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

An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow Let U be a bounded open connected set in $$\mathbb {R}^n$$ R n ( $$n\ge 1$$ n ≥ 1 ). We refer to the unique weak solution of the Poisson problem $$-\Delta u = \chi _A$$ - Δ u = χ A on U with Dirichlet boundary conditions as $$u_A$$ u A for any measurable set A in U. The function $$\psi :=u_U$$ ψ : = u U is the torsion function of U. Let V be the measure $$V:=\psi \,\mathscr {L}^n$$ V : = ψ L n on U where $$\mathscr {L}^n$$ L n stands for n-dimensional Lebesgue measure. We study the variational problem \begin{aligned} I(U,p):=\sup \Big \{ J(A)-V(U)\,p^2\,\Big \} \end{aligned} I ( U , p ) : = sup { J ( A ) - V ( U ) p 2 } with $$p\in (0,1)$$ p ∈ ( 0 , 1 ) where $$J(A):=\int _Au_A\,dx$$ J ( A ) : = ∫ A u A d x and the supremum is taken over measurable sets $$A\subset U$$ A ⊂ U subject to the constraint $$V(A)=pV(U)$$ V ( A ) = p V ( U ) . We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case $$n=1$$ n = 1 . The proof makes use of weighted isoperimetric and Pólya–Szegö inequalities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 75 (3) – Feb 16, 2016
37 pages

/lp/springer_journal/an-unstable-two-phase-membrane-problem-and-maximum-flux-exchange-flow-4N3H9kDQym
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-016-9335-7
Publisher site
See Article on Publisher Site

### Abstract

Let U be a bounded open connected set in $$\mathbb {R}^n$$ R n ( $$n\ge 1$$ n ≥ 1 ). We refer to the unique weak solution of the Poisson problem $$-\Delta u = \chi _A$$ - Δ u = χ A on U with Dirichlet boundary conditions as $$u_A$$ u A for any measurable set A in U. The function $$\psi :=u_U$$ ψ : = u U is the torsion function of U. Let V be the measure $$V:=\psi \,\mathscr {L}^n$$ V : = ψ L n on U where $$\mathscr {L}^n$$ L n stands for n-dimensional Lebesgue measure. We study the variational problem \begin{aligned} I(U,p):=\sup \Big \{ J(A)-V(U)\,p^2\,\Big \} \end{aligned} I ( U , p ) : = sup { J ( A ) - V ( U ) p 2 } with $$p\in (0,1)$$ p ∈ ( 0 , 1 ) where $$J(A):=\int _Au_A\,dx$$ J ( A ) : = ∫ A u A d x and the supremum is taken over measurable sets $$A\subset U$$ A ⊂ U subject to the constraint $$V(A)=pV(U)$$ V ( A ) = p V ( U ) . We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case $$n=1$$ n = 1 . The proof makes use of weighted isoperimetric and Pólya–Szegö inequalities.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 16, 2016

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