The calibration method for the Thermal Insulation functional

We provide minimality criteria by construction of calibrations for functionals arising in the theory of Thermal Insulation.Comment: We added an appendix to justify the method. We added more explanations and an additional appendix to show how to build the vector field in Theorem 3.

A stability result for nonlinear neumann problems in reifenberg flat domains in Rn

In this paper we prove that if Ωk is a sequence of Reifenberg-flat domains in RN that converges to Ω for the complementary Hausdorff distance and if in addition the sequence Ωk has a "uniform size of holes", then the solutions uk of a Neumann problem of the form (...) converge to the solution u of the same Neumann problem in Ω. The result is obtained by proving the Mosco convergence of some Sobolev spaces, that follows from the extension property of Reifenberg-flat domains

Higher integrability of the gradient for the Thermal Insulation problem

We prove the higher integrability of the gradient for minimizers of the thermal insulation problem, an analogue of De Giorgi's conjecture for the Mumford-Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than $n-1$

Systems of Fully Nonlinear Degenerate Elliptic Obstacle problems with Dirichlet boundary conditions

In this paper we prove existence and uniqueness of viscosity solutions of elliptic systems associated to fully nonlinear operators for minimization problems that involve interconnected obstacles. This system appears, among other, in the theory of the so-called optimal switching problems on bounded domains.Comment: Minor corrections throughout the tex