Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents
type: Item
instance of: scholarly article
title: Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents
zbMATH Open document ID: 1421.35165
author: Maria-Magdalena Boureanu
author: Alejandro Vélez-Santiago
published in: Journal of Differential Equations
publication date: 2019-03-21T00:00:00Z
review text: This paper deals with classes of quasilinear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded domains. The corresponding equations contain the $\vec p(\cdot)$-Laplace operator, namely \[\Delta_{\vec p(\cdot)} u= \sum^N_1\partial_{x_i}(|\partial_{x_i}|^{p_i(\cdot)- 2}\partial_{x_i}),\quad x\in\Omega\subset\mathbb{R}^N.\] $\Omega$ can be a domain of non-Lipschitz type as some domain with fractal boundary. The considerations are done in the anisotropic Sobolev space with variable exponent $W^{1,\vec p}$, where $\vec p$ is a $L^\infty$ vector function from $\overline\Omega$ into $[1,\infty]^N$. Solvability and global regularity results for the elliptic and parabolic Robin problems are proved in Theorem 1 and Theorem 3.\par In the elliptic case the weak solution $u$ turns out to be Hölder contiuous in $\overline\Omega$. An a-priori estimate for $u$ is given by formula (1.7). The unique strong solution of the well-posed over $C(\overline\Omega)$ parabolic problem is written in explicit form by the Markovian $C_0$-semigroup generated by $-\Delta_{\vec p(\cdot)}$ acting on the Cauchy data. In Theorem 3 it is shown that, if in the elliptic case the right-hand sides of the Robin problem are nonnegative, then the weak solution is nonnegative in $\overline\Omega$. The difference of two strong solutions of the parabolic problem provides the ultracontractivity property (Theorem 4).
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35K92
zbMATH DE Number: 7039974
zbMATH Keywords: $\vec p(\cdot)$-Laplace operator, quasilinear elliptic equation
zbMATH Keywords: quasilinear parabolic equation
zbMATH Keywords: Robin boundary value problem
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: j.jde.2018.12.026
OpenAlex ID: W2908187489
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Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents (Wikidata QID)
Wikidata QID: Q128641160

Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents (DOI)
DOI: 10.1016/J.JDE.2018.12.026