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Publication - Dr Ivor McGillivray

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

    Citation

    McGillivray, IE, 2017, ‘An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow’. Applied Mathematics and Optimization, vol 75., pp. 365?401

    Abstract

    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

    I(U,p):=sup{J(A)−V(U)p2}

    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

    Full details in the University publications repository