Some remarks on a shape optimization problem

Abstract

Given a bounded open set Ω of Rⁿ, n ≥ 2, and α ∈ R, let us consider
μ(Ω,α) = $\min_{\substack{v\in W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\ds\int_{\Omega} |\nabla v|^{2}dx+\alpha \left|\ds\int_{\Omega}|v|v\,dx \right|}{\ds\int_{\Omega} |v|^{2}dx}$.
We study some properties of μ(Ω,α) and of its minimizers, and, depending on α, we determine the sets Ω_α among those of fixed measure such that μ(Ω_α,α) is the smallest possible.