Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gradient continuity for the parabolic $(1,\,p)$-Laplace system

Shuntaro Tsubouchi

TL;DR

The paper proves spatial-temporal continuity of the gradient for parabolic $(1,p)$-Laplace systems by introducing a gradient truncation to bypass nonuniform ellipticity on the facet $oldsymbol Doldsymbol u=0$. It constructs approximate, $oldsymbol o$-regularized systems and derives robust a priori estimates via De Giorgi–Moser techniques, energy methods, and a weak maximum principle, followed by a careful passage to the limit. The main novelty lies in treating the external force term under an optimal parabolic Lebesgue condition and incorporating a generalized Uhlenbeck structure to obtain gradient continuity for all $p eq1$, with a detailed degenerate/nondegenerate analysis that yields Hölder continuity of the truncated gradient and hence of the full gradient. The results contribute a parabolic counterpart to elliptic T-system regularity, enabling gradient $C^0$ regularity for a broad class of nonuniformly elliptic parabolic systems and providing a rigorous framework for Dirichlet problems in this setting.

Abstract

This paper deals with the parabolic $(1,\,p)$-Laplace system, a parabolic system that involves the one-Laplace and $p$-Laplace operators with $p\in(1,\,\infty)$. We aim to prove that a spatial gradient is continuous in space and time. An external force term is treated under the optimal regularity assumption in the parabolic Lebesgue spaces. We also discuss a generalized parabolic system with the Uhlenbeck structure. A main difficulty is that the uniform ellipticity of the $(1,\,p)$-Laplace operator is violated on a facet, or the degenerate region of a spatial gradient. The gradient continuity is proved by showing local Hölder continuity of a truncated gradient, whose support is far from the facet. This is rigorously demonstrated by considering approximate parabolic systems and deducing various regularity estimates for approximate solutions by classical methods such as De Giorgi's truncation, Moser's iteration, and freezing coefficient arguments. A weak maximum principle is also utilized when $p$ is not in the supercritical range.

Gradient continuity for the parabolic $(1,\,p)$-Laplace system

TL;DR

The paper proves spatial-temporal continuity of the gradient for parabolic -Laplace systems by introducing a gradient truncation to bypass nonuniform ellipticity on the facet . It constructs approximate, -regularized systems and derives robust a priori estimates via De Giorgi–Moser techniques, energy methods, and a weak maximum principle, followed by a careful passage to the limit. The main novelty lies in treating the external force term under an optimal parabolic Lebesgue condition and incorporating a generalized Uhlenbeck structure to obtain gradient continuity for all , with a detailed degenerate/nondegenerate analysis that yields Hölder continuity of the truncated gradient and hence of the full gradient. The results contribute a parabolic counterpart to elliptic T-system regularity, enabling gradient regularity for a broad class of nonuniformly elliptic parabolic systems and providing a rigorous framework for Dirichlet problems in this setting.

Abstract

This paper deals with the parabolic -Laplace system, a parabolic system that involves the one-Laplace and -Laplace operators with . We aim to prove that a spatial gradient is continuous in space and time. An external force term is treated under the optimal regularity assumption in the parabolic Lebesgue spaces. We also discuss a generalized parabolic system with the Uhlenbeck structure. A main difficulty is that the uniform ellipticity of the -Laplace operator is violated on a facet, or the degenerate region of a spatial gradient. The gradient continuity is proved by showing local Hölder continuity of a truncated gradient, whose support is far from the facet. This is rigorously demonstrated by considering approximate parabolic systems and deducing various regularity estimates for approximate solutions by classical methods such as De Giorgi's truncation, Moser's iteration, and freezing coefficient arguments. A weak maximum principle is also utilized when is not in the supercritical range.

Paper Structure

This paper contains 38 sections, 41 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Some specific mathematical sources in fluid mechanics
  3. Literature overview and our strategy
  4. Notations
  5. Structural assumptions and the definition of a weak solution
  6. Main result and plan of the paper
  7. Preliminaries
  8. Approximate systems and a convergence lemma
  9. Basic structures of the approximate operators and some related mappings
  10. Composite mappings
  11. Basic lemmata for parabolic regularity
  12. Boundedness of a solution for $p\in(1,\,2)$
  13. A weak maximum principle
  14. A priori local bounds for the parabolic $(1,\,p)$-Laplace system
  15. Regularity estimates and weak formulations
  16. ...and 23 more sections

Key Result

Theorem 1.2

Let $p\in(1,\,\infty)$, $\mathbf{f}\in L^{p^{\prime}}(0,\,T;\,V_{0}^{\prime})^{N}\cap L^{r}(0,\,T;\,L^{q}(\Omega))^{N}$ with $(q,\,r)\in(n,\,\infty\rbrack\times (2,\,\infty\rbrack$ satisfying (Eq (Section 1): Condition for q,r). Assume that $\mathbf{u}\in X^{p}(0,\,T;\,\Omega)^{N}\cap C(\lbrack 0,\

Theorems & Definitions (83)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 73 more