Dirichlet and Zaremba Type Eigenvalue Optimization Problems On Annular Domains

Abstract

In this thesis, we introduce a series of eigenvalue problems of the Laplacian defined on annular domains in n-dimensional Euclidean space where the inner ball undergoes translation from the concentric configuration. The common goal in each of the problems is to find a domain which maximizes the eigenvalue. The first problem deals with optimising the fundamental eigenvalue with Dirichlet boundary conditions. The second problem also optimises the fundamental eigenvalue but with mixed boundary conditions, in particular, Dirichlet conditions on the inner boundary and Neumann on the outer boundary. After this, we introduce some results which are useful in dealing with degenerate eigenvalues. Finally, we apply these results in optimising the second Dirichlet eigenvalue on the same collection of domains as was considered in the previous optimisation problems.