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arXiv · 2506.23961

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

Fernando Ballesta-Yagüe·María J. Carro
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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.

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Fernando Ballesta-Yagüe, María J. Carro. 2025-06-30. Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty}$-measures on the boundary. https://doi.org/10.1016/j.aim.2025.110734

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