Resolutivity and invariance for the Perron method for degenerate equations of divergence type

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Introduction
We consider the Dirichlet problem for quasilinear elliptic equations of the form div A(x, ∇u) = 0 (1.1) in a bounded nonempty open subset Ω of the n-dimensional Euclidean space R n .The mapping A : Ω × R n → R n satisfies the standard ellipticity assumptions with a parameter 1 < p < ∞ and a p-admissible weight as in Heinonen-Kilpeläinen-Martio [7, Chapter 3].
The Dirichlet problem amounts to finding a solution of the partial differential equation in Ω with prescribed boundary data on the boundary of Ω.One of the most useful approaches to solving the Dirichlet problem in Ω with arbitrary boundary data f is the Perron method.This method was introduced by Perron [11] and independently Remak [12] in 1923 for the Laplace equation ∆u = 0 in a bounded domain Ω ⊂ R n .It gives an upper and a lower Perron solution (see Definition 3.2) and when the two coincide, we get a suitable solution P f of the Dirichlet problem and f is called resolutive.
The Perron method for linear equations in Euclidean domains was studied by Brelot [5], where a complete characterization of resolutive functions was given in terms of the harmonic measure.The Perron method was later extended to nonlinear equations.Granlund-Lindqvist-Martio [6] were the first to use the Perron method to study the nonlinear equation div(∇ q F (x, ∇u)) = 0 (where ∇ q F stands for the gradient of F with respect to the second variable).This is a special type of equation (1.1), including the p-Laplace equation ∆ p u := div(|∇u| p−2 ∇u) = 0. (1.2) Lindqvist-Martio [10] studied boundary regularity of (1.1) in the unweighted case and also showed that continuous boundary data f are resolutive when p > n − 1.
Kilpeläinen [8] extended the resolutivity to general p, which in turn was extended to weighted R n by Heinonen-Kilpeläinen-Martio [7].More recently, the Perron method was used to study p-harmonic functions in the metric setting, see [1]- [4].
In this paper, we consider the weighted equation div A(x, ∇u) = 0 and show that arbitrary perturbations on sets of (p, w)-capacity zero of continuous boundary data f are resolutive and that the Perron solution for f and such perturbations coincide, see Theorem 3.9.In Proposition 3.8, we also obtain, as a by-product, that Perron solutions of perturbations of Lipschitz boundary data f are the same as the Sobolev solution of f .This perturbation result, as well as the equality of the Perron and Sobolev solutions, holds also for quasicontinuous representatives of Sobolev functions, see Theorem 4.2.Moreover, we prove in Theorem 3.12 that the Perron solution for the equation (1.1) with continuous boundary data is the unique bounded solution of (1.1) that takes the required boundary data outside a set of (p, w)-capacity zero.A somewhat weaker uniqueness result is proved for quasicontinuous Sobolev functions in Corollary 4.5.
Much as we use Heinonen-Kilpeläinen-Martio [7] as the principal literature for this paper, our proof of resolutivity for continuous boundary data is quite different from the one considered in [7].In particular, we do not use exhaustions by regular domains.The obstacle problem for the operator div A(x, ∇u) and a convergence theorem for obstacle problems play a crucial role in the proof of our main results.
For p-harmonic functions, i.e. solutions of the p-Laplace equation (1.2), most of the results in this paper follow from Björn-Björn-Shanmugalingam [2], [3], where this was proved for p-energy minimizers in metric spaces.The proofs here have been inspired by [2] and [3], but have been adapted to the usual Sobolev spaces to make them more accessible for people not familiar with the nonlinear potential theory on metric spaces and Sobolev spaces based on upper gradients.They also apply to the more general A-harmonic functions, defined by equations rather than minimization problems.project 316-2014 "Capacity building in Mathematics and its applications" under the SIDA bilateral program with the Makerere University 2015-2020, contribution No. 51180060.

