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Title: Weak solutions for elliptic systems with variable growth in Clifford analysis (English)
Author: Fu, Yongqiang
Author: Zhang, Binlin
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 63
Issue: 3
Year: 2013
Pages: 643-670
Summary lang: English
Category: math
Summary: In this paper we consider the following Dirichlet problem for elliptic systems: $$ \begin {aligned} \overline {DA(x,u(x),Du(x))}=&B(x,u(x),Du(x)),\quad x\in \Omega ,\cr u(x)=&0,\quad x\in \partial \Omega , \end {aligned} $$ where $D$ is a Dirac operator in Euclidean space, $u(x)$ is defined in a bounded Lipschitz domain $\Omega $ in $\mathbb {R}^{n}$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space $W_{0}^{1,p(x)}(\Omega , {\rm C}\ell _{n})$ under appropriate assumptions. (English)
Keyword: elliptic system
Keyword: Clifford analysis
Keyword: variable exponent
Keyword: Dirichlet problem
MSC: 30G35
MSC: 35D30
MSC: 35J60
MSC: 46E35
idZBL: Zbl 06282103
idMR: MR3125647
DOI: 10.1007/s10587-013-0045-x
Date available: 2013-10-07T12:01:49Z
Last updated: 2020-07-03
Stable URL:
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