Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Fernando Ballesta-Yagüe, María J. Carro

TL;DR

This work develops a comprehensive framework for Dirichlet, Neumann, and Regularity boundary value problems for the Laplacian in graph Lipschitz domains in the plane with boundary measures in $A_{\\infty}$. By employing a conformal mapping to the upper half-plane, the authors translate the problems into weighted settings on $\\mathbb{R}$ and obtain sharp endpoint solvability results, including $L^{p,1}$ Dirichlet solvability and $L^{p}$ Neumann solvability with explicit weight conditions. The results reveal a nuanced range of solvability for the Neumann and Regularity problems, sometimes empty and sometimes an interval, with endpoint theories in Lorentz spaces and atomic Hardy spaces, and they introduce two-weight Sawyer-type inequalities arising from these endpoint analyses. Concrete cone examples with power weights illustrate the range of possibilities, including cases where endpoint solvability fails in $L^{p}$ but holds in $L^{p,1}$ or $H_{at}^{1}$. The paper also outlines open problems regarding optimality of weight conditions and the full scope of endpoint solvability for the Regularity problem.

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.

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

TL;DR

This work develops a comprehensive framework for Dirichlet, Neumann, and Regularity boundary value problems for the Laplacian in graph Lipschitz domains in the plane with boundary measures in . By employing a conformal mapping to the upper half-plane, the authors translate the problems into weighted settings on and obtain sharp endpoint solvability results, including Dirichlet solvability and Neumann solvability with explicit weight conditions. The results reveal a nuanced range of solvability for the Neumann and Regularity problems, sometimes empty and sometimes an interval, with endpoint theories in Lorentz spaces and atomic Hardy spaces, and they introduce two-weight Sawyer-type inequalities arising from these endpoint analyses. Concrete cone examples with power weights illustrate the range of possibilities, including cases where endpoint solvability fails in but holds in or . The paper also outlines open problems regarding optimality of weight conditions and the full scope of endpoint solvability for the Regularity problem.

Abstract

We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering -measures on the boundary. More specifically, we study the -solvability for the Dirichlet problem, complementing results of Kenig (1980) and Carro and Ortiz-Caraballo (2018). Then, we study -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 and atomic Hardy space solvability when . 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.

Paper Structure

This paper contains 39 sections, 46 theorems, 222 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation and preliminaries
  4. Graph Lipschitz Domains
  5. Conformal Mapping
  6. Dirichlet, Neumann and Regularity problems
  7. Function spaces and weights on
  8. Sparse operators
  9. Weights on and conformal map
  10. The Atomic Hardy Space
  11. Hardy spaces over graph Lipschitz domains
  12. Poisson integrals
  13. Poisson integrals and logarithms
  14. The Neumann integral
  15. The class
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a graph Lipschitz domain and let $\Phi\colon \mathbb{R}_{+}^{2}\to \Omega$ be the conformal mapping specified in Section sec:notation. Let $\nu\in A_{\infty}(\Lambda)$ and let where $\Phi(\nu)$ denotes $(\nu\circ \Phi)|\Phi'|$. If $p\in (p_{\Phi}(\nu),\infty)$, given any $f \in L^p(\Lambda, \nu)$, there exists $u$ harmonic in $\Omega$ so that $u$ converges to $f$ non-tangentially

Figures (2)

  • Figure 1: Transference of solutions of the Neumann problem from $\mathbb{R}_{+}^{2}$ to $\Omega$.
  • Figure 2: A cone of angle $\alpha\pi$.

Theorems & Definitions (96)

  • Theorem 1.1: Theorem 4.4 in kenigWeighted
  • Theorem 1.2: Theorem 2.4 in CarroOrtiz
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7: Solvability in $L^{p_{\pm},1}(\Lambda,\nu)$
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 86 more