Fabrizio Cuccu

Optimization problems for an elastic plate

This paper concerns optimization problems related to bi-harmonic equations subject to either Navier or Dirichlet homogeneous boundary conditions. Physically, in dimension two, our equation models the deformation of an elastic plate which is either hinged or clamped along the boundary, under load. We discuss existence, uniqueness, and properties of the optimizers.

Optimization of the first eigenvalue of equations with indefinite weights

Abstract We investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.

Symmetry of solutions to optimization problems related to partial differential equations

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

Symmetry breaking in the minimization of the second eigenvalue for composite membranes

Let Ω ⊂ ℝ N be an open bounded connected set. We consider the eigenvalue problem –Δ u = λ ρu in Ω with Dirichlet boundary condition, where ρ is an arbitrary function that assumes only two given values 0 α β and is subject to the constraint ∫ Ω ρ d x = αγ + β (| Ω | – γ ) for a fixed 0 γ Ω |. Cox and McLaughlin studied the optimization of the map ρ ⟼ λ k ( ρ ), where λ k is the k th eigenvalue. In this paper we focus our attention on the case when N ≥ 2, k = 2 and Ω is a ball. We show that, under suitable conditions on α, β and γ , the minimizers do not inherit radial symmetry.