuity of p-superharmonic functions on open sets in much the same way as for the Laplacian. Following this nonlinear development, we define the fine topology on metric spaces using the notion of thinness based on a Wiener type integral, see Definition 3.1.
In this paper we continue our study of fine potential theory on metric spaces, carried through in [6]- [8], and initiate the study of fine p-(super)minimizers with 1 < p < ∞. We consider a complete metric space X equipped with a doubling measure supporting a p-Poincaré inequality. The function space naturally associated with p-energy minimizers on such metric spaces is the Sobolev type space N 1,p , called the Newtonian space.
The following regularity result for solutions of the Dirichlet problem on finely open sets is our main result, which we obtain as a by-product of more general pointwise results. Even in unweighted R n and for ∆ p u = 0, it is more general than the similar Theorem 5.3 in Kilpeläinen-Malý [30].
Here U p is the fine closure of U . In the linear axiomatic setting, i.e. for p = 2, finely (super)harmonic functions and the Dirichlet problem on finely open sets have been rigorously investigated, see the monographs [21] and [43]. As pointed out in [43, p. 389], even in the linear setting the fine boundary can be too small for a fruitful theory of the Dirichlet problem. Thus the use of the metric boundary in Theorem 1.1 is perhaps less unnatural than it may at first seem.
Obviously, some of the linear tools used in [21] and [43] are not available to us, nor in the nonlinear setting of unweighted R n and p = 2. Already the notion of fine p-(super)harmonic functions is not straightforward, and it is an open question whether p-(super)minimizers on finely open sets have finely continuous representatives. There are other open problems concerning important properties of such functions, see Section 9 for further discussion.
In metric spaces, there is (in general) no equation to work with (such as the p-Laplace equation). Therefore our theory relies on p-fine (super)minimizers defined through p-energy integrals and upper gradients. This makes our approach essentially independent of the theory in Kilpeläinen-Malý [30] and Latvala [39], [40], even though our main result was inspired by the proof of Theorem 5.3 in [30].
The key arguments in both proofs rely on pasting lemmas and the fine continuity of p-superharmonic functions on open sets.
Finely open sets and fine topology are closely related to quasiopen sets and quasitopology, as shown by Fuglede [20]. A similar study on metric spaces is more recent, but the metric space approach seems suitable since it makes it easy to consider the Sobolev type spaces N 1,p on nonopen sets, such as finely open and quasiopen sets. These Newtonian spaces were shown in [7] and [9] to coincide with the Sobolev spaces developed on quasiopen and finely open sets in R n by Kilpeläinen-Malý [30]. Moreover, functions in the spaces N 1,p are automatically quasicontinuous, and consequently finely continuous outside a set of zero capacity, both on open and quasiopen sets, see [7], [9], [12], [14] and [34]. Several of these results play a crucial role in this paper. On unweighted R n and for nonlinear fine potential theory, they can be found in the monograph by Malý-Ziemer [46]. See also Heinonen-Kilpeläinen-Martio [28] for many of these results on weighted R n , as well as [5], [9], [35]- [38] for further results.
Obstacle problems, and thereby (super)minimizers, on nonopen sets in metric spaces were studied in [4] and it was shown therein ( [4,Theorem 7.3]) that the theory of obstacle problems is not natural beyond finely open (or quasiopen) sets. In Proposition 5.9, we show that this true also for the theory of p-fine (super)minimizers. Additional fine properties of (super)harmonic functions on open sets were derived in [6] and [8].
The outline of the paper is as follows: In Section 2, we recall some definitions from first-order analysis on metric spaces, while the fine topology is introduced in Section 3. Therein, we also give two new characterizations of quasiopen sets, which are probably known to the experts in the field.
In order to be able to study p-fine (super)minimizers and the Dirichlet problem on quasiopen sets U , we need the appropriate local Newtonian (Sobolev) space N 1,p fine-loc (U ). We study this space in Section 4, where we also establish a density result that plays a crucial role in later sections. In Sections 5 and 6, we develop the basic theory of p-fine (super)minimizers, obstacle and Dirichlet problems on quasiopen sets.
