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Hölder regularity for a class of doubly non linear PDEs

Filippo Maria Cassanello, Eurica Henriques

TL;DR

The paper addresses local Hölder regularity for the doubly nonlinear parabolic equation $\partial_t(u^q)-\operatorname{div}(|Du|^{p-2}Du)=0$ with $p>2$ and $0<q<p-1$. It develops an intrinsic geometric framework and a pair of measure-theoretic alternatives, leveraging expansion of positivity and an exponential shift to handle the intrinsic scaling. The main result proves that nonnegative, locally bounded local weak solutions are locally Hölder continuous in $\Omega_T$, with a quantitative decay of oscillation governed by a $(p,q)$-dependent distance. This work extends the regularity theory for doubly nonlinear parabolic equations and provides a robust methodology based on intrinsic geometry and De Giorgi-type arguments.

Abstract

We prove local Hölder continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations $\partial_t (u_q) - \text{div} (|Du|^{p-2} Du) = 0$ for $p > 2$, $ 0 < q < p-1$. The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated to the problem.

Hölder regularity for a class of doubly non linear PDEs

TL;DR

The paper addresses local Hölder regularity for the doubly nonlinear parabolic equation with and . It develops an intrinsic geometric framework and a pair of measure-theoretic alternatives, leveraging expansion of positivity and an exponential shift to handle the intrinsic scaling. The main result proves that nonnegative, locally bounded local weak solutions are locally Hölder continuous in , with a quantitative decay of oscillation governed by a -dependent distance. This work extends the regularity theory for doubly nonlinear parabolic equations and provides a robust methodology based on intrinsic geometry and De Giorgi-type arguments.

Abstract

We prove local Hölder continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations for , . The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated to the problem.
Paper Structure (6 sections, 12 theorems, 142 equations)

This paper contains 6 sections, 12 theorems, 142 equations.

Table of Contents

  1. introduction
  2. Setting the framework
  3. The study of the two alternatives
  4. The first alternative
  5. The second alternative
  6. Hölder Continuity

Key Result

Theorem 1.1

Let $u$ be a non negative, locally bounded, local weak solution to EQ in $\Omega_T$. Then, $u$ is locally Hölder continuous in $\Omega_T$. Moreover, there exist constants $b$, $\bar{\gamma} >1$ and $\alpha \in (0,1)$, depending only on the data, such that for every compact set $U_{\tau} \subset \Ome for every pair of point $(x_1,t_1)$, $(x_2,t_2) \in U_\tau$, where $\bar{M}= \max\{M,(2b)^{\frac{1}

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more