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Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

Verena Bögelein, Frank Duzaar, Ugo Gianazza, Naian Liao

TL;DR

This work develops a quantitative regularity theory for parabolic $p$-Laplacian-type systems with measurable coefficients in the sub-critical regime $1<p\le\frac{2N}{N+2}$. It introduces a mixed intrinsic scaling framework and sharp sup-estimates to overcome the failure of standard energy-based reverse Hölder arguments, establishing local boundedness and a Gehring-type self-improvement for $|Du|$, i.e., $|Du|\in L^{p(1+\varepsilon)}_{\text{loc}}$ for some $\varepsilon>0$ depending only on the data and sources. The approach combines Moser and De Giorgi iterations with a detailed intrinsic cylinder construction, Gluing, and Sobolev–Poincaré inequalities, culminating in a robust reverse Hölder theory on intrinsic cylinders and a Vitali-type covering to obtain higher integrability. The results rigorously quantify regularity in a regime previously open and provide sharp, data-dependent bounds that extend prior super-critical results to the sub-critical setting, with implications for quantitative partial regularity of parabolic systems.

Abstract

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \tfrac{2N}{N+2}$ and obtain two main results. \emph{Local boundedness:} starting from an $L^{\boldsymbol{\mathsf r}}$-control of $u$ with ${\boldsymbol{\mathsf r}}>\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\mathrm{loc}}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms.

Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

TL;DR

This work develops a quantitative regularity theory for parabolic -Laplacian-type systems with measurable coefficients in the sub-critical regime . It introduces a mixed intrinsic scaling framework and sharp sup-estimates to overcome the failure of standard energy-based reverse Hölder arguments, establishing local boundedness and a Gehring-type self-improvement for , i.e., for some depending only on the data and sources. The approach combines Moser and De Giorgi iterations with a detailed intrinsic cylinder construction, Gluing, and Sobolev–Poincaré inequalities, culminating in a robust reverse Hölder theory on intrinsic cylinders and a Vitali-type covering to obtain higher integrability. The results rigorously quantify regularity in a regime previously open and provide sharp, data-dependent bounds that extend prior super-critical results to the sub-critical setting, with implications for quantitative partial regularity of parabolic systems.

Abstract

A quantitative regularity theory is developed for weak solutions to the parabolic system which features the -Laplacian with measurable coefficients. We focus on the sub-critical range and obtain two main results. \emph{Local boundedness:} starting from an -control of with , we derive sharp, scale-invariant -estimates. \emph{Higher integrability of the gradient:} self-improves from to for some depending only on the data. The same results still hold given proper source terms.
Paper Structure (22 sections, 19 theorems, 393 equations)

This paper contains 22 sections, 19 theorems, 393 equations.

Table of Contents

  1. Introduction and results
  2. Novelty and significance
  3. Sharpness of the result
  4. Strategy of the proof
  5. Notation, definitions and tools
  6. Local Linfinity-control in the sub-critical regime
  7. Energy and Caccioppoli estimates
  8. Moser's iteration: from Lr to Linfinity
  9. De Giorgi's iteration: proof of Theorem \ref{['thm:sup-quant']}
  10. Reverse Hölder-type inequality
  11. Energy inequality
  12. Gluing lemma
  13. Parabolic Sobolev–Poincaré-type inequalities
  14. Estimates on (sub-)intrinsic cylinders
  15. Reverse Hölder-type inequality
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $p\in\bigl(1,\tfrac{2N}{N+2}\bigr]$, $q>\tfrac{N+p}{p}$, and ${\boldsymbol{\mathsf r}}>2$ be such that Assume that $u\in L^{{\boldsymbol{\mathsf r}}}_{\mathrm{loc}}(E_T,\mathbb{R}^k)$ is a weak solution in $E_T$ to eq-par-gen under growth-a*, with ${\boldsymbol{\mathsf F}}\in L^{qp}_{\mathrm{loc}}(E_T,\mathbb{R}^{kN})$ and ${\boldsymbol{\mathsf f}}\in L^{qp'}_{\mathrm{loc}}(E_T,\mathbb{R}^k)$

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 30 more