Finally, in Section 7, we are ready to develop the necessary framework enabling us to obtain Theorem 1.1. We also deduce corresponding pointwise results. In Section 8, we use some of our results to give some more information on fine Newtonian spaces. The final Section 9 is devoted to open problems. Acknowledgement. A. B. and J. B. were supported by the Swedish Research Council, grants 2016-03424 and 2020-04011 resp. 621-2014-3974 and 2018-04106. Part of this research was done during several visits of V. L. to Linköping University.

Notation and preliminaries
We assume throughout the paper that X = (X, d, µ) is a metric space equipped with a metric d and a positive complete Borel measure µ such that 0 < µ(B) < ∞ for all balls B ⊂ X. We also assume that 1 < p < ∞.
In this section, we introduce the necessary metric space concepts used in this paper. For brevity, we refer to Björn-Björn-Latvala [6], [8] for more extensive introductions, and references to the literature. See also the monographs Björn-Björn [3] and Heinonen-Koskela-Shanmugalingam-Tyson [29], where the theory is thoroughly developed with proofs.
A curve is a continuous mapping from an interval, and a rectifiable curve is a curve with finite length. We will only consider curves which are nonconstant, compact and rectifiable. A curve can thus be parameterized by its arc length ds. A property holds for p-almost every curve if the curve family Γ for which it fails has zero p-modulus, i.e. there is ρ ∈ L p (X) such that γ ρ ds = ∞ for every γ ∈ Γ.
where the left-hand side is ∞ whenever at least one of the terms therein is infinite.
If f has a p-weak upper gradient in L p loc (X), then it has a minimal p-weak upper gradient g f ∈ L p loc (X) in the sense that g f ≤ g a.e. for every p-weak upper gradient where the infimum is taken over all p-weak upper gradients of f . The Newtonian space on X is , is a Banach space and a lattice. In this paper we assume that functions in N 1,p (X) are defined everywhere (with values in R), not just up to an equivalence class in the corresponding function space.
For a measurable set E ⊂ X, the Newtonian space N 1,p (E) is defined by considering (E, d| E , µ| E ) as a metric space in its own right. We say that f ∈ N 1,p loc (E) if for every x ∈ E there exists a ball B x ∋ x such that f ∈ N 1,p (B x ∩ E). If f, h ∈ N 1,p loc (X), then g f = g h a.e. in {x ∈ X : f (x) = h(x)}, in particular g min{f,c} = g f χ {f <c} a.e. in X for c ∈ R.
The Sobolev capacity of an arbitrary set E ⊂ X is where the infimum is taken over all f ∈ N 1,p (X) such that f ≥ 1 on E. A property holds quasieverywhere (q.e.) if the set of points for which it fails has capacity zero. The capacity is the correct gauge for distinguishing between two Newtonian functions. If f ∈ N 1,p (X), then h ∼ f if and only if h = f q.e. Moreover, if f, h ∈ N 1,p (X) and f = h a.e., then f = h q.e. For A ⊂ U ⊂ X, where U is assumed to be measurable, we let If U = X, we write N 1,p 0 (A) = N 1,p 0 (A, X). Functions from N 1,p 0 (A, U ) can be extended by zero in U \ A and we will regard them in that sense if needed.
If E ⊂ A are bounded subsets of X, then the variational capacity of E with respect to A is where the infimum is taken over all f ∈ N 1,p 0 (A) such that f ≥ 1 on E. If no such function f exists then cap p (E, A) = ∞. Definition 2.3. X supports a p-Poincaré inequality if there exist constants C > 0 and λ ≥ 1 such that for all balls B ⊂ X, all integrable functions f on X and all p-weak upper gradients g of f , In R n equipped with a doubling measure dµ = w dx, where dx denotes Lebesgue measure, the p-Poincaré inequality (2.2) is equivalent to the p-admissibility of the weight w in the sense of Heinonen-Kilpeläinen-Martio [28], see Corollary 20.9 in [28] and Proposition A.17 in [3]. Moreover, in this case g u = |∇u| a.e. if u ∈ N 1,p (R n ).

