Geodesic
Weak solutions for elliptic systems with variable growth in Clifford analysis
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 643-670.

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In this paper we consider the following Dirichlet problem for elliptic systems: $$ \begin {aligned} \overline {DA(x,u(x),Du(x))}=(x,u(x),Du(x)),\quad x\in \Omega ,\cr u(x)=0,\quad x\in \partial \Omega , \end {aligned} $$ 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.
DOI : 10.1007/s10587-013-0045-x
Classification : 30G35, 35D30, 35J60, 46E35
Keywords: elliptic system; Clifford analysis; variable exponent; Dirichlet problem 
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Fu, Yongqiang; Zhang, Binlin. Weak solutions for elliptic systems with variable growth in Clifford analysis. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 643-670. doi : 10.1007/s10587-013-0045-x. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0045-x/

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