Notation and preliminaries
In this section, we present the basic notation and definitions that will be needed in this paper.Throughout, we assume that Ω is a bounded nonempty open subset of the n-dimensional Euclidean space R n , n ≥ 2, and 1 < p < ∞.We use ∂Ω and Ω to denote the boundary and the closure of Ω, respectively.
We write x to mean a point x = (x 1 , ... , x n ) ∈ R n and for a function v which is infinitely many times continuously differentiable, i.e. v ∈ C ∞ (Ω), we write ∇v = (∂ 1 v, ... , ∂ n v) for the gradient of v.We follow Heinonen-Kilpeläinen-Martio [7] as the primary reference for the material in this paper.
First, we give the definition of a weighted Sobolev space, which is crucial when studying degenerate elliptic differential equations, see [7] and Kilpeläinen [9].Definition 2.1.The weighted Sobolev space H 1,p (Ω, w) is defined to be the completion of the set of all v ∈ C ∞ (Ω) such that with respect to the norm v H 1,p (Ω,w) , where w is the weight function which we define later.
The space H 1,p 0 (Ω, w) is the completion of C ∞ 0 (Ω) in H Throughout the paper, the mapping A : Ω × R n → R n , defining the elliptic operator (1.1), satisfies the following assumptions with a parameter 1 < p < ∞, a p-admissible weight w(x) and for some constants α, β > 0, see [7, (3.3)-(3.7)]: First, assume that A(x, q) is measurable in x for every q ∈ R n , and continuous in q for a.e.x ∈ R n .Also, for all q ∈ R n and a.e.x ∈ R n , the following hold A function u is a subsolution of (1.1) if −u is a supersolution of (1.1).
The sum of two (super)solutions is in general not a (super)solution.However, if u and v are two (super)solutions, then min{u, v} is a supersolution, see [7, Theorem 3.23].If u is a supersolution and a, b ∈ R, then au + b is a supersolution provided that a ≥ 0.
It is rather straightforward that u is a solution if and only if it is both a suband a supersolution, see [7, bottom p. 58].By [7, Theorems 3.70 and 6.6], every solution u has a Hölder continuous representative v (i.e.v = u a.e.).
We remark that A-harmonic functions do not in general form a linear space.
Definition 2.4.A weight w on R n is a nonnegative locally integrable function.We say that a weight w is p-admissible with p ≥ 1 if the associated measure dµ = w dx is doubling and supports a p-Poincaré inequality, see [7,Chapters 1 and 20].
For instance, weights belonging to the Muckenhoupt class A p are p-admissible as exhibited for example by Heinonen-Kilpeläinen-Martio [7] and Kilpeläinen [9].By a weight w ∈ A p we mean that there exists a constant C > 0 such that for all balls where |B| is the n-dimensional Lebesgue measure of B.
We follow [7, Section 2.35] defining the Sobolev capacity as follows.
Definition 2.5.Let E be a subset of R n .The Sobolev (p, w)-capacity of E is where the infimum is taken over all u ∈ H 1,p (R n , w) such that u = 1 in an open set containing E.
The Sobolev (p, w)-capacity is a monotone, subadditive set function.It follows directly from the definition that for all In particular, if C p,w (E) = 0 then there exist open sets U j ⊃ E with C p,w (U j ) → 0 as j → ∞.For details, we refer the interested reader to [7, Section 2.1].A property is said to hold quasieverywhere (abbreviated q.e.), if it holds for every point outside a set of Sobolev (p, w)-capacity zero.

Perron solutions and resolutivity
In order to discuss the Perron solutions for (1.1), we first recall the following basic results from Heinonen-Kilpeläinen-Martio [7, Chapters 7 and 9].
Let u and v be A-superharmonic.Then au + b and min{u, v} are A-superharmonic whenever a ≥ 0 and b are real numbers, but in general u + v is not A-superharmonic, see [7, Lemmas 7.1 and 7.2].