Fine topology and Newtonian functions on finely open sets
Throughout the rest of the paper, we assume that X is complete and supports a p-Poincaré inequality, that µ is doubling, and that 1 < p < ∞.
To avoid pathological situations we also assume that X contains at least two points. In this section we recall the basic facts about the fine topology associated with Newtonian functions.
In the definition of thinness, we make the convention that the integrand is 1 whenever cap p (B(x, r), B(x, 2r)) = 0. It is easy to see that the finely open sets give rise to a topology, which is called the fine topology. Every open set is finely open, but the converse is not true in general. A function u : V → R, defined on a finely open set V , is finely continuous if it is continuous when V is equipped with the fine topology and R with the usual topology. See Björn-Björn [3, Section 11.6] and Björn-Björn-Latvala [6] for further discussion on thinness and the fine topology in metric spaces. The fine interior, fine boundary and fine closure of E are denoted fine-int E, ∂ p E and E p , respectively.
The following characterization of the fine boundary is from Corollary 7.8 in Björn-Björn [4]. We will mainly use it for finely open sets.
The following definition will also be important in this paper.
A function u defined on a set E ⊂ X is quasicontinuous if for every ε > 0 there is an open set G ⊂ X such that C p (G) < ε and u| E\G is finite and continuous.
The quasiopen sets do not in general form a topology, see Remark 9.1 in Björn-Björn [4]. However it follows easily from the countable subadditivity of C p that countable unions and finite intersections of quasiopen sets are quasiopen. Quasiopen sets have recently been characterized in several ways. Here we summarize the known and some new characterizations. Note in particular the close connection between quasiopen and finely open sets.
Theorem 3.4. Let U ⊂ X be arbitrary. Then the following conditions are equivalent : (i) U is quasiopen; (ii) U is a union of a finely open set and a set of capacity zero; Proof. (i) ⇔ (ii) This follows from Theorem 1.4 (a) in Björn-Björn-Latvala [8].
Quasiopen, and thus finely open, sets are measurable. If U is finely open and C p (E) = 0, then U \ E is finely open, from which it follows that fine limits do not see sets of capacity zero.
For any measurable set E ⊂ X the notion of q.e. in E can either be taken with respect to the global capacity C p on X or with respect to the capacity C E p determined by E as the underlying space. However, for a quasiopen set U , the capacities C p and C U p have the same zero sets, and C p -quasicontinuity in U is equivalent to C U p -quasicontinuity, by Propositions 3.4 and 4.2 in [9]. Here we collect some facts on quasicontinuity from [7,Theorem 4.4], [8, Theorem 1.4] and [9, Theorem 1.3]. For further characterizations of quasiopen sets and quasicontinuous functions see [9] and also Theorem 7.2 below.

N 1,p fine-loc (U ) and p-strict subsets
From now on we always assume that U is a nonempty quasiopen set.
In the next section, we will start developing the basic theory of fine superminimizers. For this purpose, we first need to define appropriate fine Sobolev spaces.
Here p-strict subsets will play a key role, as a substitute for relatively compact subsets. Recall Equivalently, in the definition of p-strict subsets it can in addition be required that 0 ≤ η ≤ 1, as in Kilpeläinen-Malý [30]. [7], V has a base of fine neighbourhoods consisting only of p-strict subsets of V . We recall that functions in N 1,p fine-loc (U ) are finite q.e., finely continuous q.e. and quasicontinuous, by Theorem 4.4 in [7].
