Sign changing solutions of Poisson's equation

2018/04/03 by M. van den Berg, Berg, Michiel van den, Dorin Bucur +1
Computer Science · Mathematics · #35B09 #35J25 #35P99 #58J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.1804.00903

openalex publication_date 2018/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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_\Om,A,γ of -Δv=γ\bf 1_{Ω∖ A}-(1-γ)\bf 1_A with v=0 on ∂ Ω 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,γ≥ 0. We show that \rm essinf v_\Om,A,γ<0 for any measurable set A, provided |A| >γ|\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,γ. Surprisingly, if \Om is a ball, a symmetry breaking phenomenon occurs.

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