Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains
type: Item
instance of: scholarly article
title: Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains
zbMATH Open document ID: 1170.35027
author: Sun-Sig Byun
published in: Journal of Differential Equations
publication date: 2009-05-19T00:00:00Z
review text: The interesting paper under review establishes global Hessian estimates in Orlicz spaces for fourth-order parabolic systems of the form \[ \begin{cases} \displaystyle {{\partial u^i}\over {\partial t}}+D_{\alpha\beta}(A_{ij}^{\alpha\beta ab}(x,t) D_{ab} u^j)= D_{\alpha\beta} f^i_{\alpha\beta} & \text{in}\;\Omega_T,\\ |u^i|+|Du^i|=0 & \text{on}\;\partial_P \Omega_T. \end{cases} \] Here \(\Omega_T\) is the cylinder \(\Omega\times(0,T)\) with a bounded base \(\Omega\subset \mathbb R^n\) the boundary of which is \((\delta,R)\)-Reifenberg flat and \(\partial_P\Omega_T\) stands for the parabolic boundary of \(\Omega_T.\) The tensor coefficients \(A_{ij}^{\alpha\beta ab}(x,t)\) are supposed to be bounded, uniformly elliptic and to have small weak BMO semi-norms while \(\mathbf{f}=\{f^i_{\alpha\beta}\}\) is a given tensor matrix with \(|\mathbf{f}|^2\in L^\varphi(\Omega_T)\) where \(\varphi\) is a Young function satisfying some moderate growth condition. The main result of the paper asserts that under the above conditions the unique weak solution \(u\) to the above system satisfies \(|D^2u|^2\in L^\varphi(\Omega_T)\) with the estimate \[ \int_{\Omega_T} \varphi(|D^2u|^2)\,dx\,dt \leq C \int_{\Omega_T} \varphi(|\mathbf{f}|^2)\,dx\,dt. \]
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 5555268
zbMATH Keywords: BMO space
zbMATH Keywords: Reifenberg domains
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: j.jde.2009.01.023
OpenAlex ID: W2050135899
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DOI: 10.1016/J.JDE.2009.01.023
description: scientific article; zbMATH DE number 5555268