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

TL;DR

This paper investigates the resolutivity and invariance of Perron solutions for degenerate quasilinear elliptic equations with weighted capacities, establishing uniqueness and stability under boundary data perturbations of capacity zero.

Contribution

It demonstrates that Perron solutions are invariant under boundary data perturbations on sets of zero weighted capacity, extending resolutivity results to degenerate equations with weights.

Findings

Perturbations on sets of zero weighted capacity do not affect Perron solutions.

Perron solutions with continuous boundary data are unique outside capacity-zero sets.

The results apply to degenerate elliptic equations with $p$-admissible weights.

Abstract

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u) = 0$ in a bounded open set $\Omega\subset\mathbf{R}^n$. The vector-valued function $\mathcal{A}$ satisfies the standard ellipticity assumptions with a parameter $1<p<\infty$ 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.