An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions

James Kennedy

Abstract

We prove that the second eigenvalue of the Laplacian with Robin boundary conditions is minimised amongst all bounded Lipschitz domains of fixed volume by the domain consisting of the disjoint union of two balls of equal volume.

Keywords: Isoperimetric inequality, Laplacian, Robin boundary conditions, elastically supported membrane.

AMS Subject Classification: Primary 35P15; secondary 35J25.