变指数Clifford值函数空间及其应用
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摘要
Clifford分析通常在经典的Sobolev空间中研究Dirac算子方程或广义Cauchy-Riemann系统的解,其中的解是定义在欧氏空间中取值在Clifford代数上的函数.随着自然科学和工程技术中许多非线性问题的不断出现,经典的Sobolev空间表现出其应用范围的局限性,如对一类具有变指数增长性条件的非线性问题.对于这类非线性问题,变指数Lebesgue空间和Sobolev空间发挥着重要的作用.因此,取值在Clifford代数上的变指数函数空间理论的研究就成为必要了.
本文主要研究变指数Clifford值函数空间理论以及在这个背景下一些Dirac型方程组解的存在性及唯一性问题,主要内容如下:
(ⅰ)建立了加权变指数Clifford值函数空间理论.引入了加权变指数Clifford值函数Lebesgue空间L~(p(x))(Ω, Cl_n, ω)和加权变指数Clifford值函数Sobolev空间W~(D,p(x))(Ω, Cl_n, ω)及W(Ω, Cl_n, ω),讨论了这些空间的一些基本性质,如完备性,自反性,可分性,嵌入定理等.研究了变指数Clifford值函数空间中的有关算子理论,如Teodorescu算子和Dirac算子的有界性.证明了变指数Clifford值函数Lebesgue空间L~(p(x))(Ω, Cl_n)的一个直和分解，并讨论了其相关性质.
(ⅱ)基于变指数Clifford值函数空间中的有关算子理论以及Kinderlehrer-Stampacchia定理,证明了关于齐次A-Dirac方程组数量部分的障碍问题解的存在性,从而证明了A-Dirac方程组DA(x,Du)=0数量部分在空间W_0~(1,p(x))(Ω, Cl_n)中弱解的存在性.证明了关于非齐次A-Dirac方程组数量部分的障碍问题解的存在唯一性,从而证明了非齐次A-Dirac方程组DA(x,Du)+B(x, u)=0数量部分在空间W~(D,p(x))(Ω, Cl_n, ω)中弱解的存在唯一性.基于直和分解和Minty-Browder定理,证明了具有变指数增长条件的齐次A-Dirac方程组DA(Du)=0在空间W_0~(1,p(x))(Ω, Cl_n)中存在唯一的弱解.在适当的结构条件下,证明了一般的椭圆型方程组DA(x,u,Du)=B(x, u, Du)数量部分在空间W_0~(1,p(x))(Ω, Cl_n)中弱解的存在性.
(ⅲ)根据变指数Clifford值函数空间中的算子理论以及直和分解的性质,证明了Stokes方程组在空间W_0~(1,p(x))Ω, Cl_n)×L~(p(x))中存在唯一解,给出了解的表示.进而对变指数Clifford值函数空间中的Navier-Stokes问题进行了研究.构造了一个迭代,其中每一步都要用到相应的Stokes方程组解的存在唯一性,根据压缩映射原理可得迭代的收敛性,从而证明了当外部压力满足适当条件时,稳态的Navier-Stokes方程组在空间W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω)中存在唯一解.进一步,运用了类似处理稳态Navier-Stokes问题的方法,通过迭代技巧以及压缩映射原理,证明了在适当条件下带有热传导的Navier-Stokes方程组在空间W_0~(1,p(x))(Ω, Cτ_n)×W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω)中存在唯一解.
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 W_0~(1,p(x))(Ω, Cτ_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 W_0~(1,p(x))(Ω, Cτ_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 W_0~(1,p(x))(Ω, Cτ_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 spacesW_0~(1,p(x))(Ω, Cτ_n)×W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω) to the Navier-Stokes equations with heat con-duction are obtained under certain assumptions.
本文主要研究变指数Clifford值函数空间理论以及在这个背景下一些Dirac型方程组解的存在性及唯一性问题,主要内容如下:
(ⅰ)建立了加权变指数Clifford值函数空间理论.引入了加权变指数Clifford值函数Lebesgue空间L~(p(x))(Ω, Cl_n, ω)和加权变指数Clifford值函数Sobolev空间W~(D,p(x))(Ω, Cl_n, ω)及W(Ω, Cl_n, ω),讨论了这些空间的一些基本性质,如完备性,自反性,可分性,嵌入定理等.研究了变指数Clifford值函数空间中的有关算子理论,如Teodorescu算子和Dirac算子的有界性.证明了变指数Clifford值函数Lebesgue空间L~(p(x))(Ω, Cl_n)的一个直和分解，并讨论了其相关性质.
