Shape deformation for vibrating hinged plates

Abstract

We consider the biharmonic operator subject to homogeneous intermediate boundary conditions of Steklov-type. We prove an analyticity result for the dependence of the eigenvalues upon domain perturbation and compute the appropriate Hadamard-type formulas for the shape derivatives. Finally, we prove that balls are critical domains for the symmetric functions of multiple eigenvalues subject to volume constraint

A note on the Neumann eigenvalues of the biharmonic operator

“…We also mention , where the author considers the shape sensitivity problem for the eigenvalues of the biharmonic operator (in particular, also those of problem ) for . We note that other issues have been addressed in the literature for polyharmonic operators, such as analyticity, continuity, and stability estimates for the eigenvalues with respect to the shape; we refer to and the references therein.…”

“…1 and that coincide with the Dirichlet eigenvalues of the biharmonic operator.1 N 1 , 1OE. We note that other issues have been addressed in the literature for polyharmonic operators, such as analyticity, continuity, and stability estimates for the eigenvalues with respect to the shape; we refer to [8][9][10][11][12][13] and the references therein.We recall that problem (1) admits an infinite sequence of nonnegative eigenvalues of finite multiplicity that depend on 2 OE0, 1OE and that we denote here by 0 D 1 . / D 2 .…”

Boundary homogenization for a triharmonic intermediate problem

“…for all > 0 small enough. We plan to pass to the limit in (5) as ! 0 and prove that the limit problem is as in Theorem 1.…”

“…We note that the analysis of the cases α ≤3/2 is in spirit of the paper , which is devoted to the Navier–Stokes system. For recent results concerning domain perturbation problems for higher order operators, we refer to . We believe that our analysis could be carried out in the case of general polyharmonic operators of order 2 m subject to various types of intermediate boundary conditions.…”

“…We note that the analysis of the cases˛Ä 3=2 is in spirit of the paper [9], which is devoted to the Navier-Stokes system. For recent results concerning domain perturbation problems for higher order operators, we refer to [4][5][6][7][8]. We believe that our analysis could be…”