IUMJ

Title: Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients

Authors: Erik Lindgren and Regis Monneau

Issue: Volume 62 (2013), Issue 1, 171-199

Abstract:

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $\Delta u-u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of continuity (still in $L^p$ average) for $D^2 u$ and $\partial_tu$. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right-hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the $L^p$ average). Moreover, we get a quantitative estimate of the error in this Taylor expansion. Our method is based on decay estimates obtained by contradiction, using blow-up arguments and Liouville-type theorems. As a by-product of our approach, we deduce that the regular points of the free boundary are locally contained in a $C^1$ hypersurface for the parabolic distance $\sqrt{x^2+|t|}$.