Weak solutions for elliptic systems with variable growth in Clifford analysis

Yongqiang Fu; Binlin Zhang

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 3, page 643-670
  • ISSN: 0011-4642

Abstract

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In this paper we consider the following Dirichlet problem for elliptic systems: D A ( x , u ( x ) , D u ( x ) ) ¯ = B ( x , u ( x ) , D u ( x ) ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω , where D is a Dirac operator in Euclidean space, u ( x ) is defined in a bounded Lipschitz domain Ω in 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 ) ( Ω , C ℓ n ) under appropriate assumptions.

How to cite

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Fu, Yongqiang, and Zhang, Binlin. "Weak solutions for elliptic systems with variable growth in Clifford analysis." Czechoslovak Mathematical Journal 63.3 (2013): 643-670. <http://eudml.org/doc/260666>.

@article{Fu2013,
abstract = {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.},
author = {Fu, Yongqiang, Zhang, Binlin},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic system; Clifford analysis; variable exponent; Dirichlet problem; elliptic system; Clifford analysis; variable exponent; Dirichlet problem},
language = {eng},
number = {3},
pages = {643-670},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weak solutions for elliptic systems with variable growth in Clifford analysis},
url = {http://eudml.org/doc/260666},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Fu, Yongqiang
AU - Zhang, Binlin
TI - Weak solutions for elliptic systems with variable growth in Clifford analysis
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 643
EP - 670
AB - 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.
LA - eng
KW - elliptic system; Clifford analysis; variable exponent; Dirichlet problem; elliptic system; Clifford analysis; variable exponent; Dirichlet problem
UR - http://eudml.org/doc/260666
ER -

References

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