Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $L^p$ regularity problem for parabolic operators

Martin Dindoš, Linhan Li, Jill Pipher

TL;DR

This work resolves the parabolic Regularity problem for divergence-form operators with coefficients governed by a parabolic Carleson measure, establishing solvability in an $L^p$ range $(1,p_0)$ for some $p_0>1$ on Lipschitz cylinders. The authors reduce the general operator to a block-form model on $ ext{R}^n_+ imes ext{R}$ and develop new parabolic tools to handle the nonlocal half-derivative in time, including a detailed study of the operator satisfied by $D_t^{1/2}u$ and associated area/non-tangential estimates. Through a duality framework with the adjoint Dirichlet problem, and a careful perturbation analysis, they obtain $L^p$ solvability for the Regularity problem, with a dual perspective on the Dirichlet problem for the adjoint operator. In the small Carleson norm regime, the results yield sharp solvability for a broad class of parabolic operators, extending the elliptic theory to the time-dependent setting and addressing substantial new challenges posed by nonlocal temporal terms. The techniques unify perturbation, block-form reduction, and advanced parabolic harmonic analysis to deliver sharp, robust solvability results for this fundamental boundary value problem.

Abstract

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

The $L^p$ regularity problem for parabolic operators

TL;DR

This work resolves the parabolic Regularity problem for divergence-form operators with coefficients governed by a parabolic Carleson measure, establishing solvability in an range for some on Lipschitz cylinders. The authors reduce the general operator to a block-form model on and develop new parabolic tools to handle the nonlocal half-derivative in time, including a detailed study of the operator satisfied by and associated area/non-tangential estimates. Through a duality framework with the adjoint Dirichlet problem, and a careful perturbation analysis, they obtain solvability for the Regularity problem, with a dual perspective on the Dirichlet problem for the adjoint operator. In the small Carleson norm regime, the results yield sharp solvability for a broad class of parabolic operators, extending the elliptic theory to the time-dependent setting and addressing substantial new challenges posed by nonlocal temporal terms. The techniques unify perturbation, block-form reduction, and advanced parabolic harmonic analysis to deliver sharp, robust solvability results for this fundamental boundary value problem.

Abstract

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE on a Lipschitz cylinder is solvable for some under the assumption that the matrix is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some the Regularity problem is solvable in the range . We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

Paper Structure

This paper contains 28 sections, 51 theorems, 609 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results and Methodology
  3. Formulation of the Regularity problem
  4. Parabolic Sobolev Spaces on $\partial\Omega$
  5. Reinforced weak and Energy solutions
  6. $(R)_p$ boundary value problem
  7. Preliminaries
  8. Overview of basic parabolic results
  9. Some real analysis results
  10. Proof of the main results
  11. Proof of Theorem \ref{['MainT']} assuming Theorem \ref{['thm.BL']}
  12. Proof of Theorem \ref{['thm.BL']}
  13. Solvability of the Regularity problem - reduction to the block form case
  14. Test function $\vec{h}$.
  15. The adjoint inhomogenous Dirichlet problem
  16. ...and 13 more sections

Key Result

Theorem 1.6

Consider the parabolic PDE $\mathcal{L} u= -\partial_t u +\mathop{\mathrm{div}}\nolimits (A \nabla u)=0$ on the domain $\Omega=\mathcal{O}\times\mathbb R$, where $\mathcal{O}\subset\mathbb R^n$ is a bounded or unbounded Lipschitz domain, the matrix $A$ is uniformly elliptic E:elliptic with bounded a Then there exists $p_0>1$ such that for all $1<p<p_0$ the $L^p$ Regularity problem for the equation

Figures (2)

  • Figure 1: Novel cones in the $(t,x_n)$ plane.
  • Figure 2: Sketch of domains in the $(x,x_n)$ plane.

Theorems & Definitions (74)

  • Theorem 1.6
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.15
  • Definition 2.18
  • Definition 2.20
  • Definition 2.23
  • ...and 64 more