Domain Perturbation for
Linear and Semi-Linear Boundary Value Problems

Abstract

This is a survey on elliptic boundary value problems on varying domains and tools needed for that. Such problems arise in numerical analysis, in shape optimisation problems and in the investigation of the solution structure of nonlinear elliptic equations. The methods are also useful to obtain certain results for equations on non-smooth domains by approximation by smooth domains.

Domain independent estimates and smoothing properties are an essential tool to deal with domain perturbation problems, especially for non-linear equations. Hence we discuss such estimates extensively, together with some abstract results on linear operators.

A second major part deals with specific domain perturbation results for linear equations with various boundary conditions. We completely characterise convergence for Dirichlet boundary conditions and also give simple sufficient conditions. We then prove boundary homogenisation results for Robin boundary conditions on domains with fast oscillating boundaries, where the boundary condition changes in the limit. We finally mention some simple results on problems with Neumann boundary conditions.

The final part is concerned about non-linear problems, using the Leray-Schauder degree to prove the existence of solutions on slightly perturbed domains. We also demonstrate how to use the approximation results to get solutions to nonlinear equations on unbounded domains.