AN ELLIPTIC EQUATION IN A DOMAIN WITH PERTURBED BOUNDARY

Abstract

We consider the Laplace equation, with different boundary conditions, on a planar domain defined as a disc with a small perturbation of its boundary. First, we find an asymptotic solution in the interior of the disc using two analytical methods: separations of variables and Laplace’s formula. A third approach consists in combining the asymptotic approach with finited ifferences in polar coordinates. Some numerical examples are given at the end of this work.