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Gradient higher integrability for singular parabolic double-phase systems

Wontae Kim, Lauri Särkiö

Abstract

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of $p$-Laplace type when $\tfrac{2n}{n+2}< p\le2$. The result is based on a reverse Hölder inequality in intrinsic cylinders combining $p$-intrinsic and $(p,q)$-intrinsic geometries. A singular scaling deficits affects the range of $q$.

Gradient higher integrability for singular parabolic double-phase systems

Abstract

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of -Laplace type when . The result is based on a reverse Hölder inequality in intrinsic cylinders combining -intrinsic and -intrinsic geometries. A singular scaling deficits affects the range of .
Paper Structure (12 sections, 18 theorems, 198 equations)

This paper contains 12 sections, 18 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Notation and main result
  3. Notation
  4. Main result
  5. Auxiliary lemmas
  6. Reverse Hölder inequality
  7. The $p$-intrinsic case
  8. The $(p,q)$-intrinsic case
  9. Proof of the main result
  10. Stopping time argument
  11. Vitali type covering argument
  12. Final proof of the gradient estimate

Key Result

Theorem 2.2

Let $u$ be a weak solution to sec1:1. There exist constants $0<\epsilon_0=\epsilon_0(\mathit{data})$ and $c=c(\mathit{data},\lVert a\rVert_{L^\infty(\Omega_T)})\ge1$, such that for every $Q_{2r}(z_0)\subset\Omega_T$ and $\epsilon\in(0,\epsilon_0)$.

Theorems & Definitions (32)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 22 more