*preprint*

**Inserted:** 8 oct 2018

**Year:** 2018

**Abstract:**

Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$,
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 $-\Delta
v=\gamma{\bf 1}_{\Omega-A}-(1-\gamma){\bf 1}_{A},\, v\in H_0^1(\Omega)$ 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.