We briefly state how supersolutions and A-superharmonic functions are related.It is proved in [7,Theorem 7.16] that if u is a supersolution of (1.1) and then u * = u a.e. and u * is A-superharmonic.Conversely, if u is an A-superharmonic function in Ω, then u * = u in Ω.If moreover, u is locally bounded from above, then u ∈ H 1,p loc (Ω, w) and u is a supersolution of (1.1) in Ω, see [7,Corollary 7.20].That is, every supersolution has an A-superharmonic representative and locally bounded A-superharmonic functions are supersolutions.
The upper Perron solution P f of f is defined by Analogously, let L f be the set of all A-subharmonic functions v on Ω bounded from above such that lim sup Ω∋y→x v(y) ≤ f (x) for all x ∈ ∂Ω.
The lower Perron solution P f of f is defined by We remark that if If P f = P f is A-harmonic, then f is said to be resolutive with respect to Ω.In this case, we write P f := P f .Continuous functions f are resolutive by [7,Theorem 9.25].
The following comparison principle shows that P f ≤ P f .We follow [7,Chapter 3] giving the following definition.
A function u ∈ K ψ,f (Ω) is a solution of the obstacle problem in Ω with obstacle ψ and boundary data f if In particular, a solution u of the obstacle problem for K ψ,u (Ω) with ψ ≡ −∞ is a solution of (1.1).By considering v = u + ϕ with 0 ≤ ϕ ∈ C ∞ 0 (Ω), it is easily seen that the solution u of the obstacle problem is always a supersolution of (1.1) in Ω.Conversely, a supersolution u in Ω is always a solution of the obstacle problem for K u,u (Ω ′ ) for all open sets Ω ′ ⋐ Ω.Moreover, a solution u of (1.1) is a solution of the obstacle problem for K ψ,u (Ω ′ ) with ψ ≡ −∞ for all Ω ′ ⋐ Ω, see [7,Section 3.19].
By [7, Theorem 3.21], there is an almost everywhere (a.e) unique solution u of the obstacle problem whenever K ψ,f (Ω) is nonempty.Furthermore, by defining u * as in (3.1) we get a lower semicontinuously regularized solution in the same equivalence class as u, see [7,Theorem 3.63].
We call u * the lower semicontinuous (lsc) regularization of u.Moreover, with ψ ≡ −∞, the lsc-regularization of the solution of the obstacle problem for K ψ,f (Ω) provides us with the A-harmonic extension Hf of f in Ω, that is, Hf − f ∈ H 1,p 0 (Ω, w) and Hf is A-harmonic.The continuity of Hf in Ω is guaranteed by [7, Theorem 3.70].
By [7,Theorem 9.20], x is regular if and only if it is Sobolev regular, we will therefore just say "regular" from now on.By the Kellogg property [7, Theorem 8.10 and 9.11], the set of irregular points on ∂Ω has Sobolev (p, w)-capacity zero.
The following result is due to Björn-Björn-Shanmugalingam [2, Lemma 5.3].Here it is slightly modified to suite our context.For completeness and the reader's convenience, the proof is included.
Then there exists a decreasing sequence of nonnegative functions {ψ j } ∞ j=1 such that for all j, m = 1, 2, ... , In particular, Replacing ϕ k by its positive part max{ϕ k , 0}, we can assume that each ϕ k is nonnegative.Define We will need the following convergence theorem due to Heinonen-Kilpeläinen-Martio [7,Theorem 3.79] in order to prove the next proposition.
