Inserted: 19 jun 2009
Last Updated: 8 jun 2011
Journal: Arch. Ration. Mech. Anal.
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 $\lambda_1$ for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of $\lambda_1$ among domains with prescribed volume.
Keywords: isoperimetric inequality, Robin eigenvalue, non-smooth domains, SBV-spaces