Structure Theory of Parabolic Nodal and Singular Sets

TL;DR

This paper develops new estimates for the size and structure of nodal and singular sets of solutions to parabolic inequalities, showing they are mostly covered by regular graphs and satisfy Minkowski estimates, advancing understanding even for the heat equation.

Contribution

It introduces novel size and structural estimates for parabolic nodal and singular sets, including regular graph coverings and Minkowski bounds, applicable to solutions with Lipschitz coefficients.

Findings

Most of the nodal and singular sets are covered by regular parabolic Lipschitz graphs.

Both sets satisfy parabolic Minkowski estimates depending on a doubling condition.

Results are new even for the classical heat equation.

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 $\mathbb{R}^{n}\times \mathbb{R}$.