A few shape optimization results for a biharmonic Steklov problem (Q2347014)

The authors study the dependence of the eigenvalues of the following Steklov problem for the biharmonic operator on \(\Omega\subset\mathbb{R}^n\): \[ \begin{cases} \triangle ^2 u - \tau\triangle u =0 \text{ in }\Omega, \;\tau>0\\ \frac{\partial ^2 u}{d\nu ^2}=0 \text{ on }\partial\Omega\\ \tau\frac{\partial u}{d\nu}-\text{div}(D^2_{u,\nu})-\frac{\partial\triangle u}{d\nu} =\lambda u\text{ on }\partial\Omega, \end{cases}\tag{1} \] where \(D^2\) is the Hessian matrix of \(u\). The purpose of the paper is to study the dependence of the eigenvalues \(\lambda_j\) of ({1}) on the domain \(\Omega\). The proof consists of several steps: First study ({1}) as the natural fourth order extension of a simpler problem. Prove that the spectrum of the eigenvalue problem for \(N=2\) is discrete. Show that the elementary symmetric functions of the eigenvalues are analytic functions, and that the ball is the critical point of these functions. Show that the ball is a maximizer for the fundamental tone. The proofs use symmetric spaces, Sobolev spaces, integration by parts, spherical coordinates, Bessel functions, Brock-Weinstock inequality.