Symmetry in the composite plate problem

Abstract

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$\underset{\rho \in \mathrm{P}}{\inf }\underset{u\in \mathcal{W}\setminus \{0\}}{\inf }\frac{{\int }_{\Omega }(\Delta u{)}^2}{{\int }_{\Omega }\rho u^2},$$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}=H_0^2(\Omega )$ for Dirichlet boundary conditions and $\mathcal{W}=H^2(\Omega )\cap H_0^1(\Omega )$ 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 $\rho$.

26 pages

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