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Very weak solutions to degenerate parabolic double-phase systems

Wontae Kim, Lauri Särkiö

TL;DR

This work addresses the regularity of parabolic double-phase systems with growth governed by two exponents $p$ and $q$ in the presence of a Hölder continuous coefficient $a(z)$. By introducing an intrinsic geometric framework and a phase analysis approach, the authors establish a local reverse Hölder inequality for the gradient of very weak solutions, enabling a self-improving gradient integrability result that is independent of the specific solution. The method hinges on a Lipschitz truncation-based Caccioppoli inequality, careful handling of intrinsic phases, and a stopping-time/Vitali covering argument to derive global gradient estimates. The results extend the parabolic double-phase regularity theory to the very weak setting and provide a rigorous pathway to higher gradient integrability in degenerate nonlinear PDEs with nonstandard growth.

Abstract

We prove a local self-improving property for the gradient of very weak solutions to degenerate parabolic double-phase systems. The result is based on a reverse Hölder inequality with constants that are independent of the solution. Delicate methods are required to avoid a self-referential argument. In particular, we develop a new phase analysis method.

Very weak solutions to degenerate parabolic double-phase systems

TL;DR

This work addresses the regularity of parabolic double-phase systems with growth governed by two exponents and in the presence of a Hölder continuous coefficient . By introducing an intrinsic geometric framework and a phase analysis approach, the authors establish a local reverse Hölder inequality for the gradient of very weak solutions, enabling a self-improving gradient integrability result that is independent of the specific solution. The method hinges on a Lipschitz truncation-based Caccioppoli inequality, careful handling of intrinsic phases, and a stopping-time/Vitali covering argument to derive global gradient estimates. The results extend the parabolic double-phase regularity theory to the very weak setting and provide a rigorous pathway to higher gradient integrability in degenerate nonlinear PDEs with nonstandard growth.

Abstract

We prove a local self-improving property for the gradient of very weak solutions to degenerate parabolic double-phase systems. The result is based on a reverse Hölder inequality with constants that are independent of the solution. Delicate methods are required to avoid a self-referential argument. In particular, we develop a new phase analysis method.

Paper Structure

This paper contains 18 sections, 34 theorems, 404 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and setup of the proof
  3. Notation
  4. Setup of the proof
  5. Auxiliary lemmas
  6. Caccioppoli inequality
  7. Lipschitz truncation
  8. Lipschitz regularity
  9. Estimates on the bad set
  10. Time-slice estimate
  11. Proof of the Caccioppoli inequality
  12. Reverse Hölder inequality
  13. The $p$-intrinsic case
  14. $(p,q)$-intrinsic case
  15. Proof of Theorem \ref{['HI_theorem']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Suppose $p,q$ satisfy eq_range_q. Then there exists $\delta_0=\delta_0(\mathit{data})\in(0,1)$ with $2-q(1-\delta_0)>0$ such that if $u$ is a very weak solution to 11 with integrability deficit $\delta\in(\delta_0,1)$ and $u\in C(0,T;L^2(\Omega_T,\mathbb{R}^N))$, we have for every $Q_{2R}\subset\Omega_T$ where $c\ge 1$ depending on $\mathit{data},\|a\|_{L^\infty(\Omega_T)},\| (H(z,|\nabla u|)+H(z

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Proposition 3.2
  • proof : Proof of \ref{['UUi']}
  • proof : Proof of \ref{['viii']}
  • Proposition 3.3
  • Proposition 3.4
  • ...and 51 more