## Abstract

Let

with

*U*be a bounded open connected set in R*(*^{n}*n*≥1). We refer to the unique weak solution of the Poisson problem −Δu=*χ*on_{A}*U*with Dirichlet boundary conditions as*u*for any measurable set_{A}*A*in*U*. The function*ψ*:=*is the torsion function of*^{u}U*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*):=∫*and the supremum is taken over measurable sets*_{A}u_{A}dx*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 inequalitiesOriginal language | English |
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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