An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow

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Let U be a bounded open connected set in Rn (n≥1). We refer to the unique weak solution of the Poisson problem −Δu=χA on U with Dirichlet boundary conditions as uA for any measurable set A in U. The function ψ:=uU is the torsion function of U. Let V be the measure V:=ψLn on U where Ln stands for n-dimensional Lebesgue measure. We study the variational problem


with p∈(0,1) where J(A):=∫AuAdx and the supremum is taken over measurable sets AU subject to the constraint V(A)=pV(U). We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case n=1. The proof makes use of weighted isoperimetric and Pólya–Szegö inequalities
Original languageEnglish
Pages (from-to)365–401
Number of pages37
JournalApplied Mathematics and Optimization
Issue number3
Early online date16 Feb 2016
Publication statusPublished - Jun 2017


  • Two-phase membrane problem
  • Isoperimetric inequality
  • Pólya–Szegö inequality
  • Spherical cap symmetrisation

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