Symmetry in the composite plate problem

TL;DR

This paper investigates the composite plate eigenvalue problem, establishing existence, regularity, explicit optimal densities, and proving symmetry and positivity of solutions in spherical domains.

Contribution

It provides new results on existence, regularity, explicit optimal densities, and symmetry of solutions for the composite plate eigenvalue problem.

Findings

Existence and regularity of optimal pairs (u, ρ)

Explicit expression for the optimal density ρ

Radial symmetry and positivity of solutions in spherical domains

Abstract

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$ \inf_{\rho \in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho u^2}, $$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W= H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,\rho)$. In particular, we prove existence and regularity and we find the explicit expression of $\rho$. When $\Omega$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $\rho$.