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Resolutivity and Invariance for the Perron Method for Degenerate

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Resolutivity and Invariance for the Perron Method for Degenerate Resolutivity and invariance for the Perron method for degenerate equations of divergence type Anders Bj¨orn Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden [email protected], ORCID: 0000-0002-9677-8321 Jana Bj¨orn Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden [email protected], ORCID: 0000-0002-1238-6751 Abubakar Mwasa Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden Department of Mathematics, Busitema University, P.O.Box 236, Tororo, Uganda [email protected], [email protected], ORCID: 0000-0003-4077-3115 Abstract We consider Perron solutions to the Dirichlet problem for the quasilinear n elliptic equation div A(x, ∇u) = 0 in a bounded open set Ω ⊂ R . The vector-valued function A satisfies the standard ellipticity assumptions with a parameter 1 <p< ∞ and a p-admissible weight w. We show that arbitrary perturbations on sets of (p,w)-capacity zero of continuous (and certain quasi- continuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Per- ron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p,w)-capacity zero. Key words and phrases: capacity, degenerate quasilinear elliptic equation of divergence type, Dirichlet problem, Perron solution, quasicontinuous function, resolutive. Mathematics Subject Classification (2020): Primary: 35J66, Secondary: 31C45, 35J25, 35J92. arXiv:2008.00883v1 [math.AP] 3 Aug 2020 1. 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 Rn. The mapping A :Ω × Rn → Rn satisfies the standard ellipticity assumptions with a parameter 1 <p< ∞ and a p-admissible weight as in Heinonen–Kilpel¨ainen– 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 1 2 Anders Bj¨orn, Jana Bj¨orn and Abubakar Mwasa 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 ⊂ Rn. It gives an upper and a lower Perron solution (see Definition 3.2) and when the two coincide, we get a suitable solution Pf 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 ∇qF 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−2 ∆pu := div(|∇u| ∇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. Kilpel¨ainen [8] extended the resolutivity to general p, which in turn was extended to weighted Rn by Heinonen–Kilpel¨ainen–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 continu- ous 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 some- what weaker uniqueness result is proved for quasicontinuous Sobolev functions in Corollary 4.5. Much as we use Heinonen–Kilpel¨ainen–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 Bj¨orn–Bj¨orn–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. Acknowledgement. A. B and J. B. were partially supported by the Swedish Research Council grants 2016-03424 resp. 621-2014-3974 and 2018-04106. A. M. was supported by the SIDA (Swedish International Development Cooperation Agency) Resolutivity and invariance for the Perron method for degenerate equations 3 project 316-2014 “Capacity building in Mathematics and its applications” under the SIDA bilateral program with the Makerere University 2015–2020, contribution No. 51180060. 2. 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 Rn,n ≥ 2, and 1 <p< ∞. We use ∂Ω and Ω to denote the boundary and the closure of Ω, respectively. n We write x to mean a point x = (x1, ... , xn) ∈ R and for a function v which is infinitely many times continuously differentiable, i.e. v ∈ C∞(Ω), we write ∇v = (∂1v, ... , ∂nv) for the gradient of v. We follow Heinonen–Kilpel¨ainen–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 Kilpel¨ainen [9]. Definition 2.1. The weighted Sobolev space H1,p(Ω, w) is defined to be the com- pletion of the set of all v ∈ C∞(Ω) such that 1/p p p kvkH1,p(Ω,w) = (|v| + |∇v| )w dx < ∞ ZΩ with respect to the norm kvkH1,p(Ω,w), where w is the weight function which we define later. 1,p ∞ 1,p The space H0 (Ω, w) is the completion of C0 (Ω) in H (Ω, w) while a function 1,p 1,p ′ ′ ⋐ v is in Hloc (Ω, w) if and only if it belongs to H (Ω , w) for every open set Ω Ω. As usual, E ⋐ Ω if E is a compact subset of Ω and ∞ ∞ n ⋐ C0 (Ω) = {v ∈ C (R ) : supp v Ω}. Throughout the paper, the mapping A : Ω × Rn → Rn, 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 ∈ Rn, and continuous in q for a.e. x ∈ Rn. Also, for all q ∈ Rn and a.e. x ∈ Rn, the following hold A(x, q) · q ≥ αw(x)|q|p and |A(x, q)|≤ βw(x)|q|p−1, (2.1) n (A(x, q1) − A(x, q2)) · (q1 − q2) > 0 for q1, q2 ∈ R , q1 6= q2, A(x, λq)= λ|λ|p−2A(x, q) for λ ∈ R, λ 6=0. 1,p Definition 2.2. A function u ∈ Hloc (Ω, w) is said to be a (weak) solution of (1.1) ∞ in Ω if for all test functions ϕ ∈ C0 (Ω), the following integral identity holds A(x, ∇u) · ∇ϕ dx =0. (2.2) ZΩ 1,p A function u ∈ Hloc (Ω, w) is said to be a supersolution of (1.1) in Ω if for all ∞ nonnegative functions ϕ ∈ C0 (Ω), A(x, ∇u) · ∇ϕ dx ≥ 0. ZΩ A function u is a subsolution of (1.1) if −u is a supersolution of (1.1). 4 Anders Bj¨orn, Jana Bj¨orn and Abubakar Mwasa 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, The- orem 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 sub- and a supersolution, see [7, bottom p. 58]. By [7, Theorems 3.70 and 6.6], every solution u has a H¨older continuous representative v (i.e.
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