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On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

Junyuan Fang, Tuoc Phan

TL;DR

This work develops a weighted regularity theory for parabolic equations in non-divergence form with singular and/or degenerate leading coefficients given by $\mathcal{L}u = u_t - \omega(x) a_{ij}(x,t) D_{ij}u$, where the weight $\omega$ lies in the Muckenhoupt class $A_{1+\frac{1}{n}}$ and exhibits small weighted mean oscillation. By constructing weighted parabolic cylinders $C_{r,\omega}$ and $Q_{r,\omega}$ and working in the weighted Sobolev space $\mathcal{W}^{2,1}_{n+1}(U,\omega)$, the paper proves Evans-type quantitative lower bounds (mean sojourn time estimates) and leverages weighted ABP and Krylov-Safonov type covering to establish Lin-type $W^{2,\varepsilon}$ estimates: for $p\in(0,p_0]$, $\|D^2u\|_{L^p(C_{1,\omega},\omega)}$ is controlled by boundary data and $\mathcal{L}u$ data. The results are first proved for smooth coefficients and then extended to general measurable coefficients via truncation/regularization of weights and coefficients. These findings provide foundational ingredients for $L^p$-theory and regularity of fully nonlinear parabolic equations with singular-degenerate coefficients in weighted settings. The approach combines barrier constructions, weighted ABP, a perturbation by freezing the weight $\omega$, and a robust stability analysis of weights under regularization. Overall, the work significantly advances weighted parabolic regularity theory in degenerate and singular media and lays groundwork for nonlinear extensions.

Abstract

We study a class of parabolic equations in non-divergence form with measurable coefficients that are singular, degenerate, or both singular and degenerate through a weight belonging to the $A_{1+\frac{1}{n}}$ -Muckenhoupt class of weights. Under some smallness assumption on a weighted mean oscillation of the weight, F.-H. Lin type weighted $W^{2,\varepsilon}$-estimates are proved. To prove the result, we establish a result on local quantitative lower estimates of solutions to the class of equations, which are known as the mean sojourn times of sample paths within sets. This type of estimate was proved by L. C. Evans for the class of linear elliptic equations in non-divergence form with uniformly elliptic and bounded measurable coefficients. A class of weighted parabolic cylinders intrinsically suitable for the class of equations is introduced. The parabolic ABP estimates, and a perturbation method are used to overcome the singularity and degeneracy of the coefficients. Careful analysis on regularization and truncation of the weights is performed. The paper provides foundational ingredients and estimates for the study of fully nonlinear parabolic equations with singular-degenerate coefficients.

On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

TL;DR

This work develops a weighted regularity theory for parabolic equations in non-divergence form with singular and/or degenerate leading coefficients given by , where the weight lies in the Muckenhoupt class and exhibits small weighted mean oscillation. By constructing weighted parabolic cylinders and and working in the weighted Sobolev space , the paper proves Evans-type quantitative lower bounds (mean sojourn time estimates) and leverages weighted ABP and Krylov-Safonov type covering to establish Lin-type estimates: for , is controlled by boundary data and data. The results are first proved for smooth coefficients and then extended to general measurable coefficients via truncation/regularization of weights and coefficients. These findings provide foundational ingredients for -theory and regularity of fully nonlinear parabolic equations with singular-degenerate coefficients in weighted settings. The approach combines barrier constructions, weighted ABP, a perturbation by freezing the weight , and a robust stability analysis of weights under regularization. Overall, the work significantly advances weighted parabolic regularity theory in degenerate and singular media and lays groundwork for nonlinear extensions.

Abstract

We study a class of parabolic equations in non-divergence form with measurable coefficients that are singular, degenerate, or both singular and degenerate through a weight belonging to the -Muckenhoupt class of weights. Under some smallness assumption on a weighted mean oscillation of the weight, F.-H. Lin type weighted -estimates are proved. To prove the result, we establish a result on local quantitative lower estimates of solutions to the class of equations, which are known as the mean sojourn times of sample paths within sets. This type of estimate was proved by L. C. Evans for the class of linear elliptic equations in non-divergence form with uniformly elliptic and bounded measurable coefficients. A class of weighted parabolic cylinders intrinsically suitable for the class of equations is introduced. The parabolic ABP estimates, and a perturbation method are used to overcome the singularity and degeneracy of the coefficients. Careful analysis on regularization and truncation of the weights is performed. The paper provides foundational ingredients and estimates for the study of fully nonlinear parabolic equations with singular-degenerate coefficients.
Paper Structure (13 sections, 22 theorems, 369 equations)

This paper contains 13 sections, 22 theorems, 369 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Muckenhoupt weights and weighted parabolic cylinders
  4. Bounded mean oscillations with respect to weights and their stability
  5. Parabolic Aleksandrov-Bakelman-Pucci estimates
  6. Krylov-Safonov covering lemma
  7. Prop-up lemmas
  8. Evans type estimates and Proof of Theorem \ref{['M-thm']}
  9. Auxiliary results
  10. An auxiliary function
  11. Proof of Theorem \ref{['M-thm']}.
  12. Parabolic weighted $W^{2,\varepsilon}$-Lin type estimates and Proof of Theorem \ref{['Lin thm']}
  13. Proof of Lemma \ref{['covering-1']}

Key Result

Theorem 1.1

For every $K_0\in [1,\infty)$ and $\nu \in (0,1)$, there exist sufficiently small positive constants $\delta=\delta(n,\nu, K_0)$ and $p_0=p_0(n,\nu, K_0)$ such that the following assertion holds. Suppose that elliptic holds, and $\omega \in A_{1+\frac{1}{n}}$ satisfying Then, for every $u\in \mathcal{W}^{2,1}_{n+1}(C_{1,\,\omega},\, \omega) \cap C(\overline{C}_{1,\,\omega})$, and $p\in(0,p_0]$, w

Theorems & Definitions (46)

  • Theorem 1.1: Weighted $W^{2,\varepsilon}$-Lin type estimates
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • ...and 36 more