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

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 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 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. Key words and phrases: capacity, degenerate quasilinear elliptic equation of divergence type, Dirichlet problem, Perron solution, quasicontinuous function, resolutive.