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Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity

Se-Chan Lee, Yuanyuan Lian, Hyungsung Yun, Kai Zhang

TL;DR

This work proves that solutions to the parabolic $p$-Laplace equation with $p>2$ have bounded time derivatives, $u_t \in L^{\infty}_{\text{loc}}$, by combining a regularized, nondivergence-structure Bernstein argument with a carefully chosen auxiliary function. This time-derivative bound enables a reduction of the parabolic problem to an elliptic one with a nonhomogeneous term $f=-u_t$, establishing a direct link to the elliptic $C^{p'}$-regularity conjecture and suggesting optimal regularity in time that aligns with spatial optimality. The authors extend the approach to fully nonlinear and general quasilinear degenerate parabolic equations, obtaining analogous time-derivative estimates and translating these into sharp interior and boundary regularity results, including up to the boundary. Collectively, the results provide a unified Bernstein-based methodology for time-derivative control and optimal regularity across a broad class of degenerate parabolic problems, with significant implications for elliptic–parabolic regularity theory and boundary behavior.

Abstract

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal regularity result with a connection to the well-known $C^{p'}$-conjecture in the elliptic setting. Finally, we extend our method to treat global regularity results for both fully nonlinear and general quasilinear degenerate parabolic problems.

Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity

TL;DR

This work proves that solutions to the parabolic -Laplace equation with have bounded time derivatives, , by combining a regularized, nondivergence-structure Bernstein argument with a carefully chosen auxiliary function. This time-derivative bound enables a reduction of the parabolic problem to an elliptic one with a nonhomogeneous term , establishing a direct link to the elliptic -regularity conjecture and suggesting optimal regularity in time that aligns with spatial optimality. The authors extend the approach to fully nonlinear and general quasilinear degenerate parabolic equations, obtaining analogous time-derivative estimates and translating these into sharp interior and boundary regularity results, including up to the boundary. Collectively, the results provide a unified Bernstein-based methodology for time-derivative control and optimal regularity across a broad class of degenerate parabolic problems, with significant implications for elliptic–parabolic regularity theory and boundary behavior.

Abstract

We establish the boundedness of time derivatives of solutions to parabolic -Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal regularity result with a connection to the well-known -conjecture in the elliptic setting. Finally, we extend our method to treat global regularity results for both fully nonlinear and general quasilinear degenerate parabolic problems.

Paper Structure

This paper contains 14 sections, 20 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Optimal regularity
  3. Strategy of the proof
  4. Historical background
  5. Applications
  6. Overview
  7. Preliminaries
  8. Regularity results for p-Laplace equations
  9. Regularity results for fully nonlinear equations
  10. Parabolic p-Laplace equations
  11. Applications
  12. Fully nonlinear equations
  13. Time derivative estimates on the boundary
  14. General quasilinear degenerate equations

Key Result

Theorem 1.1

Let $u\in C(Q_1)$ be a viscosity solution to eq-plaplace in $Q_1$ with $p \in (2, \infty)$. Then $u$ is weakly differentiable in time and $u_t \in L^{\infty}_{\mathrm{loc}}(Q_1)$ with the uniform estimate where $C>0$ is a constant depending only on $n$, $p$ and $\|u\|_{L^{\infty}(Q_1)}$.

Theorems & Definitions (37)

  • Conjecture : $C^{p'}$-regularity conjecture
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Viscosity solutions for $p$-Laplace equations
  • Definition 2.2: Viscosity solutions for fully nonlinear equations
  • Remark 2.3
  • Lemma 2.4: Uniform Lipschitz estimates, IJS19
  • Lemma 2.5: Uniform global modulus of continuity, IJS19
  • Theorem 2.6: Solvability of Cauchy--Dirichlet problem, LSU68
  • ...and 27 more