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Hölder regularity for degenerate parabolic double-phase equations

Wontae Kim, Kristian Moring, Lauri Särkiö

TL;DR

This work establishes local Hölder continuity for bounded weak solutions to parabolic double-phase equations with nonstandard growth $\partial_t u - \operatorname{div}\mathcal{A}(x,t,u,\nabla u)=0$, where $2\le p<q\le p+\alpha$ and $a\in C^{\alpha,\alpha/2}$ satisfies $q\le p+\alpha$. The authors introduce an intrinsic, phase-based framework that identifies whether the local behavior aligns with a $p$-Laplace or a $q$-Laplace regime, and then apply De Giorgi-type energy estimates (Caccioppoli and logarithmic) in phase-appropriate cylinders. A measure-theoretic alternative, combined with a phase criterion and embedding theorems, yields two complementary routes to reduce oscillation; a recursive scheme then provides geometric decay of oscillation and a Hölder modulus with exponent $\beta$ depending on the data, including $n,p,q,\alpha,[a]_{\alpha}$, and $\|u\|_{\infty}$. The results extend parabolic regularity theory to doubly nonlinear, nonstandard-growth diffusion and offer a robust framework for further exploration of Harnack-type inequalities and gradient regularity in the double-phase setting.

Abstract

We prove that bounded weak solutions to degenerate parabolic double-phase equations of $p$-Laplace type are locally Hölder continuous. The proof is based on phase analysis and methods for the $p$-Laplace equation. In particular, the phase analysis determines whether the double-phase equation is locally similar to the $p$-Laplace or the $q$-Laplace equation.

Hölder regularity for degenerate parabolic double-phase equations

TL;DR

This work establishes local Hölder continuity for bounded weak solutions to parabolic double-phase equations with nonstandard growth , where and satisfies . The authors introduce an intrinsic, phase-based framework that identifies whether the local behavior aligns with a -Laplace or a -Laplace regime, and then apply De Giorgi-type energy estimates (Caccioppoli and logarithmic) in phase-appropriate cylinders. A measure-theoretic alternative, combined with a phase criterion and embedding theorems, yields two complementary routes to reduce oscillation; a recursive scheme then provides geometric decay of oscillation and a Hölder modulus with exponent depending on the data, including , and . The results extend parabolic regularity theory to doubly nonlinear, nonstandard-growth diffusion and offer a robust framework for further exploration of Harnack-type inequalities and gradient regularity in the double-phase setting.

Abstract

We prove that bounded weak solutions to degenerate parabolic double-phase equations of -Laplace type are locally Hölder continuous. The proof is based on phase analysis and methods for the -Laplace equation. In particular, the phase analysis determines whether the double-phase equation is locally similar to the -Laplace or the -Laplace equation.
Paper Structure (9 sections, 19 theorems, 191 equations)

This paper contains 9 sections, 19 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and definition of solution
  4. Auxiliary results
  5. Energy bounds
  6. Reduction of oscillation
  7. First Alternative
  8. Second Alternative
  9. Recursive argument

Key Result

Theorem 1.1

Let $u$ be a bounded weak solution to double-phase according to Definition def:weak-sol such that as:structure and as:range-a are in force. Then $u$ is locally Hölder continuous in $\Omega_T$. Moreover, there exist $c > 0$ and $\beta \in(0,1)$ depending only on $n,p,q,\alpha,C_0,C_1,[a]_\alpha$ and holds for every pair of points $(x_1,t_1),(x_2,t_2)\in K$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 26 more