Free Boundary Problems for Harmonic and Caloric Measure

In this paper we consider two free boundary problems, which we solve using a combination of techniques and tools from harmonic analysis, geometric measure theory and partial differential equations. The first problem is a two-phase problem for harmonic measure, initially studied by Kenig and Toro [KT06]. The central difficulty in that problem is the possibility of degeneracy; losing geometric information at a point where both phases vanish. We establish non-degeneracy by proving that the Almgren frequency formula, applied to an appropriately constructed function, is "almost monotone". In this way, we prove a sharp Holder regularity result (this work was originally published in [Eng14]). The second problem is a one-phase problem for caloric measure, initially posed by Hofmann, Lewis and Nystrom [HLN04]. Here the major difficulty is to classify the "flat blowups". We do this by adapting work of Andersson and Weiss [AW09], who analyzed a related problem arising in combustion. This classification allows us to generalize results of [KT03] to the parabolic setting and answer in the affirmative a question left open in the aforementioned paper of Hofmann et al. (this work was originally published in [Eng15]).