Variable Exponent Spaces Of Clifford-valued Functions And Their Applications

In Clifford analysis usually the Dirac operators or generalized Cauchy-Riemann sys-tem are studied in classic Sobolev spaces, in which solutions are defined on domains inEuclidean spaces and taken values in Clifford algebras. With the appearance of nonlinearproblems in nature science and engineering, Sobolev spaces demonstrate limitations inapplications, for instance, the study on a class of nonlinear problems with variable ex-ponent growth. In the research of this kinds of nonlinear problems, variable exponentLebesgue spaces and Sobolev spaces play an important role. Therefore, it is natural tostudy variable exponent spaces of Clifford-valued functions.In this dissertation, variable exponent spaces of Clifford-valued functions are in-vestigated with applications to the existence and uniqueness of solutions to Dirac typeequations. The main contents are as follows:（ⅰ） Weighted variable exponent spaces of Clifford-valued functions are introduced.Weighted variable exponent Lebesgue spaces of Clifford-valued functions and weight-ed variable exponent Sobolev spaces of Clifford-valued functions W^{D,p（x）}（Ω, Cl_n, ω） andvariable exponent Sobolev spaces of Clifford-valued functions W^{1,p（x）}（Ω, Cl_n, ω） are in-troduced, then the properties of these spaces, for example, completeness, reflexivity, sep-arability, embedding theorem etc, are discussed. Operator theory in variable exponentspaces of Clifford-valued functions is studied, especially the boundedness of Teodores-cu operators and Dirac operators are investigated. A direct decomposition for the spaceL^{p（x）}（Ω, Cl_n） is established, then the related properties are discussed.（ⅱ） Based on related operator theory in variable exponent spaces of Clifford-valuedfunctions and Kinderlehrer-Stampacchia theorem, the existence of weak solutions for ob-stacle problems for the scalar parts of homogeneous A-Dirac equations is obtained, andthen the existence of weak solutions to the scalar part of homogeneous A-Dirac equa-tions DA（x,Du）=0in spaces W^{1,p（x）} is obtained. Furthermore, the existenceof weak solutions for obstacle problems for the scalar parts of nonhomogeneous A-Diracequations is obtained, and then the existence of weak solutions to the scalar parts of non-homogeneous A-Dirac equations DA（x,Du）+B（x, u）=0in spaces W^{D,p（x）}（Ω, Cl_n, ω） isobtained. Using the direct decomposition and Minty-Browder theorem, the existence of weak solutions to A-Dirac equations DA（Du）=0with variable growth is obtained. More-over, the existence of weak solutions to the scalar parts of elliptic systems DA（x,u,Du）=B（x,u,Du） in spaces W₀^{1,p（x）}（Ω, CÏ„_n） is obtained under certain conditions.（ⅲ） By operator theory in variable exponent spaces of Clifford-valued functions andthe properties of the direct decomposition, the existence and uniqueness and represen-tation of solutions in spaces W₀^{1,p（x）}（Ω, CÏ„_n）×L^p（x）（Ω） to the Stokes equations are ob-tained. Furthermore, the Navier-Stokes questions are studied in variable exponent spacesof Clifford-valued functions. Constructing an iteration in which the unique solvability ofthe corresponding Stokes equation is used at each step, the convergence of the iterationis proved by the contraction mapping principle, and then the existence and uniquenessof solutions in spaces W₀^{1,p（x）}（Ω, CÏ„_n）×L^p（x）（Ω） to the steady Navier-Stokes equationsare obtained under certain condition on the external force. Furthermore, by the itera-tive technique and the contraction mapping principle which are similar to those of thestationary Navier-Stokes equations, the existence and uniqueness of solutions in spacesW₀^{1,p（x）}（Ω, CÏ„_n）×W₀^{1,p（x）}（Ω, CÏ„_n）×L^p（x）（Ω） to the Navier-Stokes equations with heat con-duction are obtained under certain assumptions.