(ⅱ)基于变指数Clifford值函数空间中的有关算子理论以及Kinderlehrer-Stampacchia定理,证明了关于齐次A-Dirac方程组数量部分的障碍问题解的存在性,从而证明了A-Dirac方程组DA(x,Du)=0数量部分在空间W_0~(1,p(x))(Ω, Cl_n)中弱解的存在性.证明了关于非齐次A-Dirac方程组数量部分的障碍问题解的存在唯一性,从而证明了非齐次A-Dirac方程组DA(x,Du)+B(x, u)=0数量部分在空间W~(D,p(x))(Ω, Cl_n, ω)中弱解的存在唯一性.基于直和分解和Minty-Browder定理,证明了具有变指数增长条件的齐次A-Dirac方程组DA(Du)=0在空间W_0~(1,p(x))(Ω, Cl_n)中存在唯一的弱解.在适当的结构条件下,证明了一般的椭圆型方程组DA(x,u,Du)=B(x, u, Du)数量部分在空间W_0~(1,p(x))(Ω, Cl_n)中弱解的存在性.
(ⅲ)根据变指数Clifford值函数空间中的算子理论以及直和分解的性质,证明了Stokes方程组在空间W_0~(1,p(x))Ω, Cl_n)×L~(p(x))中存在唯一解,给出了解的表示.进而对变指数Clifford值函数空间中的Navier-Stokes问题进行了研究.构造了一个迭代,其中每一步都要用到相应的Stokes方程组解的存在唯一性,根据压缩映射原理可得迭代的收敛性,从而证明了当外部压力满足适当条件时,稳态的Navier-Stokes方程组在空间W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω)中存在唯一解.进一步,运用了类似处理稳态Navier-Stokes问题的方法,通过迭代技巧以及压缩映射原理,证明了在适当条件下带有热传导的Navier-Stokes方程组在空间W_0~(1,p(x))(Ω, Cτ_n)×W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω)中存在唯一解.
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 W_0~(1,p(x))(Ω, Cτ_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 W_0~(1,p(x))(Ω, Cτ_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 W_0~(1,p(x))(Ω, Cτ_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 spacesW_0~(1,p(x))(Ω, Cτ_n)×W_0~(1,p(x))(Ω, Cτ_n)×L~p(x)(Ω) to the Navier-Stokes equations with heat con-duction are obtained under certain assumptions.
引文
[1] Orlicz W.U¨ber Konjugierte Exponentenfolgen[J]. Studia Mathematica,1931,3:200–212.
[2] Sharapudinov I I. On the Topology of the Space Lp(t)([0,1])[J]. MathematicalNotes,1979,26:796–806.
[3] Zhikov V V. Averaging of Functionals of the Calculus Variations and Elastici-ty Theory[J]. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya,1986,877:675–710.
[4] Zhikov V V. On Lavrentiev’s Phenomenon[J]. Journal of Mathematical Physics,1995,3:249–269.
[5] Zhikov V V. Solvability of the Three-dimensional Thermistor Problem[J]. TrudyMatematicheskogo Instituta Imeni V. A. Steklova,2008,261:101–114.
[6] Kova′cˇik O, Ra′kosn′k J. On Spaces Lp(x)and Wk,p(x)[J]. Czechoslovak Mathemati-cal Journal,1991,41:592–618.
[7] Fan X, Shen J, Zhao D. Sobolev Embedding Theorems for Spaces Wk,p(x)()[J].Journal of Mathematical Analysis and Applications,2001,262:749–760.
[8] Fan X, Zhao D. On the Spaces Lp(x)() and Wm,p(x)()[J]. Journal of MathematicalAnalysis and Applications,2001,263:424–446.
[9] Fan X, Zhao Y, Zhao D. Compact Imbedding Theorems with Symmetry of Strauss-Lions Type for the Space W1,p(x)()[J]. Journal of Mathematical Analysis andApplications,2001,255:333–348.
[10] Fan X. Boundary Trace Embedding Theorems for Variable Exponent SobolevSpaces[J]. Journal of Mathematical Analysis and Applications,2008,339:1395–1412.
