Title: Geometric Optimization of the Robin Eigenvalue Problem in the Complement of a Bounded Set

Abstract: We consider the Laplace operator under Robin boundary conditions in the complement of a compact set. In contrast to bounded domains, the spectrum here is not purely discrete. We characterize the discrete spectrum using an appropriate Steklov Eigenvalue problem, with the peculiarity that the eigenfunctions are not necessarily square-integrable.

Assuming that the lowest point of the spectrum is a discrete eigenvalue, we consider the following geometric optimization problem: Among all sets with a given measure, the complement of which set provides the largest first Robin Eigenvalue?

To this end, we present the idea of Domain Variations. Using this approach, we show that the ball is a local maximizer of the first Robin Eigenvalue among all domains with prescribed measure. In general, however, the ball is not the global maximizer.