Theorem 3.7.Let {ψ j } ∞ j=1 be an a.e.decreasing sequence of functions in H 1,p (Ω, w) such that ψ j → ψ in H 1,p (Ω, w).Let u j ∈ H 1,p (Ω, w) be a solution of the obstacle problem for K ψj ,ψj (Ω).Then there exists a function u ∈ H 1,p (Ω, w) such that the sequence u j decreases a.e. in Ω to u and u is a solution of the obstacle problem for K ψ,ψ (Ω).Proposition 3.8.Let the function f be Lipschitz on Ω and h : ∂Ω → [−∞, ∞] be such that h = 0 q.e. on ∂Ω.Then both f and f + h are resolutive and Proof.Since f is Lipschitz and Ω bounded, we get that f ∈ H 1,p (Ω, w).First, we assume that f ≥ 0. Let I p ⊂ ∂Ω be the set of all irregular points.Let E = {x ∈ ∂Ω : h(x) = 0}.Then by the Kellogg property [7, Theorem 8.10], we have Consider the decreasing sequence of nonnegative functions {ψ j } ∞ j=1 given in Lemma 3.6.Let u j be the lsc-regularized solution of the obstacle problem with obstacle and boundary data f j = Hf + ψ j , see [7,Theorems 3.21 and 3.63].Let m be a positive integer.By the comparison principle [7, Lemma 3.18], we have that Hf ≥ 0 and hence by Lemma 3.6, In particular, u j ≥ f j ≥ m a.e. in U j+m ∩ Ω and since u j is lsc-regularized, we have that Let ε > 0 and x ∈ ∂Ω be arbitrary.If x / ∈ U j+m , then x is a regular point and thus Hf is continuous at x. Hence, there is a neighbourhood V x of x such that As ψ j ≥ 0, we have that f j = Hf + ψ j ≥ Hf .So, Since u j is lsc-regularized, we get Consequently, for all x ∈ ∂Ω, we have Letting ε → 0 and m → ∞ yields lim inf Ω∋y→x u j (y) ≥ (f + h)(x) for all x ∈ ∂Ω.
Since u j is A-superharmonic and nonnegative, we conclude that u j ∈ U f +h (Ω), and thus u j ≥ P (f + h).As Hf is the solution of the obstacle problem for K Hf,Hf (Ω), we get by Theorem 3.7 that the sequence u j decreases a.e. to Hf in Ω.Thus, Hf ≥ P (f + h) a.e. in Ω.But Hf and P (f + h) are continuous, so we have that for all Lipschitz functions f ≥ 0, Next, let f be an arbitrary Lipschitz function on Ω.Since f is bounded, there exists a constant c ∈ R such that f + c ≥ 0. By the definition of Hf and of Perron solutions we see that This together with (3.4) shows that i.e. (3.4) holds for arbitrary Lipschitz functions f .Applying it to −f and −h gives us that Together with the inequality P (f + h) ≤ P (f + h), implied by Theorem 3.3, we get that and thus P (f + h) = Hf and f + h is resolutive.Finally, letting h = 0, it follows directly that f is resolutive and P f = Hf .
It is now possible to extend the resolutivity results to continuous functions.This gives us an alternative way of solving the Dirichlet problem with prescribed continuous boundary data.Theorem 3.9.Let f ∈ C(∂Ω) and h : ∂Ω → [−∞, ∞] be such that h = 0 q.e. on ∂Ω.Then both f and f + h are resolutive and P (f + h) = P f .Proof.Since continuous functions can be approximated uniformly by Lipschitz functions, we have that there exists a sequence {f k } ∞ k=1 of Lipschitz functions such that From Definition 3.2 it follows that i.e. the functions P f k converge uniformly to P f in Ω as k → ∞.Using (3.5), we also obtain similar inequalities for P f, P (f + h) and P (f + h) in terms of P f k , P (f k + h) and P (f k + h), respectively.By Proposition 3.8, we have that f k and f k + h are resolutive and moreover P (f k + h) = P f k .Using the resolutivity of f k + h, we have from which it follows that where C Ω is a constant which depends on Ω.Since u j is a solution and A satisfies the ellipticity conditions (2.1), testing (2.2) with ϕ = u j − f j yields , where C is a constant depending on the structure constants α and β in (2.1).Therefore, since the sequence {f j } ∞ j=1 is bounded in H 1,p (Ω, w).This shows that u j − f j is bounded in H 1,p (Ω, w).Consequently, by [7,Lemma 1.32], u − f ∈ H 1,p 0 (Ω, w) and u = Hf by uniqueness, cf.[7, Theorem 3.17].
