Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty}$-measures on the boundary

TL;DR

This paper investigates boundary value problems for Laplace's equation in graph Lipschitz domains with $A_{}$-measures, providing new solvability results in various function spaces and establishing conditions for endpoint cases.

Contribution

It extends solvability results for Dirichlet, Neumann, and Regularity problems in graph Lipschitz domains, including endpoint and Lorentz space cases, with new inequalities and conditions.

Findings

$L^{p,1}$-solvability for Dirichlet problem established.

Range of Neumann problem solvability characterized, including endpoint conditions.

A two-weight Sawyer-type inequality is derived for endpoint Lorentz solvability.

Abstract

We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering $A_{\infty}$-measures on the boundary. More specifically, we study the $L^{p,1}$-solvability for the Dirichlet problem, complementing results of Kenig (1980) and Carro and Ortiz-Caraballo (2018). Then, we study $L^p$-solvability of the Neumann problem, obtaining a range of solvability which is empty in some cases, a clear difference with the arc-length case. When it is not empty, it is an interval, and we consider solvability at its endpoints, establishing conditions for Lorentz space solvability when $p>1$ and atomic Hardy space solvability when $p=1$. Solving the Lorentz endpoint leads us to a two-weight Sawyer-type inequality, for which we give a sufficient condition. Finally, we show how to adapt to the Regularity problem the results for the Neumann problem.