Sign changing solutions of Poisson's equation

Let Ω be an open, possibly unbounded, set in Euclidean space R m with boundary ∂ Ω , let A be a measurable subset of Ω with measure | A | and let γ ∈ ( 0 , 1 ) . We investigate whether the solution v Ω , A , γof − Δ v = γ 1 Ω ∖ A − ( 1 − γ ) 1 Awith v = 0 on ∂ Ω changes sign. Bounds are obtained for | A | in terms of geometric characteristics of Ω (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure or R ‐smoothness of the boundary) such that essinf v Ω , A , γ ⩾ 0 . We show that essinf v Ω , A , γ < 0 for any measurable set A , provided | A | > γ | Ω | . 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 Ω , A , γ . Surprisingly, if Ω is a ball, a symmetry breaking phenomenon occurs.