Structure Theory of Parabolic Nodal and Singular Sets

Abstract

We establish new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions $u$ to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost all of the nodal and singular sets are covered by regular parabolic Lipschitz graphs with estimates, and that both sets satisfy parabolic Minkoswki estimates depending only on a doubling quantity at a point. Many of our results are new even for the heat equation on $\mathbb{R}^{n}\times \mathbb{R}$.