Claudia Anedda

Second-order estimates for boundary blowup solutions of special elliptic equations

We find a second-order approximation of the boundary blowup solution of the equation, with, in a bounded smooth domain. Furthermore, we consider the equation. In both cases, we underline the effect of the geometry of the domain in the asymptotic expansion of the solutions near the boundary.

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.