Some remarks on a shape optimization problem

Abstract

Given $\Omega$ bounded open set of ${\mathbb{R}}^n$ and $\alpha \in \mathbb{R}$, 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 $\mu (\Omega ,\alpha )$ and of its minimizers, and, depending on $\alpha$, we determine the set ${\Omega }_{\alpha }$ among those of fixed measure such that $\mu ({\Omega }_{\alpha },\alpha )$ is the smallest possible.

arXiv commentThis paper has been written for possible publication in a special volume dedicated to the 3rd Italian-Japanese workshop "Geometric Properties for Parabolic and Elliptic PDE's", held in Tokyo in September 2013