Throughout the paper, we consider minimal p-weak upper gradients in U . The following fact is then convenient: If u ∈ N 1,p loc (X) then the minimal p-weak upper gradients g u,U and g u with respect to U and X, respectively, coincide a.e. in U , see Björn-Björn [4,Corollary 3.7] or [7,Lemma 4.3]. For this reason we drop U from the notation and simply write g u .  To see this let U = B(0, 2) \ {0} ⊂ R n , with 1 < p < n, in which case it is easy to see that Since u is p-harmonic in U also Corollary 5.6 would fail.
fine-loc (G). Conversely, assume that that f ∈ N 1,p fine-loc (G) and x ∈ G. Then there is r x such that B(x, r x ) ⋐ G. It is straightforward to see that B(x, r x ) is a p-strict subset of G, and thus f ∈ N 1,p (B(x, r x )). Hence f ∈ N 1,p loc (G).
The following density result will play a crucial role.
Proposition 4.5. Let E ⊂ X be an arbitrary set and 0 ≤ u ∈ N 1,p 0 (E). Then there exist finely open p-strict subsets V j ⋐ E and bounded functions u j ∈ N 1,p We may also require that u j ≡ 0 outside V j .
Proof. Let U = fine-int E. By Theorem 7.3 in Björn-Björn [4], u ∈ N 1,p 0 (U ) and u = 0 q.e. in X \ U . In the rest of the proof we therefore replace E by U , which is quasiopen by Theorem 3.4. Modifying u in a set of zero capacity, we can also assume that u ≡ 0 in X \ U .
By truncating and multiplying by a constant and by a cutoff function, we may assume that 0 ≤ u ≤ 1 and that u has bounded support, see the proof of Lemma 5.43 in [3]. As U is quasiopen and u is quasicontinuous on X (by Theorem 3 As ϕ j and u are bounded it follows from the Leibniz rule [3, Theorem 2.15] that v j ∈ N 1,p (X), and thus also u j ∈ N 1,p (X). Hence, by Theorem 3.4, W j is quasiopen and there is a set E j with zero capacity such that W j \ E j is finely open. Let Then u j ∈ N 1,p 0 (V j ) and u j ≤ u. By the continuity of u| X\Gj and since u j = 0 in the open set G j , we see that Note that supp u j is bounded since supp u is bounded.
from which we conclude that V j is a p-strict subset of U as well as of E.
We next want to show that (4.1) Since g ϕj → 0 in L p (X), ϕ j → 0 a.e., and g u ∈ L p (X), the right-hand side in (4.1) tends to 0 in L p (X), by dominated convergence. Also x. We thus conclude that u − u j N 1,p (X) → 0 as j → ∞. By construction, V j ⊂ V j+1 and 0 ≤ u j ≤ u j+1 ≤ u for j = 1, 2, ... . It then follows from Corollary 1.72 in [3], that u j (x) → u(x) for q.e. x ∈ X, as j → ∞. After replacing u j by u j χ Vj one can also require that u j ≡ 0 on X \ V j .

Fine (super)minimizers
for every finely open p-strict subset V ⋐ U and for every (resp. every nonnegative) ϕ ∈ N 1,p 0 (V ). Moreover, u is a fine subminimizer if −u is a fine superminimizer.
By Remark 4.2, we may equivalently consider quasiopen p-strict subsets V ⋐ U in Definition 5.1.
Remark 5.2. It follows from Proposition 5.9 below that if u ∈ N 1,p fine-loc (U ) then u is a fine (super)minimizer in U if and only if it is a fine (super)minimizer in fine-int U . On the other hand, this equivalence is not true if we drop the assumption u ∈ N 1,p fine-loc (U ) as seen in Example 8.2 below. For the reader's convenience, let us first look at the Euclidean case considered in Kilpeläinen-Malý [30]. By Remark 4.2 and [7, Theorem 1.1] the spaces N 1,p (U ), N 1,p fine-loc (U ) and N 1,p 0 (U ) are equal (up to a.e.-equivalence) to the spaces W 1,p (U ), W 1,p loc (U ) and W 1,p 0 (U ) defined for quasiopen subsets of (unweighted) R n in [30]. See also Theorem 7.2 below and [30, Theorem 2.10]. This is in particular true for open U , in which case N 1,p (U ) also agrees with the Sobolev space H 1,p (U ) in Heinonen-Kilpeläinen-Martio [28] (up to refined equivalence classes) also on weighted R n .
