Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior

Liao, Naian

doi:10.1016/j.matpur.2021.01.008

Cited by 22 publications

(18 citation statements)

References 17 publications

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“…Indeed, as a byproduct of our analysis, by applying the ideas of [19] to the parabolic setting, we show that lower semi-continuity of super-solutions is a sole consequence of these general invariants.…”

mentioning

confidence: 86%

On a particular scaling for the prototype anisotropic p-Laplacian

Ciani¹,

Guarnotta²,

Vespri³

2022

Preprint

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In this brief note we show that under a volume non-preserving scaling it is possible to recover the basics for a regularity theory regarding local weak solutions to the fully anisotropic equationWe characterize self-similar solutions regarding this particular scaling and we show that semi-continuity for solutions to this equation is a consequence of a simple property that is itself invariant under scaling.

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mentioning

confidence: 86%

On a particular scaling for the prototype anisotropic p-Laplacian

Ciani¹,

Guarnotta²,

Vespri³

2022

Preprint

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“…Proof. The De Giorgi iteration has been performed in [12,Lemma 2.2] for super-solutions, whereas the proof for sub-solutions is analogous. In order to obtain the present formulation, choose a = 1 2 and replace M by ξω.…”

Section: Preliminary Toolsmentioning

confidence: 99%

“…More precisely, one has to choose a = 1 2 , replace M by ξω and note that the constant max{L q−1 , M q−1 } is controlled by max{1, 9 q−1 }(ξω) q−1 whenever |µ − | ≤ 8ξω. This allows to choose the parameter θ in [12,Lemma 3.1] in the form ν o (ξω) q+1−p for some ν o depending only on the data.…”

Section: Preliminary Toolsmentioning

confidence: 99%

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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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“…Based on expansion of positivity, Liao [25] derived the Hölder continuity of locally bounded solutions to parabolic equations with structures modeled after the parabolic p-Laplace equation and the porous medium equation. We refer the readers to [1,4,14,[26][27][28]30] and references therein for more related results. For the nonlocal p-Laplace equation…”

Section: Introductionmentioning

confidence: 99%