Boundary value problems for degenerate elliptic equations and systems

TL;DR

This paper investigates boundary value problems for degenerate elliptic equations with $A_2$ weights, establishing new quadratic estimates, boundary trace representations, and solvability results that extend and improve previous unweighted analyses.

Contribution

It introduces novel quadratic estimates for weighted degenerate elliptic equations, enabling boundary trace representations and solvability results in weighted settings.

Findings

Established quadratic estimates for weighted degenerate elliptic equations

Derived boundary trace representations for solutions

Proved solvability and perturbation results in weighted contexts

Abstract

We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.