Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Bögelein, Verena; Herán, Andreas; Schätzler, Leah; Singer, Thomas

doi:10.1007/s00526-021-02044-z

Cited by 7 publications

(7 citation statements)

References 30 publications

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“…Indeed, when 0 < q ≤ p − 1, weak solutions are locally bounded, cf. [19]. The optimality of the range of p and q given in Corollary 1.7 is shown in the following counterexamples, built on the analogous ones for the fast diffusion equation given in [53,67] or for the parabolic p-Laplacian considered in [10].…”

Section: Optimalitymentioning

confidence: 94%

“…In particular, our Theorem 1.10 includes Theorems 1.2 & 16.2 in [31,Chapter 6] as special cases and offers a unified perspective in a larger framework. A Harnack inequality for the complete slow diffusion range can be found in [19].…”

Section: Harnack's Inequalitymentioning

confidence: 99%

“…Applying the Dominated Convergence Theorem as δ ↓ 0 in (4. 19), and taking into account that both v and w are non-negative yield (4.20)…”

Section: Comparison Principle IImentioning

confidence: 99%

“…There is always a root, since we assumed Y n > ν below (6. 19) and impose a smallness condition on c, that is,…”

Section: 313mentioning

confidence: 99%

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Gradient Hölder regularity for degenerate parabolic systems

Bögelein¹,

Duzaar²,

Liao³

et al. 2022

Nonlinear Analysis

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Section: Optimalitymentioning

confidence: 94%

Section: Harnack's Inequalitymentioning

confidence: 99%

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Gradient Hölder regularity for degenerate parabolic systems

Bögelein¹,

Duzaar²,

Liao³

et al. 2022

Nonlinear Analysis

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“…1 2 and properly adjusting the Hölder exponent, we obtain the power-like decay of oscillation ess osc…”

Section: Consequently Iterating the Above Recursive Inequality We Obt...mentioning

confidence: 99%

On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II

Bögelein¹,

Duzaar²,

Liao³

et al. 2021

Preprint

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We demonstrate two proofs for the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype isThe first proof takes advantage of the expansion of positivity for the degenerate, parabolic p-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.

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A comparison principle for doubly nonlinear parabolic partial differential equations

Bögelein,

Strunk

2023

Annali di Matematica

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In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is $$\begin{aligned} \partial _t u^q - {{\,\textrm{div}\,}}{\big (|\nabla u|^{p-2}\nabla u \big )}=0 \qquad \text{ in } \Omega _T, \end{aligned}$$ ∂ t u q - div ( | ∇ u | p - 2 ∇ u ) = 0 in Ω T , with $$q>0$$ q > 0 and $$p>1$$ p > 1 and $$\Omega _T:=\Omega \times (0,T)\subset \mathbb {R}^{n+1}$$ Ω T : = Ω × ( 0 , T ) ⊂ R n + 1 . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain $$\Omega _T$$ Ω T , we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.

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Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Abstract: In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t … Show more

Cited by 7 publications

References 30 publications

Gradient Hölder regularity for degenerate parabolic systems

Gradient Hölder regularity for degenerate parabolic systems

On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II

A comparison principle for doubly nonlinear parabolic partial differential equations

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