Some remarks on a shape optimization problem

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

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