Higher integrability of the gradient for the thermal insulation problem

Camille Labourie (1) , Emmanouil Milakis (1)

1 UCY - University of Cyprus [Nicosia]

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 $n-1$.

Mots clés: Calculus of variations; Free discontinuity problems; Thermal insulation; Higher integrability

Domaines: Equations aux dérivées partielles [math.AP]

Camille Labourie, Emmanouil Milakis. Higher integrability of the gradient for the thermal insulation problem. Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, 2022, ⟨10.4171/IFB/481⟩. ⟨hal-03207514v3⟩