Abstract
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 Rsmoothness of theboundary) such that essinfvΩ,A,γ ≥ 0. We show that essinfvΩ,A,γ < 0 forany measurable set A, provided A > γΩ. 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.
Original language  English 

Pages (fromto)  513536 
Number of pages  24 
Journal  Proceedings of the London Mathematical Society 
Volume  121 
Issue number  3 
Early online date  29 Apr 2020 
DOIs  
Publication status  Published  1 Sep 2020 
Keywords
 Torsion function
 Dirichlet boundary condition
 Poisson’s equation
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Professor Michiel van den Berg
 School of Mathematics  Emeritus Professor
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member, Honorary and Visiting Academic