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Sign changing solutions of Poisson's equation
Author(s) -
Berg M. van den,
Bucur D.
Publication year - 2020
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms.12334
Subject(s) - mathematics , infimum and supremum , measure (data warehouse) , mathematical analysis , laplace operator , ball (mathematics) , open set , obstacle problem , combinatorics , boundary (topology) , database , computer science
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.