Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

JB Kennedy

Abstract

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in R^N of given volume M. When k=2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p=2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant α in the boundary condition, or equivalently of the volume M. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions Δ u + β ∂ u/∂ ν + γ u = 0.

Keywords: Laplacian, p-Laplacian,isoperimetric problem, shape optimisation, Robin boundary conditions, Wentzell boundary conditions.

AMS Subject Classification: Primary 35P15; secondary (35J25, 35J60).