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

TL;DR

This paper extends the theory of boundary value problems for the Laplace equation to Robin boundary conditions in multiply connected domains, using layer potential methods to represent solutions.

Contribution

It introduces a double layer potential approach for Robin problems, complementing the classical single layer potential method in multiply connected domains.

Findings

Double layer potential effectively represents Robin boundary solutions.

The approach generalizes existing methods for Dirichlet and Neumann problems.

Provides theoretical foundation for Robin boundary value problems.

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.