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Structure Theory of Parabolic Nodal and Singular Sets

Max Hallgren, Robert Koirala, Zilu Ma

TL;DR

This work develops a comprehensive parabolic geometric-measure framework for nodal and singular sets of solutions to parabolic inequalities with parabolic Lipschitz coefficients. It introduces a robust frequency-based analysis, quantitative symmetry, and a multi-scale neck-decomposition that captures the transition from singular to regular behavior. The authors prove that $Z(u)$ and $S(u)$ are parabolic rectifiable up to negligible sets and satisfy sharp parabolic Minkowski bounds that depend only on a doubling quantity, with sharper caloric-case bounds tied to the parabolic frequency $N$. The approach hinges on a new strong neck structure, a uniform plane of symmetry across scales, and Carleson-type control of parabolic $\beta$- and $\kappa$-numbers, enabling finite-resolution decompositions and time-slice regularity results that extend classical heat-equation insights to parabolic inequalities.

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

Structure Theory of Parabolic Nodal and Singular Sets

TL;DR

This work develops a comprehensive parabolic geometric-measure framework for nodal and singular sets of solutions to parabolic inequalities with parabolic Lipschitz coefficients. It introduces a robust frequency-based analysis, quantitative symmetry, and a multi-scale neck-decomposition that captures the transition from singular to regular behavior. The authors prove that and are parabolic rectifiable up to negligible sets and satisfy sharp parabolic Minkowski bounds that depend only on a doubling quantity, with sharper caloric-case bounds tied to the parabolic frequency . The approach hinges on a new strong neck structure, a uniform plane of symmetry across scales, and Carleson-type control of parabolic - and -numbers, enabling finite-resolution decompositions and time-slice regularity results that extend classical heat-equation insights to parabolic inequalities.

Abstract

We establish new estimates for the size and structure of the nodal set and the singular set of solutions 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 .

Paper Structure

This paper contains 42 sections, 100 theorems, 729 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. General setting
  5. Caloric setting
  6. Outline of the proof
  7. Frequency
  8. Conjugate heat kernel
  9. Energy functionals
  10. Monotonicity and quantitative uniqueness in caloric setting
  11. Symmetry and Cone Splitting
  12. Quantitative symmetry
  13. Propagation of symmetry
  14. Sharp cone splitting inequality
  15. Existence of a plane of symmetry
  16. ...and 27 more sections

Key Result

Theorem 1

Suppose $u$ is a solution to eq-parabolic-equation on $P(\mathbf{0},5)$ satisfying eq-intro-doubling-condition. For any $k\in \{n,n+1\}$ and any $\epsilon\in (0,1]$, there exist points $\mathbf{x}_i \in P(\mathbf{0},\tfrac{1}{2})$, vertical planes $V_i \in \operatorname{Aff}_{\mathcal{P}}(k)$, radii and the graphs $G_i \coloneqq \{ \mathbf{x}+ F_i(\mathbf{x})\colon \mathbf{x} \in V_i \cap P(\mathb

Figures (1)

  • Figure 1: Suppose the plane of symmetry is $V=L^k\times \{0\}$. Bottom right: If we are close to the $t=0$ time slice but away from $V$, then the characteristic curves appear horizontal under parabolic scaling. Top right: If we are away from $t=0$ and $V$, then the characteristic curves appear vertical under parabolic scaling.

Theorems & Definitions (275)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Remark 6
  • Definition 7
  • Theorem 8
  • Remark 9
  • Corollary 10
  • ...and 265 more