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Calderón-Zygmund type estimate for the singular parabolic double-phase system

Wontae Kim

TL;DR

The paper tackles a Calderón–Zygmund type gradient estimate for parabolic double-phase systems in the singular regime $p<2$, where nonstandard growth is driven by a phase function $H(z,s)=s^p+a(z)s^q$. By employing phase analysis, the authors introduce two intrinsic geometries—$p$-intrinsic and $(p,q)$-intrinsic cylinders—to align regularity with local growth behavior. They develop robust comparison estimates across these geometries, constructing auxiliary solutions with higher integrability and Lipschitz control, and implement stopping-time arguments together with a Vitali covering to perform a localized-to-global analysis. Under a VMO condition on the coefficient $b$ and a positive lower bound on $a(z)$, the main result establishes that if $|F|^p+a|F|^q abla ext{locally} \

Abstract

This paper discusses the local Calderón-Zygmund type estimate for the singular parabolic double-phase system. The proof covers the counterpart $p<2$ of the result in [23]. Phase analysis is employed to determine an appropriate intrinsic geometry for each phase. Comparison estimates and scaling invariant properties for each intrinsic geometry are the main techniques to obtain the main estimate.

Calderón-Zygmund type estimate for the singular parabolic double-phase system

TL;DR

The paper tackles a Calderón–Zygmund type gradient estimate for parabolic double-phase systems in the singular regime , where nonstandard growth is driven by a phase function . By employing phase analysis, the authors introduce two intrinsic geometries—-intrinsic and -intrinsic cylinders—to align regularity with local growth behavior. They develop robust comparison estimates across these geometries, constructing auxiliary solutions with higher integrability and Lipschitz control, and implement stopping-time arguments together with a Vitali covering to perform a localized-to-global analysis. Under a VMO condition on the coefficient and a positive lower bound on , the main result establishes that if $|F|^p+a|F|^q abla ext{locally} \

Abstract

This paper discusses the local Calderón-Zygmund type estimate for the singular parabolic double-phase system. The proof covers the counterpart of the result in [23]. Phase analysis is employed to determine an appropriate intrinsic geometry for each phase. Comparison estimates and scaling invariant properties for each intrinsic geometry are the main techniques to obtain the main estimate.

Paper Structure

This paper contains 9 sections, 26 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Notations and main result
  3. Notations
  4. Main result
  5. Comparison estimates
  6. $p$-intrinsic case.
  7. $(p,q)$-intrinsic case.
  8. Stopping time arguments
  9. Proof of Theorem \ref{['main_theorem']}

Key Result

Theorem 2.2

Let $u$ be a weak solution to eq. Then there exist $\varepsilon_0=\varepsilon_0(\mathit{data})\in (0,1)$ and $c=c(\mathit{data},\|a\|_{L^\infty(\Omega_T)})$ such that for any $Q_{2\rho}(z_0)\subset \Omega_T$ and $\varepsilon\in(0,\varepsilon_0]$, there holds

Theorems & Definitions (51)

  • Definition 2.1
  • Theorem 2.2: MR4718687, Higher integrability
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 41 more