Eigenvalues of polyharmonic operators on variable domains

Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35126 Padova, Italy
dbuoso@math.unipd.it; lamberti@math.unipd.it

Received: 4 April 2012
Revised: 13 November 2012

Abstract

We consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains.