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Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients

Jehan Oh, Abhrojyoti Sen

TL;DR

The paper proves gradient higher integrability for weak solutions of degenerate parabolic double-phase systems with two modulating coefficients by introducing an intrinsic parabolic geometry that separates $p$-phase, $q$-phase, and $(p,q)$-phase behaviors. It establishes reverse Hölder inequalities on each phase and then assembles them via a stopping-time argument and a Vitali-type covering to obtain a global $L^{1+\varepsilon}$-gain for $H_1(z,|Du|)$. The main result shows that under the structural assumptions and $a(z)+b(z)\ge\delta>0$, there exist $\varepsilon_0>0$ and $c\ge1$ such that the interior gradient of solutions enjoys higher integrability, quantified by an explicit intrinsic-norm bound involving $F$. The framework generalizes parabolic regularity for elliptic double-phase problems to the parabolic setting with two modulating coefficients, enabling better modeling of strongly anisotropic materials and composites.

Abstract

We establish an interior gradient higher integrability result for weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. To be more precise, we study systems of the form \begin{align*} u_t-\operatorname{div} \left(a(z)|Du|^{p-2}Du+ b(z)|Du|^{q-2}Du\right)=-\operatorname{div} \left(a(z)|F|^{p-2}F+ b(z)|F|^{q-2}F\right), \end{align*} where $2\leq p\leq q < \infty$ and the modulating coefficients $a(z)$ and $b(z)$ are non-negative, with $a(z)$ being uniformly continuous and $b(z)$ being Hölder continuous. We further assume that the sum of two modulating coefficients is bounded from below by some positive constant. To establish the gradient higher integrability result, we introduce a suitable intrinsic geometry and develop a delicate comparison scheme to separate and analyze the different phases--namely, the $p$-phase, $q$-phase and $(p,q)$-phase. To the best of our knowledge, this is the first regularity result in the parabolic setting that addresses general double phase systems.

Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients

TL;DR

The paper proves gradient higher integrability for weak solutions of degenerate parabolic double-phase systems with two modulating coefficients by introducing an intrinsic parabolic geometry that separates -phase, -phase, and -phase behaviors. It establishes reverse Hölder inequalities on each phase and then assembles them via a stopping-time argument and a Vitali-type covering to obtain a global -gain for . The main result shows that under the structural assumptions and , there exist and such that the interior gradient of solutions enjoys higher integrability, quantified by an explicit intrinsic-norm bound involving . The framework generalizes parabolic regularity for elliptic double-phase problems to the parabolic setting with two modulating coefficients, enabling better modeling of strongly anisotropic materials and composites.

Abstract

We establish an interior gradient higher integrability result for weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. To be more precise, we study systems of the form \begin{align*} u_t-\operatorname{div} \left(a(z)|Du|^{p-2}Du+ b(z)|Du|^{q-2}Du\right)=-\operatorname{div} \left(a(z)|F|^{p-2}F+ b(z)|F|^{q-2}F\right), \end{align*} where and the modulating coefficients and are non-negative, with being uniformly continuous and being Hölder continuous. We further assume that the sum of two modulating coefficients is bounded from below by some positive constant. To establish the gradient higher integrability result, we introduce a suitable intrinsic geometry and develop a delicate comparison scheme to separate and analyze the different phases--namely, the -phase, -phase and -phase. To the best of our knowledge, this is the first regularity result in the parabolic setting that addresses general double phase systems.

Paper Structure

This paper contains 17 sections, 26 theorems, 303 equations, 1 table.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. Notations
  4. Basic results and useful lemmas
  5. Reverse Hölder inequalities
  6. Reverse Hölder inequality for $p$-phase
  7. Reverse Hölder inequality for $q$-phase
  8. Reverse Hölder inequality for $(p, q)$-phase
  9. Gradient higher integrability
  10. Stopping time argument
  11. Existence of $p$-intrinsic case \ref{['eq: p-phase condition']}
  12. Existence of $q$-intrinsic case \ref{['eq: q-phase condition']}
  13. Existence of $(p,q)$-intrinsic case \ref{['Eq 3.8']}--\ref{['Eq 3.9']}
  14. Vitali covering lemma
  15. Space inclusion
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $u$ be a weak solution of eq: main equation with eq: growth condition of A--eq:assumption of a in force. Then there exist positive constants $\varepsilon_0 = \varepsilon_0(\textnormal{data})\in (0,1)$ and $c=c (\textnormal{data})\geq 1$ such that the following estimate holds for every $Q_{2r}(z_0)\subset \Omega_T$ and any $\varepsilon \in (0, \varepsilon_0)$, where $Q_{(\cdot)}(z_0)$, $H_1(z,

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1: Parabolic metric
  • Definition 2.2: Scaled parabolic metric
  • Lemma 2.1: Hasto_2021, Lemma 2.12
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 46 more