\(L^ p\) theory and partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth
type: Item
instance of: scholarly article
title: \(L^ p\) theory and partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth
zbMATH Open document ID: 0573.35050
DOI: 10.1007/BF01766852
author: Mario Marino
author: Antonino Maugeri
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1985-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded open set in \({\mathbb{R}}^ n\) \((n>2)\) with smooth boundary. \(X=(x,t)\) is a variable in \(Q=\Omega \times (-T,0)\), where \(T>0\). We put \(p\equiv \{p^{\alpha}\}_{| \alpha | \leq m}\), \(p^*\equiv \{p^{\alpha}\}_{| \alpha | \leq m-1}\), \(p^{\alpha}\in {\mathbb{R}}^ n\). Then \(p\in {\mathcal R}\equiv \prod_{| \alpha | \leq m}{\mathbb{R}}^ n_{\alpha}\), \(p^*\in {\mathcal R}^*\equiv \prod_{| \alpha | \leq m-1}{\mathbb{R}}^ n_{\alpha}\). Consider in Q the following quasilinear system \[ (1)\quad (-1)^ m\sum_{| \alpha | =m}\sum_{| \beta | =m}D^{\alpha}(A_{\alpha \beta}(x,\delta u)D^{\beta}u)+\partial u/\partial t=F\equiv \] \[ \equiv (-1)^ m\sum_{| \alpha | =m}D^{\alpha}f^{\alpha}(X,\delta u)+\sum_{| \alpha | \leq m-1}(-1)^{| \alpha |}D^{\alpha}f^{\alpha}(X,Du) \] where \(A_{\alpha \beta}(x,p^*)\), \(| \alpha | =m\), \(| \beta | =m\), are \(n\times n\) matrices, uniformly continuous and bounded in \(\bar Q\times {\mathcal R}^*\) and such that \[ (2)\quad \sum_{| \alpha | =m}\sum_{| \beta | =m}(A_{\alpha \beta}(X,p^*)\xi^{\beta}| \xi^{\alpha})\geq \sum_{| \alpha | =m}\| \xi^{\alpha}\|^ 2,\quad \nu >0, \] for \((X,p^*)\in \bar Q\times {\mathcal R}^*\) and \(\{\xi^{\alpha}\}_{| \alpha | =m}\), \(\xi^{\alpha}\in {\mathbb{R}}^ n\), \(f^{\alpha}(X,p^*)\), \(| \alpha | =m\), and \(f^{\alpha}(X,p)\), \(| \alpha | \leq m-1\), are vectors in \({\mathbb{R}}^ n\), measurable in \(X\in Q\), continuous in \(p^*\in {\mathcal R}^*\) and \(p\in {\mathcal R}\) respectively, which satisfy the conditions: \[ (3)\quad \| f^{\alpha}(X,p^*)\| \leq g^{\alpha}(X)+c\sum_{| \beta | \leq m-1}\| p^{\beta}\|^{Q(m,| \beta |)},\quad | \alpha | =m, \] \[ \| f^{\alpha}(X,p)\| \leq g^{\alpha}(X)+c\sum_{| \beta | \leq m}\| p^{\beta}\|^{Q(| \alpha |,| \beta |)},\quad | \alpha | \leq m-1, \] with \[ (4)\quad 1\leq \theta (m,| \beta |)<(n+2m/n+2| \beta |),\quad | \beta | \leq m-1, \] \[ 1\leq \theta (| \alpha |,| \beta |)<(n+4m-2| \alpha | /n+2| \beta |),\quad | \alpha | \…
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35K35
zbMATH DE Number: 3914833
zbMATH Keywords: partial Hölder continuity
zbMATH Keywords: higher order
zbMATH Keywords: strictly controlled growth
zbMATH Keywords: quasilinear system
reviewed by: Yukiyoshi Ebihara
MaRDI profile type: Publication
cites work: The spaces \(L^ p\), with mixed norm
cites work: Q3900391
cites work: Q3905778
cites work: Partial Hölder continuity of solutions of quasilinear parabolic systems of second order with linear growth
cites work: \(L^ p \)regularity and partial Hölder continuity for solutions of second order parabolic systems with strictly controlled growth
cites work: Spazi \({\mathcal L}^{(p,\vartheta)} (\Omega,\delta)\) e loro proprieta
cites work: Partial regularity for the solutions to nonlinear parabolic systems
cites work: On the partial regularity of weak solutions on nonlinear parabolic systems
cites work: Q3667285
description: scientific article; zbMATH DE number 3914833