Optimal regularity for supercritical parabolic obstacle problems

Abstract

We study the obstacle problem for parabolic operators of the type (Formula presented.), where L is an elliptic integro-differential operator of order 2s, such as (Formula presented.), in the supercritical regime (Formula presented.). The best result in this context was due to Caffarelli and Figalli, who established the (Formula presented.) regularity of solutions for the case (Formula presented.), the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the (Formula presented.) regularity of the free boundary. Moreover, at all free boundary points (Formula presented.), we establish the following expansion: (Formula presented.) with (Formula presented.), (Formula presented.) and (Formula presented.). © 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.