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

Bögelein, Verena; Duzaar, Frank; Liao, Naian

doi:10.1016/j.jfa.2021.109173

Cited by 35 publications

(34 citation statements)

References 23 publications

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“…N.Liao in [30] has effected a direct proof of Hölder regularity for parabolic p-Laplace equation by using the expansion of positivity. It has also been used to give the first proofs of Hölder regularity for signchanging solutions of doubly nonlinear equations of porous medium type in [4,9,31].…”

Section: Rnmentioning

confidence: 99%

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Adimurthi¹,

Harsh²,

Tewary³

2022

Preprint

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Section: Rnmentioning

confidence: 99%

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Adimurthi¹,

Harsh²,

Tewary³

2022

Preprint

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“…The condition requires some changes in the classical proof for the prototype porous medium equation u t = ∆u m in [7]. It is more general than what we would need in view of the application to the biofilm model (2) we have in mind.…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

“…we recognize that the first equation of ( 2) is a particular case of (1). In (2) the actual biofilm is the subregion of Ω where M is positive, that is,…”

Section: Introductionmentioning

confidence: 99%

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Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

Muller¹

2022

Preprint

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We show interior Hölder continuity for a class of quasi-linear degenerate reactiondiffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyse is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solutions of regularized problems, which is a common assumption.

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“…However, it seems that a complete picture about the long-time behaviour of solutions to (1.2) is still missing, in the case of sign-changing solutions and for more general boundary data. We refer to [6,Section 1.3] and the references therein, for a discussion on the physical relevance of equation (1.2). 1 We notice that [40] deals with the (apparently) different equation…”

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confidence: 99%

A comparison principle for the Lane-Emden equation and applications to geometric estimates

Brasco¹,

Prinari²,

Chiara³

2021

Preprint

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We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the p−Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets. Contents 1. Introduction 1.1. The Lane-Emden equation 1.2. Main results 1.3. Some comments 1.4. Plan of the paper 2. Preliminaries 2.1. Notation 2.2. Sobolev spaces 2.3. Weak solutions 2.4. Hidden convexity 3. Some properties of the energy functional 4. A comparison principle 5. Applications to geometric estimates 5.1. Solutions of the Lane-Emden equation 5.2. Localization of maximum points 5.3. Generalized principal frequencies Appendix A. Quantified convexity of power functions Appendix B. Some subsolutions Appendix C. Asymptotics of the positive solution in a slab-type sequence References

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

Cited by 35 publications

References 23 publications

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

A comparison principle for the Lane-Emden equation and applications to geometric estimates

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