Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary value problems for holomorphic functions on Lipschitz planar domains

William Gryc, Loredana Lanzani, Jue Xiong, Yuan Zhang

Abstract

We study the $\bar\partial$ equation subject to various boundary value conditions on bounded simply connected Lipschitz domains $D\subset\mathbb C$: for the Dirichlet problem with datum in $L^p(bD, σ)$, this is simply a restatement of the fact that members of the holomorphic Hardy spaces are uniquely and completely determined by their boundary values. Here we identify the maximal data spaces and obtain estimates in the maximal $p$-range for the Dirichlet, Regularity-for-Dirichlet, Neumann, and Robin boundary conditions for $\bar\partial$.

Boundary value problems for holomorphic functions on Lipschitz planar domains

Abstract

We study the equation subject to various boundary value conditions on bounded simply connected Lipschitz domains : for the Dirichlet problem with datum in , this is simply a restatement of the fact that members of the holomorphic Hardy spaces are uniquely and completely determined by their boundary values. Here we identify the maximal data spaces and obtain estimates in the maximal -range for the Dirichlet, Regularity-for-Dirichlet, Neumann, and Robin boundary conditions for .
Paper Structure (10 sections, 15 theorems, 103 equations)

This paper contains 10 sections, 15 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boundary value problems for the Laplace operator
  4. The Dirichlet problem for $\overline\partial$: proof of Theorem \ref{['T:D-Dbar']}
  5. Holomorphic Sobolev-Hardy spaces
  6. The Regularity problem for $\overline\partial$: proof of Theorem \ref{['T:D-Dbar-reg']}
  7. The Neumann problem for $\overline\partial$: proof of Theorem \ref{['T:NDbar']}
  8. The Robin problem for $\overline\partial$
  9. The smoothing operator $\mathcal{T}_b$
  10. Proof of Theorem \ref{['T:RDbar']}

Key Result

Theorem 1.1

Let $D\subset \mathbb C$ be a bounded simply connected Lipschitz domain and let $p>0$. Given $f\in L^p(bD, \sigma)$, we have that E:D-Dbar is solvable if and only if $f\in h^p(bD)$, in which case E:D-Dbar has a unique solution and it lies in $\mathcal{H}^p(D)$. Furthermore,

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • ...and 23 more