Sign changing solutions of Poisson's equation

Abstract

Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partial\Omega,$ let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$. We investigate whether the solution $v_{\Om,A,\gamma}$ of $-\Delta v=\gamma{\bf 1}_{\Omega \setminus A}-(1-\gamma){\bf 1}_{A}$ with $v=0$ on $\partial \Omega$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of $\Om$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that ${\rm essinf} v_{\Om,A,\gamma}\ge 0$. We show that ${\rm essinf} v_{\Om,A,\gamma}<0$ for any measurable set $A$, provided $|A| >\gamma |\Om|$. This value is sharp. We also study the shape optimisation problem of the optimal location of $A$ (with prescribed measure) which minimises the essential infimum of $v_{\Om,A,\gamma}$. Surprisingly, if $\Om$ is a ball, a symmetry breaking phenomenon occurs.