We next show that the fine supersolutions of [30] coincide with our fine superminimizers in R n . Recall that, for any v ∈ N 1,p fine-loc (U ), with U ⊂ R n quasiopen, we have |∇v| = g v a.e. in U, where ∇v is as defined in [30]; see [7,Theorem 5.7]. The proof and the details above apply equally well if R n is equipped with a p-admissible measure. for all p-strict subsets V ⋐ U and all bounded nonnegative ϕ ∈ N 1,p 0 (V ).
Proof. First, let u be a fine supersolution of (5.3) in U and let V ⋐ U be a p-strict subset of U . Let ϕ ∈ N 1,p 0 (V ), ϕ ≥ 0. Assuming also that ϕ is bounded, we obtain from (5.4) that Since u ∈ N 1,p (V ), the first integral on the right-hand side is finite, and dividing by it shows that If ϕ is not bounded, then dominated convergence implies that V |∇(u + ϕ)| p dx = lim k→∞ V |∇(u + min{ϕ, k})| p dx.
Using also (5.2) shows that u is a fine superminimizer in the sense of Definition 5.1.
For the converse implication, assume that u is a fine superminimizer in U . Let V ⋐ U be a p-strict subset of U and let ϕ ∈ N 1,p 0 (V ) be bounded and nonnegative. Using (5.2), we have for any 0 < ε < 1 that From this the inequality U |∇u| p−2 ∇u · ∇ϕ dx ≥ 0 follows in the same way as in the proof of Theorem 5.13 in Heinonen-Kilpeläinen-Martio [28].

Lemma 5.4. A function u is a fine minimizer in U if and only if it is both a fine subminimizer and a fine superminimizer in U .
Proof. Assume that u is both a fine subminimizer and a fine superminimizer in U . Let V ⋐ U be a finely open p-strict subset and let ϕ ∈ N 1,p 0 (V ). We may assume that ϕ = 0 everywhere in X \ V . Since {ϕ± = 0} are quasiopen p-strict subsets of U (by Theorem 3.4), testing (5.1) with ϕ± implies that g p u+ϕ dµ, see Remark 4.2. Adding V ∩{ϕ=0} g p u dµ = V ∩{ϕ=0} g p u+ϕ dµ to both sides shows that u is a fine minimizer. The converse implication is trivial.
The following characterization is quite convenient. It also shows that condition (5.1) in Definition 5.1 can equivalently be required to hold for arbitrary V ⊂ U .
for every (nonnegative) ϕ ∈ N 1,p 0 (U ). Note that for some ϕ the integrals in (5.5) may be infinite, but then they are always infinite simultaneously. The characterization in Lemma 5.5 is in contrast to the definition (5.4) of supersolutions, where V = U is allowed only if u ∈ N 1,p (U ).
Proof. Assume first that u is a fine superminimizer and that ϕ ∈ N 1,p 0 (U ) is nonnegative. By Proposition 4.5, there are finely open p-strict subsets V j ⋐ U and functions ϕ j ∈ N 1,p 0 (V j ) such that 0 ≤ ϕ j ≤ ϕ and lim j→∞ ϕ j − ϕ N 1,p (X) = 0. (5.6) Since u is a fine superminimizer, we see that As u ∈ N 1,p (V j ) the last term is finite, and we can thus subtract it from both sides in the inequality obtaining which together with (5.6) shows that (5.5) holds. Conversely, let V ⋐ U be a finely open p-strict subset and ϕ ∈ N 1,p 0 (V ) be nonnegative. It then follows from (5.5) and the fact that g u = g u+ϕ on {x : ϕ(x) = 0}, that (5.1) holds and thus u is a fine superminimizer. The claim for fine minimizers follows from Lemma 5.4. Here we define (super)minimizers as in Definition 7.7 in [3].
If u ∈ N 1,p fine-loc (U 2 ), then u is a fine superminimizer in U 2 .
