# Article

 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: http://hdl.handle.net/10338.dmlcz/143482 . Reference: [1] Abłamowicz, R., Fauser, B., eds.: Clifford Algebras and Their Applications in Mathematical Physics.Proceedings of the 5th Conference, Ixtapa-Zihuatanejo, Mexico, June 27--July 4, 1999. Volume 1: Algebra and Physics. Progress in Physics 18 Birkhäuser, Boston (2000). MR 1783520 Reference: [2] Abreu-Blaya, R., Bory-Reyes, J., Delanghe, R., Sommen, F.: Duality for harmonic differential forms via Clifford analysis.Adv. Appl. 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