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Parabolic Lipschitz truncation for multi-phase problems: the degenerate case

Bogi Kim, Jehan Oh, Abhrojyoti Sen

TL;DR

The paper addresses energy estimates for weak solutions to parabolic multi-phase problems with degenerate (nonlinear) growth governed by $H(z,\kappa)=\kappa^p+a(z)\kappa^q+b(z)\kappa^s$ and modulated coefficients $a(z), b(z)$. It introduces a parabolic Lipschitz truncation framework based on a Whitney decomposition and a phase-sensitive Vitali covering to construct Lipschitz test functions $v_h^\Lambda$, enabling a Caccioppoli-type inequality in a smaller cylinder controlled by data in a larger one plus a source term $F$. Key contributions include establishing Lipschitz regularity for the truncated test functions, deriving a parabolic Poincaré-type inequality, and obtaining robust energy bounds without requiring gradient higher integrability, under Hölder regularity of $a$ and $b$ and the condition $q\le p+\frac{2\alpha}{n+2}$, $s\le p+\frac{2\beta}{n+2}$. The framework extends elliptic double-phase Lipschitz truncation to the parabolic, degenerate regime, providing tools for existence and regularity theory in heterogeneous multi-phase media.

Abstract

This article is devoted to exploring the Lipschitz truncation method for parabolic multi-phase problems. The method is based on Whitney decomposition and covering lemmas with a delicate comparison scheme of appropriate alternatives to distinguish phases, as introduced by the first and the second author in [24].

Parabolic Lipschitz truncation for multi-phase problems: the degenerate case

TL;DR

The paper addresses energy estimates for weak solutions to parabolic multi-phase problems with degenerate (nonlinear) growth governed by and modulated coefficients . It introduces a parabolic Lipschitz truncation framework based on a Whitney decomposition and a phase-sensitive Vitali covering to construct Lipschitz test functions , enabling a Caccioppoli-type inequality in a smaller cylinder controlled by data in a larger one plus a source term . Key contributions include establishing Lipschitz regularity for the truncated test functions, deriving a parabolic Poincaré-type inequality, and obtaining robust energy bounds without requiring gradient higher integrability, under Hölder regularity of and and the condition , . The framework extends elliptic double-phase Lipschitz truncation to the parabolic, degenerate regime, providing tools for existence and regularity theory in heterogeneous multi-phase media.

Abstract

This article is devoted to exploring the Lipschitz truncation method for parabolic multi-phase problems. The method is based on Whitney decomposition and covering lemmas with a delicate comparison scheme of appropriate alternatives to distinguish phases, as introduced by the first and the second author in [24].
Paper Structure (15 sections, 33 theorems, 239 equations, 1 table)

This paper contains 15 sections, 33 theorems, 239 equations, 1 table.

Table of Contents

  1. Introduction
  2. Whitney decomposition and covering lemmas
  3. Auxiliary definitions
  4. Some preliminary estimates
  5. Vitali covering and their properties
  6. Partition of unity
  7. Construction of test function via Lipschitz truncation
  8. Definition of test function
  9. Preliminary lemmas
  10. Poincaré type inequality for the test function
  11. Bounds on Lipschitz truncation and its derivatives
  12. Lipschitz regularity of $v^{\Lambda}_h$
  13. Some more properties of Lipschitz truncation
  14. Proof of energy estimate
  15. Proof of Theorem \ref{['thm : the Caccioppoli inequality']}

Key Result

Theorem 1.2

Let $u$ be a weak solution to eq: main equation. Then, for $U_{R_2,S_2}(z_0)\subset \Omega_T$, $R_1\in [R_2/2,R_2)$ and $S_1\in[S_2/2^2,S_2)$, there exists a constant $c$ depending on $n,p,q,s,\nu$ and $L$ such that the following inequality holds:

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 53 more