Corollary 5.8. If u and v are fine superminimizers in U , then min{u, v} is also a fine superminimizer in U .
Assume that E is an arbitrary measurable set. Then the space N 1,p fine-loc (E) as well as fine minimizers and fine superminimizers in E can be defined in the same way as in Definitions 4.1 and 5.1 (just replacing U be E). The following characterization suggests that the notions of fine superminimizers and minimizers might not be very interesting beyond quasiopen sets. Proposition 5.9. Let E be measurable and assume that u ∈ N 1,p fine-loc (E). Then u is a fine (super )minimizer in E if and only if it is a fine (super )minimizer in V := fine-int E.
Proof. Assume that u is a fine superminimizer in V , and let ϕ ∈ N 1,p 0 (E) be nonnegative. By Theorem 7.3 in Björn-Björn [4] we see that ϕ ∈ N 1,p 0 (V ). By Lemma 5.5, Since Proposition 4.5 holds for E, so does Lemma 5.5, from which it follows that u is a fine superminimizer in E. The converse implication is clear and the proof for fine minimizers is similar.

The obstacle and Dirichlet problems
The obstacle problem will be a fundamental tool for studying fine minimizers.
Definition 6.1. Assume that U is bounded and C p (X \ U ) > 0. Let f ∈ N 1,p (U ) and ψ : U → R. Then we define The Dirichlet problem is a special case of the obstacle problem, with the trivial obstacle ψ ≡ −∞. Note that the boundary data f are only required to belong to N 1,p (U ), i.e. f need not be defined on ∂U or the fine boundary ∂ p U . Theorem 6.2. Assume that U is bounded and C p (X \ U ) > 0. Let f ∈ N 1,p (U ) and ψ : U → R, and assume that K ψ,f (U ) = ∅. Then there exists a solution u of the K ψ,f (U )-obstacle problem, and this solution is unique q.e. Moreover, u is a fine superminimizer in U .
If ψ ≡ −∞ in U or if ψ is a fine subminimizer in U , then u is a fine minimizer in U .
To show that u is a fine (super)minimizer in U , let V ⋐ U be a finely open p-strict subset and let ϕ ∈ N 1,p 0 (V ). If ψ is not a fine subminimizer and ψ ≡ −∞, then we also require ϕ to be nonnegative.
It is easily verified that v := max{u + ϕ, ψ} ∈ K ψ,f (U ). Hence, as u is a solution of the K ψ,f (U )-obstacle problem, we get that where the second inequality is justified by Lemma 5.5 if ψ is a fine subminimizer, and is trivial otherwise as u + ϕ ≥ ψ q.e. in U in that case.
Since u ∈ N 1,p (U ), we see that the last integral in (6.1) is finite and subtracting it from both sides of (6.1) yields (5.1) in Definition 5.1 for the above choices of V and ϕ ∈ N 1,p 0 (V ). As V was arbitrary, it follows that u is a fine superminimizer in U . When ϕ is not required to be nonnegative, we conclude that u is a fine minimizer in U .
Note that there is a comparison principle for solutions of obstacle problems, see

Fine continuity for solutions of the Dirichlet problem
In this section we assume that U is a nonempty finely open set. Except for Theorem 7.2, we also assume that U is bounded and that C p (X \ U ) > 0.
We do not know in general if fine minimizers have finely continuous representatives. However in this section we obtain sufficient conditions for the fine continuity of solutions of the (fine) Dirichlet problem, and deduce Theorem 1.1. The proof of our key Lemma 7.3 below was inspired by the proof of Theorem 5.3 in Kilpeläinen-Malý [30]. As we study fine continuity in this section it is natural to consider only finely open sets U .
With continuous boundary data, the solution of the Dirichlet problem in an open set need not be continuous at an irregular boundary point. However, the solution is finely continuous. We demonstrate this by the following example using Corollary 7.7 below. Since z is strongly irregular, it follows from Theorem 13.13 in [3] that the continuous solution h of the K −∞,d (G)-obstacle problem, with d(x) = d(x, z), does not have a limit at z. However, by Corollary 7.7 below, h does have a fine limit.
