Strong barriers for weighted quasilinear equations

TL;DR

This paper develops strong barrier functions for weighted quasilinear elliptic operators, enabling solutions to boundary value problems with singular data and establishing a geometric Hardy inequality, applicable to rough domains.

Contribution

It introduces a novel construction method for barriers applicable to general divergence form elliptic operators on irregular domains.

Findings

Established solvability of Poisson equations with boundary singularities.

Derived a geometric Hardy inequality using the constructed barriers.

Applicable to a broad class of elliptic operators on rough domains.

Abstract

In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.