[11] Rajagopal K R, Ru zˇicˇka M. On the Modeling of Electrorheological Materials[J].Mechanics Research Communications,1996,23:401–407.
[12] Halsey T C. Electrorheological Fluids[J]. Science,1992,258:761–766.
[13] Ru zˇicˇka M. Electrorheological Fluids: Modeling and Mathematical Theory[M].Berlin: Springer-Verlag,2000.
[14] Acerbi G, E.and Mingione. Regularity Results for a Class of Functionals with Non-standard Growth[J]. Archive for Rational Mechanics and Analysis,2001,156:121–140.
[15] Acerbi G, E.and Mingione. Regularity Results for Stationary ElectrorheologicalFluids[J]. Archive for Rational Mechanics and Analysis,2002,164:213–259.
[16] Chen Y, Levine S, Rao M. Variable Exponent, Linear Growth Functionals in ImageRestoration[J]. SIAM Journal on Applied Mathematics,2006,66(4):1383–1406.
[17] Aboulaich R, Meskine R, Souissi D. New Diffusion Models in Image Process-ing[J]. Computers and Mathematics with Applications,2008,56:874–882.
[18] Diening L, Harjulehto P, Ha¨sto¨ P, et al. Lebesgue and Sobolev Spaces with VariableExponents[M]. New Work: Springer,2011.
[19] Diening L. Maximal Function on Generalized Lebesgue Spaces Lp(·)[J]. Mathe-matical Inequalities and Applications,2004,7:245–253.
[20] Alves C O, Souto M A. Existence of Solutions for a Class of Problems inRNIn-volving p(x)-Laplacian[J]. Progress in Nonlinear Differential Equations and TheirApplications,2005,66:17–32.
[21] Alves C O. Existence of Solution for a Degenerate p(x)-Laplacian Equation inRN[J]. Journal of Mathematical Analysis and Applications,2008,345:731–742.
[22] Miha ilescu M. Existence and Multiplicity of Solutions for an Elliptic Equation withp(x)-Growth Conditions[J]. Glasgow Mathematical Journal,2006,48:411–418.
[23] Miha ilescu M. Existence and Multiplicity of Solutions for a Neumann ProblemInvolving the p(x)-Laplace Operator[J]. Nonlinear Analysis,2007,67:1419–1425.
[24] Miha ilescu M. Elliptic Problems in Variable Exponent Spaces[J]. Bulletin of theAustralian Mathematical Society,2006,74(2):197–206.
[25] Miha ilescu M, Ra dulescu V. A Multiplicity Result for a Nonlinear DegenerateProblem Arising in the Theory of Electrorheological Fluids[J]. Proceedings of theRoyal Society A,2006,462:2625–2641.
[26] Miha ilescu M, Ra dulescu V. On a Nonhomogeneous Quasilinear Eigenvalue Prob-lem in Sobolev Spaces with Variable Exponent[J]. Proceedings of The AmericanMathematical Society,2007,135(9):2929–2937.
[27] Fan X, Han X. Existence and Multiplicity of Solutions for p(x)-Laplacian Equa-tions inRN[J]. Nonlinear Analysis,2004,59:173–188.
[28] Fan X, Zhang Q. Existence of Solutions for p(x)-Laplacian Dirichlet Problem[J].Nonlinear Analysis,2003,52:1843–1852.
[29] Fu Y, Dong Z, Yan Y. On the Existence of Weak Solutions for a Class of EllipticPartial Differential Systems[J]. Nonlinear Analysis,2002,48:961–977.
[30] Fu Y. The Existence of Solutions for Elliptic Systems with Nonuniform Growth[J].Studia Mathematica,2002,151(3):227–246.
[31] Fu Y. Weak Solution for Obstacle Problem with Variable Growth[J]. NonlinearAnalysis,2004,59:371–383.
[32] Chabrowski J, Fu Y. Existence of Solutions for p(x)-Laplacian Problems on aBounded Domain[J]. Journal of Mathematical Analysis and Applications,2005,306:604–618. Erratum in: Journal of Mathematical Analysis and Applications,2006,323:1483.
[33] Fu Y. Existence of Solutions for p(x)-Laplacian Problem on an Unbounded Do-main[J]. Topological Methods in Nonlinear Analysis,2007,30:235–249.
