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

作者：McGillivray I^*

来源：Applied Mathematics and Optimization, 2017, 75(3): 365-401.

DOI：10.1007/s00245-016-9335-7

• Two-phase membrane problem
• Isoperimetric inequality
• Polya-Szegoinequality
• Spherical cap symmetrisation

Let U be a bounded open connected set in R-n (n >= 1). We refer to the unique weak solution of the Poisson problem - Delta u = xA on U with Dirichlet boundary conditions as u (A) for any measurable set A in U. The function psi := u(U) is the torsion function of U. Let V be the measure V := psi 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 is an element of (0, 1) where J (A) := J(A) u(A) dx and the supremum is taken over measurable sets A subset of U 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 Polya-Szego inequalities.

• 出版日期2017-6