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Sign changing solutions of Poisson's equation

Research output: Contribution to journalArticle

Original languageEnglish
Number of pages27
JournalProceedings of the London Mathematical Society
DateSubmitted - 3 Apr 2018
DateAccepted/In press (current) - 20 Jan 2020


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 vΩ,A,γ of −∆v = γ1Ω\A − (1 − γ)1A with 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 essinfvΩ,A,γ ≥ 0. We show that essinfvΩ,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 vΩ,A,γ. Surprisingly, if Ω is a ball, a symmetry breaking phenomenon occurs.

    Research areas

  • Torsion function, Dirichlet boundary condition, Poisson’s equation



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