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Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

Simon Bortz, Steven Hofmann, José María Martell, Kaj Nyström

TL;DR

The paper proves that, under minimal hypotheses, solvability of the $L^p(dσ)$ Dirichlet problem for the heat equation in a parabolic domain Ω forces the boundary Σ to be parabolic uniformly rectifiable. The authors develop a parabolic analogue of the elliptic theory, introducing the parabolic weak half-space approximation (WHSA) condition, a Corona decomposition by regular Lip(1,1/2) graphs, and detailed caloric measure/Green function estimates. By first obtaining WHSA from weak-$A_∞$ caloric measure and corkscrew conditions, and then upgrading corona pieces to regular Lip graphs, they derive a semi-coherent Corona decomposition that yields parabolic UR. This work completes the program of establishing parabolic UR as the natural geometric framework governing boundary regularity and $L^p$ solvability for the heat equation, extending prior elliptic results to the parabolic setting and highlighting the role of temporal dynamics in boundary structure.

Abstract

Let $Ω\subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $Σ= \partial Ω$. Under minimal and natural background assumptions - namely, that $Σ$ is time-symmetrically parabolic Ahlfors--David regular and that $Ω$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(dσ)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $Ω$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $σ= \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_Σ$, then $Σ$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $Ω$ with boundary data in $L^p(dσ)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting.

Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

TL;DR

The paper proves that, under minimal hypotheses, solvability of the Dirichlet problem for the heat equation in a parabolic domain Ω forces the boundary Σ to be parabolic uniformly rectifiable. The authors develop a parabolic analogue of the elliptic theory, introducing the parabolic weak half-space approximation (WHSA) condition, a Corona decomposition by regular Lip(1,1/2) graphs, and detailed caloric measure/Green function estimates. By first obtaining WHSA from weak- caloric measure and corkscrew conditions, and then upgrading corona pieces to regular Lip graphs, they derive a semi-coherent Corona decomposition that yields parabolic UR. This work completes the program of establishing parabolic UR as the natural geometric framework governing boundary regularity and solvability for the heat equation, extending prior elliptic results to the parabolic setting and highlighting the role of temporal dynamics in boundary structure.

Abstract

Let be an open set in space-time with boundary . Under minimal and natural background assumptions - namely, that is time-symmetrically parabolic Ahlfors--David regular and that satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with satisfies a weak- condition with respect to the surface measure , then is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in with boundary data in , for some , implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and solvability, in the parabolic setting.
Paper Structure (39 sections, 54 theorems, 640 equations)

This paper contains 39 sections, 54 theorems, 640 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Outline of the proof
  3. Organization of the paper
  4. Notation and Definitions
  5. Conventions concerning floating constants
  6. Dyadic cubes and associated geometrical notions
  7. Dyadic cubes
  8. Whitney regions
  9. Chain regions
  10. The (parabolic) weak half-space approximation (WHSA) condition
  11. Parabolic uniform rectifiability
  12. Lip(1,1/2) and Regular Lip(1,1/2) graph domains
  13. Corona decompositions and big pieces squared
  14. Estimates for caloric measure and the Green function
  15. Proof of Theorem \ref{['main.thrm']}: the (parabolic) WHSA condition
  16. ...and 24 more sections

Key Result

Theorem 1.1

Assume that $\Omega \subset \mathbb{R}^{n+1}$ is an open set. Assume that $\Sigma=\partial\Omega$ is time symmetric Ahlfors-David regular in the sense of Definition ADR.def with constant $M$, that $\Omega$ satisfies the corkscrew condition in the sense of Definition CS.def with constant $\gamma$, an

Theorems & Definitions (131)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.2
  • Definition 2.4: Parabolic Ahlfors-David regular
  • Lemma 2.6
  • proof
  • Definition 2.7: Corkscrew condition
  • Remark 2.8
  • Remark 2.9
  • Definition 2.11: Continuous Dirichlet problem
  • ...and 121 more