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 A⊂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 Pólya–Szegö inequalities
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 A⊂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 Pólya–Szegö inequalities
| Original language | English |
|---|---|
| Pages (from-to) | 365–401 |
| Number of pages | 37 |
| Journal | Applied Mathematics and Optimization |
| Volume | 75 |
| Issue number | 3 |
| Early online date | 16 Feb 2016 |
| DOIs | |
| Publication status | Published - Jun 2017 |
Keywords
- Two-phase membrane problem
- Isoperimetric inequality
- Pólya–Szegö inequality
- Spherical cap symmetrisation