Calculus of Variations and Geometric Measure Theory

D. Bucur - A. Giacomini

A variational approach to the isoperimetric inequality for the Robin eigenvalue problem

created by bucur on 19 Jun 2009
modified by giacomini on 08 Jun 2011


Published Paper

Inserted: 19 jun 2009
Last Updated: 8 jun 2011

Journal: Arch. Ration. Mech. Anal.
Volume: 198
Pages: 927-961
Year: 2010


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