A Variational Approach to the Isoperimetric Inequality for the Robin Eigenvalue Problem

The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ1 for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ1 among domains with prescribed volume.