[34] Fu Y. The Principle of Concentration Compactness in Lp(x)Spaces and its Appli-cation[J]. Nonlinear Analysis,2009,71:1876–1892.
[35] Fu Y, Zhang X. A Multiplicity Result for p(x)-Laplacian Problem inRN[J]. Non-linear Analysis,2009,70:2261–2269.
[36] Fu Y, Zhang X. Multiple Solutions for a Class of p(x)-Laplacian Equations inRNInvolving the Critical Exponent[J]. Proceedings of the Royal Society A,2010,466:1667–1686.
[37] Lukkari T. Elliptic Equations with Nonstandard Growth Involving Measures[J].Hiroshima Mathematical Journal,2008,38:155–176.
[38] Fan X. Global C1,αRegularity for Variable Exponent Elliptic Equations in Diver-gence Form[J]. Journal Differential Equations,2007,235(2):397–417.
[39] Miha ilescu M. On a Class of Nonlinear Problems Involving a p(x)-Laplace TypeOperator[J]. Czechoslovak Mathematical Journal,2008,58(133):155–172.
[40] Diening L, Lengeler D, Ru zˇicˇka M. The Stokes and Poission Problem in VariableExponent Spaces. Complex Varibles and Elliptic Equations[J]. Complex Variblesand Elliptic Equations,2011,56:789–811.
[41] Miha ilescu M, Pucci P, Ra dulescu V. Eigenvalue Problems for Anisotropic Quasi-linear Elliptic Equations with Variable Exponent[J]. The Journal of MathematicalAnalysis and Applications,2008,340:687–698.
[42] Harjulehto P, Ha¨sto¨ P, V. Le U, et al. Nuortio. Overview of Differential Equationswith Non-standard Growth[J]. Nonlinear Analysis,2010,72:4551–4574.
[43]付永强,张夏.变指数函数空间在偏微分方程上的应用[M].科学出版社,2011.
[44] Clifford W K. Applications of Grassmann’s Extensive Algebra[J]. The AmericanJournal of Mathematics,1878,1:350–358.
[45] Dirac P A M. The quantum Theory of the Electron[J]. Proceedings of the RoyalSociety A,1928,117:610–624.
[46] Dalanghe R. Clifford Analysis: History and Perspective[J]. Computational Meth-ods and Function Theory,2001,1:107–153.
[47] Brackx F, Delanghe R, Sommen F. Clifford Analysis[M]. London: Pitman Re-search Notes in Mathematics,1982.
[48] Gu¨rlebeck K, Spro¨ ig W. Quaternionic and Clifford Calculus for Physicists andEngineers[M]. Chichester: John Wiley&Sons,1997.
[49] Gu¨rlebeck K, Spro¨ ig W. Quaternionic Analysis and Elliptic Boundary[M]. Berlin:Birha¨user,1990.
[50] Gu¨rlebeck K, Spro¨ ig W. Fluid Flow Problems with Quaternionic Analysis–AnAlternative Conception[J]. Geometric Algebra Computing,2010,2010:345–380.
[51] Gu¨rlebeck K, Spro¨ ig W. An Introduction to the Mathematical Theory of theNavier-Stokes Equations[M]. New York: Springer,2011.
[52] Teman R. Navier-Stokes Equations: Theory and Numerical Analysis[M]. Amster-dam: North–Holland Publishing Company,1977.
[53] Teman R. Navier-Stokes Equations and Nonlinear Functional Analysis[M]. Soci-ety for Industrial and Applied Mathematics,1995.
[54] Spro¨ ig W. Fluid Flow Equations with Variable Viscosity in Quternionic Setting[J].Advances in Applied Clifford Algebras,2007,17:259–272.
[55] Ka¨hler U. On a Direct Decomposition in the Space Lp()[J]. Zeitschrift fu¨r Anal-ysis und ihre Anwendungen,1999,4:839–848.
[56] Ka¨hler U. Elliptic Boundary Value Problems in Bounded and Unbounded Do-main[M]. Harlow: Addison Wesley,1998.
[57] Ka¨hler U. Clifford Analysis and the Navier-Stokes Equations over UnboundedDomains[J]. Advances in Applied Clifford Algebras,2001,11:305–318.
[58] Gu¨rlebeck K, Ka¨hler U, Ryan J. Clifford Analysis over Unbounded Domain[J].Advances in Applied Mathematics,1997,19:216–239.
