Sign changing solutions of Poisson's equation

Let Ω be an open, possibly unbounded, set in Euclidean space Rm with boundary ∂Ω, let A be a measurable subset of Ω with measure |A|, andlet γ ∈ (0, 1). We investigate whether the solution vΩ,A,γ of −∆v =γ1Ω\A − (1 − γ)1A with v = 0 on ∂Ω changes sign. Bounds are obtainedfor |A| in terms of geometric characteristics of Ω (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or R-smoothness of theboundary) such that essinfvΩ,A,γ ≥ 0. We show that essinfvΩ,A,γ < 0 forany measurable set A, provided |A| > γ|Ω|. This value is sharp. We also study the shape optimisation problem of the optimal location of A (withprescribed measure) which minimises the essential infimum of vΩ,A,γ. Surprisingly, if Ω is a ball, a symmetry breaking phenomenon occurs.