Analyticity and criticality results for the eigenvalues of the biharmonic operator

2016/03/09 by Buoso, Davide · 3 citations
#FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1603.02923

Abstract

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.

Cited by

• Extremal problems for clamped plates under tension
• Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms
• Steklov vs. Steklov: A Fourth-Order Affair Related to the Babuška Paradox

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