A Free Boundary Problem Related to Thermal Insulation: Flat Implies Smooth

Dennis Kriventsov

Published 2015-11-18Version 1

We show that for a new free boundary problem studied by Caffarelli and the author, the interface is locally the union of the graphs of two $C^{1,\alpha}$ functions at most points. Specifically, this happens at all points where the interface is trapped between two planes which are sufficiently close together. The proof combines ideas introduced by Ambrosio, Fusco, and Pallara for the Mumford-Shah functional with new arguments specific to the problem considered.