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

• Published in: Applied mathematics & optimization Vol. 75; no. 3; pp. 365 - 401
• Main Author: McGillivray, I
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2017; Springer Nature B.V
• Subjects: Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Control; Dirichlet problem; Functions (mathematics); Inequalities; Inequality; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Membranes; Numerical and Computational Physics; Partial differential equations; Simulation; Stands; Systems Theory; Texts; Theoretical; Torsion; 35J20; Two-phase membrane problem; 26D10; 35J60; Isoperimetric inequality; Pólya–Szegö inequality; Spherical cap symmetrisation
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9335-7

Let U be a bounded open connected set in R n ( n ≥ 1 ). We refer to the unique weak solution of the Poisson problem - Δ u = χ A on U with Dirichlet boundary conditions as u A for any measurable set A in U . The function ψ : = u U is the torsion function of U . Let V be the measure V : = ψ L n on U where L n stands for n -dimensional Lebesgue measure. We study the variational problem I ( U , p ) : = sup { J ( A ) - V ( U ) p 2 } with p ∈ ( 0 , 1 ) where J ( A ) : = ∫ A u A d x and the supremum is taken over measurable sets A ⊂ U subject to the constraint 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 . The proof makes use of weighted isoperimetric and Pólya–Szegö inequalities.