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Gradient regularity for widely degenerate parabolic equations

Michael Strunk

TL;DR

This work studies gradient regularity for weak solutions of widely degenerate parabolic equations $\partial_t u - \mathrm{div}\nabla \mathcal{F}(x,t,Du)=f$ in a bounded space-time cylinder, where the diffusion degenerates on a bounded convex set $E$ with $0\in\mathrm{Int}E$. The authors develop a robust regularization and De Giorgi-based framework, proving that $\mathcal{K}(Du)\in C^0(\Omega_T)$ for any continuous $\mathcal{K}$ vanishing on $E$; this extends $C^1$-regularity results from elliptic settings to the parabolic regime and allows spatially varying diffusion. The strategy combines truncation and convex augmentation of $\mathcal{F}$ to obtain uniformly elliptic approximations, Hölder continuity of $\mathcal{G}_\delta(Du_\varepsilon)$, differentiation to linear equations with controlled coefficients, and a careful non-degenerate/degenerate regime analysis via De Giorgi classes, followed by limit passages $\varepsilon\to0$ and $\delta\to0$. The results significantly broaden gradient regularity theory for widely degenerate parabolic PDEs, accommodating general degeneracy sets $E$ and non-standard growth outside $E$, with potential implications for nonlinear diffusion models and related variational problems.

Abstract

In this paper, we are interested in the regularity of weak solutions $u\colonΩ_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $Ω_T$}, \end{equation*} where $\mathcal{F}$ is only elliptic for values of $Du$ outside a bounded and convex set $E\subset \mathbb{R}^n$ with the property that $0\in \mathrm{Int}{E}$. Here, $Ω_T :=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken over a bounded domain $Ω\subset\mathbb{R}^n$ for some finite time $T>0$. The function $\mathcal{F} : Ω_T\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ present in the diffusion is assumed to satisfy: the partial mapping $ξ\mapsto \mathcal{F}(x,t,ξ)$ is regular whenever $ξ$ lies outside of $E$, and vanishes entirely whenever $ξ$ lies within this set. Additionally, the datum $f$ is assumed to be of class $L^{n+2+σ}(Ω_T)$ for some parameter $σ> 0$. As our main result we establish that \begin{equation*} \mathcal{K}(Du)\in C^0(Ω_T) \end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ that vanishes on $E$. This article aims to extend the $C^1$-regularity result for the elliptic case to the parabolic setting.

Gradient regularity for widely degenerate parabolic equations

TL;DR

This work studies gradient regularity for weak solutions of widely degenerate parabolic equations in a bounded space-time cylinder, where the diffusion degenerates on a bounded convex set with . The authors develop a robust regularization and De Giorgi-based framework, proving that for any continuous vanishing on ; this extends -regularity results from elliptic settings to the parabolic regime and allows spatially varying diffusion. The strategy combines truncation and convex augmentation of to obtain uniformly elliptic approximations, Hölder continuity of , differentiation to linear equations with controlled coefficients, and a careful non-degenerate/degenerate regime analysis via De Giorgi classes, followed by limit passages and . The results significantly broaden gradient regularity theory for widely degenerate parabolic PDEs, accommodating general degeneracy sets and non-standard growth outside , with potential implications for nonlinear diffusion models and related variational problems.

Abstract

In this paper, we are interested in the regularity of weak solutions to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in }, \end{equation*} where is only elliptic for values of outside a bounded and convex set with the property that . Here, denotes a space-time cylinder taken over a bounded domain for some finite time . The function present in the diffusion is assumed to satisfy: the partial mapping is regular whenever lies outside of , and vanishes entirely whenever lies within this set. Additionally, the datum is assumed to be of class for some parameter . As our main result we establish that \begin{equation*} \mathcal{K}(Du)\in C^0(Ω_T) \end{equation*} for any continuous function that vanishes on . This article aims to extend the -regularity result for the elliptic case to the parabolic setting.

Paper Structure

This paper contains 19 sections, 29 theorems, 306 equations.

Table of Contents

  1. Introduction and main results
  2. Literature overview
  3. Structure conditions
  4. Definition of weak solution
  5. Main result
  6. Strategy of the proof
  7. Novelties and Significance
  8. Plan of the paper
  9. Preliminaries
  10. Notation and setting
  11. Convex sets and the Minkowski functional
  12. Mollification in time
  13. Auxiliary material
  14. Proof of Theorem \ref{['hauptresultat']}
  15. Regularizing the equation
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $n\geq 2$, $f\in L^{n+2+\sigma}(\Omega_T)$ for some $\sigma>0$, and be a local weak solution to pde under structure conditions fregularity and voraussetzung, where $E\subset \mathbb{R}^n$ denotes a bounded and convex set with $0\in\mathop{\mathrm{Int}}\nolimits E$ on which $\mathcal{F}$ degenerates. Then, there holds for any $\mathcal{K}\in C^0(\mathbb{R}^n)$ with $\mathcal{K}\equiv 0$ on $E

Theorems & Definitions (50)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 40 more