A free boundary problem related to thermal insulation
Luis A. Caffarelli, Dennis Kriventsov · Communications in Partial Differential Equations · 2016
We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have “locally flat” boundaries.