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On the Robin problem for the Laplace equation in multiply connected domains

Alberto Cialdea, Vita Leonessa

TL;DR

The paper addresses the Robin boundary value problem for the Laplace equation $\Delta u=0$ in a multiply connected domain $\Omega$ with boundary condition $\partial u/\partial \nu+ h u=g$ on $\Sigma$. It develops a potential-theoretic representation using a double layer potential $u=D\psi$ and reduces solvability to a Fredholm integral equation for the density, incorporating the Robin term via an operator $H$. The results establish existence and uniqueness for $g\in L^p(\Sigma)$ (with appropriate 2D exceptional-case refinements) and provide explicit density expressions in terms of $S\varphi$ with $\varphi$ solving $-\tfrac14\varphi+H\varphi=g$. This work extends the prior Dirichlet/Neumann theory in CiLeMa2012 to Robin problems in multiply connected domains and reinforces the use of boundary integral methods in potential theory.

Abstract

This paper complements the existing theory developed in [5] for the Dirichlet and Neumann problems for the Laplace equation, in multiply connected domains. Within the framework of layer potential methods, we study the Laplace equation under Robin boundary conditions, representing the solutions by means of a double layer potential. We observe that the classical approach searches the solutions in terms of a single layer potential.

On the Robin problem for the Laplace equation in multiply connected domains

TL;DR

The paper addresses the Robin boundary value problem for the Laplace equation in a multiply connected domain with boundary condition on . It develops a potential-theoretic representation using a double layer potential and reduces solvability to a Fredholm integral equation for the density, incorporating the Robin term via an operator . The results establish existence and uniqueness for (with appropriate 2D exceptional-case refinements) and provide explicit density expressions in terms of with solving . This work extends the prior Dirichlet/Neumann theory in CiLeMa2012 to Robin problems in multiply connected domains and reinforces the use of boundary integral methods in potential theory.

Abstract

This paper complements the existing theory developed in [5] for the Dirichlet and Neumann problems for the Laplace equation, in multiply connected domains. Within the framework of layer potential methods, we study the Laplace equation under Robin boundary conditions, representing the solutions by means of a double layer potential. We observe that the classical approach searches the solutions in terms of a single layer potential.
Paper Structure (4 sections, 6 theorems, 58 equations)

This paper contains 4 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. On the Dirichlet and Neumann problems
  4. On the Robin problem

Key Result

lemma 1

The singular integral operator $J$ can be reduced on the left by the operator $J^{'}:L_{1}^{p}(\Sigma)\longrightarrow L^{p}(\Sigma)$ defined as where the symbol $\underset{{\Sigma}}{\ast}$ means that, if $w$ is an $(n-1)$-form on $\Sigma$ and $w=w_0\,\mathrm{d}\sigma$, then $\underset{{\Sigma}}{\ast}w=w_0$.

Theorems & Definitions (7)

  • lemma 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • proof