Harmonic Analysis on chord arc domains

TL;DR

This paper investigates the solvability of the Dirichlet problem for elliptic operators in complex, non-Lipschitz domains, emphasizing the development of tent space theory on such domains.

Contribution

It introduces a novel approach using tent spaces to analyze elliptic operators in rough, chord arc domains, extending classical results beyond Lipschitz boundaries.

Findings

Established solvability conditions for the Dirichlet problem in chord arc domains.

Developed the theory of tent spaces tailored for rough domains.

Showed that small perturbations of elliptic operators preserve solvability.

Abstract

In the present paper we study the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains beyond the Lipschitz category. In our approach, the development of the theory of tent spaces on these domains is essential.