Eigenvalues of polyharmonic operators on variable domains

Buoso, Davide; Lamberti, Pier Domenico

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1225-1235.

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.

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