The Bilaplacian with Robin boundary conditions

TL;DR

This paper introduces Robin boundary conditions for biharmonic operators, analyzing their spectral properties, domain dependence, and asymptotic behavior, with implications for elastically supported plates and shape optimization.

Contribution

It develops a comprehensive framework for Robin boundary conditions in biharmonic problems, including spectral analysis, asymptotic limits, and shape derivatives, which were not previously studied in detail.

Findings

Robin parameters influence eigenvalues and eigenfunctions

As parameters tend to infinity, Robin problems converge to classical biharmonic problems

Shape derivatives of eigenvalues are characterized for domain optimization

Abstract

We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. We study the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters. We show in particular that when the parameters go to plus infinity the Robin problem converges to other biharmonic problems, and obtain estimates on the rate of divergence when the parameters go to minus infinity. We also analyse the dependence of the operator on smooth perturbations of the domain, computing the shape derivatives of the eigenvalues and giving a characterisation for critical domains under volume and perimeter constraints. We include a number of open problems arising in the context of our results.