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

Abubakar Mwasa,Jana Björn,Anders Björn

doi:10.1016/j.jmaa.2021.125937

Abubakar Mwasa, Jana Björn + Show 1 more

Open Access

https://doi.org/10.1016/j.jmaa.2021.125937

Copy DOI

Abstract

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation divA(x,∇u)=0 in a bounded open set Ω⊂Rn. The vector-valued function A satisfies the standard ellipticity assumptions with 1<p<∞ and a p-admissible weight w. We show that arbitrary perturbations on sets of (p,w)-capacity zero of continuous (and certain quasicontinuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p,w)-capacity zero.

Highlights

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 Dirichlet problem amounts to finding a solution of the partial differential equation in Ω with prescribed boundary data on the boundary of Ω
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

Summary

Introduction

Granlund–Lindqvist–Martio [6] were the first to use the Perron method to study the nonlinear equation div(∇qF (x, ∇u)) = 0. 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. 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 apply to the more general A-harmonic functions, defined by equations rather than minimization problems

Notation and preliminaries

Perron solutions and resolutivity

Quasicontinuous functions