We will need the following auxiliary result, which may also be of independent interest. In what follows, the notions of fine lim, fine lim sup and fine lim inf are defined using punctured fine neighbourhoods. Note that since cap p (B(x, r) \ {x}, B(x, 2r)) = cap p (B(x, r), B(x, 2r)), there are no isolated points in the fine topology, i.e. no singleton sets are finely open.
Theorem 7.2. Let U ⊂ V ⊂ X be finely open sets. Assume that u ∈ N 1,p (U ) and extend it by 0 to V \ U . Then the following are equivalent : (a) u ∈ N 1,p 0 (U, V ), i.e. u ∈ N 1,p (V ); (b) u is quasicontinuous in V ; (c) u is finite q.e. and finely continuous q.e. in V ; (d) u is measurable, finite q.e., and u • γ is continuous for p-almost every curve γ : [0, l γ ] → V ; (e) fine lim U∋y→x u(y) = 0 for q.e. x ∈ V ∩ ∂ p U .
We will only need the equivalence (a) ⇔ (e) (when proving Lemma 7.3). However, when deducing this equivalence we will rely on several earlier results, which essentially requires us to obtain the full equivalence of (a)-(e).
(b) ⇔ (d) This follows from Theorem 1.2 in Björn-Björn-Malý [9]. (d) ⇒ (a) Let g ∈ L p (U ) be a p-weak upper gradient of u in U , extended by zero to V \ U . Consider a curve γ as in (d) such that none of its subcurves in U is exceptional in (2.1) for the pair (u, g). Lemma 1.34 (c) in [3] implies that p-almost every curve has this property. If γ ⊂ U or γ ⊂ V \ U , there is nothing to prove. Hence by splitting γ into two parts, if necessary, and possibly reversing the direction, we may assume that x = γ(0) ∈ U and y = γ(l γ ) / ∈ U . Let c = inf{t : γ(t) / ∈ U } and y 0 = γ(c). By continuity, u(y 0 ) = 0, and hence It follows that g is a p-weak upper gradient of u in V and hence u ∈ N 1,p (V ).
(c) ⇒ (e) As u is finely continuous q.e. and u ≡ 0 in V \ U , (e) follows directly.
(e) ⇒ (c) Since u ∈ N 1,p (U ), it is finely continuous q.e. and finite q.e. in U , by Theorem 3.5. Thus u is finite q.e. in V and finely continuous q.e. in V \ ∂ p U . As u ≡ 0 in V \ U and (e) holds, u is finely continuous q.e. in V ∩ ∂ p U .
We define for any function u : U → R the fine lsc-regularization u * : and the fine usc-regularization u * : U p → R of u as In this paper, we will only regularize Newtonian functions. As these are finely continuous q.e., we have u = u * = u * q.e. in U . We say that u is finely lscregularized if u = u * in U and finely usc-regularized if u = u * in U . Note that u * (resp. u * ) is finely lsc-regularized (resp. finely usc-regularized) in U . Recall also the characterization of ∂ p U in Lemma 3.2.
Lemma 7.3. Let z ∈ U , B = B(z, r), f ∈ N 1,p (U ) and let u be a fine superminimizer in B ∩ U such that u − f ∈ N 1,p 0 (B ∩ U, B). Assume that c ∈ R is such that f * ≥ c q.e. in B ∩ ∂ p U.
If u * (z) < c, then u * is finely continuous at z.
Proof. Assume that u * (z) = fine lim inf U∋y→z u(x) < c. In what follows, the C p -ess lim inf, C p -ess lim sup and C p -ess lim are taken with respect to the metric topology from X and up to sets of zero capacity in punctured neighbourhoods. For instance, for a function v defined in a set E, In particular, Corollary 7.4. Let z ∈ U , f ∈ N 1,p (U ) and let u be a fine superminimizer in U such that u − f ∈ N 1,p 0 (U ). If

3)
then u * is finely continuous at z.
