A parabolic analogue of the higher-order comparison theorem of De Silva and Savin and its slit domain counterpart

Abstract:  In this colloquium, I will talk about a higher Schauder regularity result which states that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ \alpha}$ domain is $H^{k+ \alpha}$ up to the boundary for $k \geq 1$ and not just $H^{k-1 + \alpha}$ which is implied by standard Schauder regularity theory. This can be thought of as a parabolic analogue of a recent important results of De Silva and Savin and we closely follow their ideas. We also give counterexamples to the fact that analogous results are not true at the corner and base of the parabolic boundary. Besides being an interesting regularity result in its own right, a direct application of such a result as well as its elliptic counterpart above implies $C^{\infty}$ smoothness of a priori $C^{1, \alpha}$ free boundaries without the use of the hodograph transformation of Kinderlehrer-Nirenberg-Spruck. This is a joint work with Nicola Garofalo. If time permits,I will also talk about its subsequent generalization to the case of slit like domains. Such a result implies smoothness of the free boundary in the fairly recent breakthrough paper of Danielli-Garofalo-Petrosyan-To on parabolic Signorini problem ( which is the prototype of evolutionary variational inequalities) where among other results, they established the optimal regularity of solutions in space variable by a delicate adaptation of Poon's frequency function approach. As a side note, I would like to mention as well that the frequency function approach in past has had far reaching applications in backward uniqueness of parabolic equations and liquid crystal flow and also in regularity theory of harmonic map flow which are however more global phenomenons. This later result has been obtained more recently in a joint work with Mariana Smit Vega Garcia and Andrew Zeller.