Symmetry in the composite plate problem

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $\inf_{ρ\in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_Ω(Δu)^2}{\int_Ω ρu^2},$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(Ω)$ for Dirichlet boundary conditions and $\mathcal W= H^2(Ω) \cap H^1_{0}(Ω)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $Δ^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,ρ)$. In particular, we prove existence and regularity and we find the explicit expression of $ρ$. When $Ω$ 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 $ρ$.

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