Proof. It follows directly from the definition of f * that and thus we can without loss of generality assume the latter inequality in (7.3). We can then find c > u * (z) and B = B(z, r) such that Then h U − f ∈ N 1,p 0 (U ) and by the uniqueness part of Theorem 6.2, we conclude that h U = h V q.e. in U . Theorem 7.5 and the assumption (b) imply that h U is finely continuous at z, and thus by Theorem 7.2, In terms of Perron solutions on open sets, Corollary 7.7 yields the following consequence. Here P f denotes the the Perron solution in G with boundary data f , see [3,Section 10.3]. Recall that if f ∈ C(∂G) then f is resolutive and thus P f exists, by Theorem 6.1 in Björn-Björn-Shanmugalingam [11]   exists for all z ∈ ∂G.

Removability
In this section we assume that U is a quasiopen set.
We conclude the paper by deducing some simple removability results.
is a fine (super )minimizer in U , and u is extended arbitrarily to E, then u is a fine (super )minimizer in V .
Proof. By Theorem 3.4, V is quasiopen. Let ϕ ∈ N 1,p 0 (V ). Since ϕ = 0 q.e. in X \ U , also ϕ ∈ N 1,p 0 (U ) and the statement follows directly from Lemma 5.5. which is harmonic (and thus a fine minimizer) in U . However u has no extension in N 1,2 loc (V ) = N 1,2 fine-loc (V ), with V = B(0, 1), and in particular no extension as a fine superminimizer (i.e. as a superminimizer because V is open), even though C p (V \ U ) = 0.
(b) Even if U = fine-int V , the assumption u ∈ N 1,p fine-loc (V ) cannot be replaced by u ∈ N 1,p fine-loc (U ) in Lemma 8.1. Moreover, fine (super)minimizers on a quasiopen set V can differ from those on its fine interior fine-int V .
(a) If u is a fine superminimizer in V , then u * is finely continuous in V .
(b) If u is a fine minimizer in V , then u * is continuous in V , with respect to the metric topology.
Proof. If u ∈ N 1,p (V ) then it follows from Proposition 1.48 in [3] thatũ ∈ N 1,p (G), whereũ is any extension of u to G. Thus we can assume that u ∈ N 1,p loc (G). By Lemma 8.1 and Corollary 5.6, u is a superminimizer in G. It follows from Proposition 7.4 in Kinnunen-Martio [32] (or [3, Proposition 9.4]), that u has a superharmonic representative v such that v = u q.e. in G.
In (a), v is finely continuous in G, by Björn [14,Theorem 4.4]  As v = u q.e., we have u * = v * = v in V , which proves the lemma.

Open problems
Fine superminimizers and fine supersolutions can be changed arbitrarily on sets of capacity zero. To fix a precise representative, in potential theory one usually studies pointwise defined finely (super)harmonic functions with additional regularity properties, as used in the proofs of Lemma 7.3 and Corollary 8.3. In this paper, we do not go further into making a definition of finely (super)harmonic functions in metric spaces. Even in the linear case, there have been several different suggestions for such definitions in the literature, see Lukeš-Malý-Zajíček [43, Section 12.A and Remarks 12.1]. Some definitions have been given in the nonlinear theory on R n , but the theory is even less developed and there are many open questions in this context. A few of these are listed below.
(1) Is every finely superharmonic function finely continuous? This is known in the linear case, see [21,Theorem 9.10] and [43,Theorem 12.6]. In the nonlinear case, the best known result is Corollary 7.12 in Latvala [39], which says that the finely superharmonic functions associated with the n-Laplacian on unweighted R n are approximately continuous. (4) On unweighted R n , Latvala [40] showed that U \ E is a p-fine domain if U is a p-fine domain and C p (E) = 0. As an application of this result a strong version of the minimum principle for finely superharmonic functions was obtained. We do not know if the corresponding fine connectedness result holds in our metric setting, or on weighted R n .