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H\"older Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations

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arxiv 2305.08539 v1 pith:NCOOGGMK submitted 2023-05-15 math.AP

Verena Bögelein , Frank Duzaar , Ugo Gianazza , Naian Liao , Christoph Scheven This is my paper · ORCID
classification math.AP
keywords equationestimatessolutionsgradientparabolicweakdoublynon-linear
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abstract

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time cylinder. H\"older estimates are established for the gradient of its weak solutions in the super-critical fast diffusion regime $0<p-1< q<\frac{N(p-1)}{(N-p)_+}$ where $N$ is the space dimension. Moreover, decay estimates are obtained for weak solutions and their gradient in the vicinity of possible extinction time. Two main components towards these regularity estimates are a time-insensitive Harnack inequality that is particular about this regime, and Schauder estimates for the parabolic $p$-Laplace equation.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions

    math.AP 2025-06 conditional novelty 7.0 of 10

    A generalized parabolic De Giorgi class with unbalanced growth is shown to have locally bounded members under sharp integrability assumptions, with quantitative supremum estimates.

  2. Integral Harnack estimates and the rate of extinction of singular fractional diffusion

    math.AP 2026-02 conditional novelty 6.0 of 10

    Singular fractional p-Laplacian diffusion satisfies local integral Harnack estimates that imply finite-time extinction with (T*-t)^{1/(2-p)} decay.

  3. Schauder estimates for parabolic $p$-Laplace systems

    math.AP 2025-07 conditional novelty 6.0 of 10

    Bounded weak solutions to parabolic p-Laplace systems with Hölder continuous coefficients have locally Hölder continuous spatial gradients for every p > 1, with quantitative estimates in terms of the solution's oscillation.

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