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Weak solutions for elliptic systems with variable growth in Clifford analysis. (English) Zbl 1299.30126
Summary: In this paper we consider the following Dirichlet problem for elliptic systems: \[ \begin{split} \overline {DA(x,u(x),Du(x))}& = B(x,u(x),Du(x)),\quad x\in \Omega ,\\ u(x)& = 0,\quad x\in \partial \Omega , \end{split} \] 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 \(W_{0}^{1,p(x)}(\Omega , {\text{C}}\ell _{n})\) under appropriate assumptions.

30G35 Functions of hypercomplex variables and generalized variables
35J60 Nonlinear elliptic equations
35D30 Weak solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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