We now prove Theorem 4.2 and refer the reader to closely look at the proof of Proposition 3.8 to fill in details where needed.
Proof of Theorem 4.2.First assume that f ≥ 0 and so Hf ≥ 0. Define u := Hf extended by f outside Ω.By Lemma 4.3, u is (p, w)-quasicontinuous in R n .Let {U k } ∞ k=1 be a decreasing sequence of bounded open sets in R n such that C p,w (U k ) < 2 −kp , h = 0 outside U k and u restricted to R n \ U k is continuous.Let u j be the lsc-regularized solution of the obstacle problem with the obstacle and boundary data f j = u + ψ j , where ψ j are as in Lemma 3.6, j = 1, 2, ... .As in the proof of Proposition 3.8 we get Let ε > 0 and x ∈ ∂Ω be arbitrary.If x ∈ ∂Ω \ U j+m , then by quasicontinuity, u restricted to R n \ U j+m is continuous at x. Thus, there is a neighbourhood V x of x such that Since ψ j ≥ 0, we get f j (y) ≥ u(y) = Hf (y) and so, Then by (4.1) and (4.2), we get for all x ∈ ∂Ω, Thus, (4.3) holds for any f ∈ H 1,p (Ω, w) and applying it to −f and −h together with the inequality P (f + h) ≤ P (f + h), concludes the proof.
Unlike for continuous boundary data in Theorem 3.9, for quasicontinuous boundary data it is in general impossible to have lim Ω∋y→x P f (y) = f (x) for q.e.x ∈ ∂Ω, see Example 4.6 below.However, we get the following uniqueness result as a consequence of Theorem 4.2.
Corollary 4.5.Let f : R n → [−∞, ∞] be a (p, w)-quasicontinuous function in R n such that f ∈ H 1,p (Ω, w).Assume that u is a bounded A-harmonic function in Ω and that there is a set E ⊂ ∂Ω with C p,w (E) = 0 such that lim Ω∋y→x u(y) = f (x) for all x ∈ ∂Ω \ E.
Then u = P f .Proof.Since u is a bounded A-harmonic function in Ω, we have that u ∈ L f +∞χE and u ∈ U f −∞χE .Thus by Theorem 4.2, we get that u ≤ P (f + ∞χ E ) = P f = P (f − ∞χ E ) ≤ u in Ω.
The following example shows that in many situations there is a bounded quasicontinuous function f ∈ H 1,p (R n , w) such that no function u satisfies lim Ω∋y→x u(y) = f (x) for q.e.x ∈ ∂Ω.
In particular it is impossible for the Perron solution P f to attain these quasicontinuous boundary data q.e.
Since the partial sums of f are continuous and coincide with f outside the open sets j≥k B(x j , r j ), k = 1, 2, ... , with arbitrarily small (p, w)-capacity, we see that f is quasicontinuous.For each j there is r ′ j < r j such that f j ≥ 1 2 in B(x j , r ′ j ).Thus But this violates the assumption that ũ(x) = f (x) = 0 q.e. in ∂Ω\G, since C p,w (∂Ω\ G) > 0. Hence there is no function u satisfying (4.4).Replacing f by min{f, 1} yields a similar bounded counterexample.

Theorem 3 . 3 .
([7, Comparison principle 7.6]) Assume that u is A-superharmonic and that v is A-subharmonic in Ω.If lim sup Ω∋y→x v(y) ≤ lim inf Ω∋y→x u(y) for all x ∈ ∂Ω, and if both sides are not simultaneously ∞ or −∞, then v ≤ u in Ω.
and consequently, (3.3) follows.Letting ε → 0 and m → ∞, we conclude that u j ∈ U f +h (Ω).Continuing as in Proposition 3.8, we can conclude that