[59] Cerejeiras P, Ka¨hler U. Elliptic Boundary Value Problems of Fluid Dynamics overUnbounded Domains[J]. Mathemetical Methods in the Applied Sciences,2000,23:81–101.
[60] Borchers W, Miyakawa T. On Stability of Exterior Stationary Navier-StokesFlows[J]. Acta Mathematica,1995,174:311–382.
[61] Novotny A, Padula M. Note on Decay of Solutions of Steady Navier-Stokes Equa-tions in3-D Exterior Domains[J]. Differential Integral Equations,1995,8:1833–1842.
[62] Kozono H, Sohr H. New A-priori-estimates for the Stokes Equations in ExteriorDomains[J]. Indiana University Mathematics Journal,1991,40:1–27.
[63] Kozono H, Sohr H. On a New Class of Generalized of Solutions of the StokesEquations in Exterior Domains[J]. Annali della Scuola Normale Superiore di Pisa,1992,19:155–181.
[64] Varnhorn W. The Stokes Equations[M]. Berlin: Akademie–Verlag,1994.
[65] Nolder C A. A-Harmonic Equations and the Dirac Operator[J]. Journal of Inequal-ity and Applications,2010,2010:1–9.
[66] Nolder C A. Nonlinear A-Dirac Equations[J]. Advances in Applied Clifford Alge-bras,2011,21:429–440.
[67] Nolder C A, Ryan J. p-Dirac Operators[J]. Advances in Applied Clifford Algebras,2009,19:391–402.
[68] Nolder C A. Conjungate Harmonic Functions and Clifford Algebras[J]. Journal ofMathematical Analysis and Applications,2005,302:137–142.
[69] Abreu-Blaya R, Bory-Reyes J, Delanghe R, et al. Duality for Harmonic DifferentialForms Via Clifford Analysis[J]. Advances in Applied Clifford Algebras,2007,17:589–610.
[70] Brackx F, Delanghe R, Sommen F. Diffrential Forms and/or Multi-vector Functionsin Hermitean Clifford Analysis[J]. Cubo,2011,13:85–117.
[71] Gilbert J, Murray M A M. Clifford Algebra and Dirac Oprators in HarmonicAnalysis[M]. Oxford: Oxford University Press,1993.
[72] Delanghe R, Sommen F, Soucek V. Clifford Algebra and Spinor-valued Func-tion[M]. Dordrecht: Kluwer Academic Publishers,1992.
[73] Gu¨rlebeck K, Spro¨ ig W. Holomorphic Functions in the Plane and n-dimensionalSpace[M]. Boston: Birkha¨user,2008.
[74] Huang S, Qiao Y, Wen G. Real and Complex Clifford Analysis[M]. New Work:Springer,2006.
[75] Du J, Zhang Z. A Cauchy’s Integral Formula for Functions with Values in a Uni-versal Clifford Algebra and its Applications[J]. Complex Variables,2002,47:915–928.
[76] Edmunds D, Ra′kosn′k J. Sobolev Embedding with Variable Exponent[J]. StudiaMathematica,2000,143:267–293.
[77] Edmunds D, Ra′kosn′k J. Sobolev Embedding with Variable Exponent, II[J]. Math-ematische Nachrichten,2002,246:53–67.
[78] Hudzik H. The Problems of Separability, Duality, Reflexivity and of Comparisonfor Generalized Orlicz-Sobolev Spaces Wkm()[J]. Commentationes Mathematicae(Prace Matematyczne),1979,21:315–324.
[79] Doran C, Lasenby A. Geometric Algebra for Physicists[M]. Cambridge: Cam-bridge University Press,2003.
[80] Dubinskii J, Reissig M. Variational Problems in Clifford Analysis[J]. MathematicalMethods in the Applied Sciences,2002,25:1161–1176.
[81] Kravchenko V, Oviedo H. On a Quaternionic Reformulation of Maxwell’s Equa-tions[J]. Zeitschrift fu¨r Analysis and ihre Anwendungen,2003,22:569–589.
[82] Ablamowicz R. Clifford Algebras and Their Applications in Mathematical Physics,Volume1: Algebra and Physics[M]. Boston: Birkha¨user,2000.
[83] Ryan J, Spro¨ ig W. Clifford Algebras and Their Applications in MathematicalPhysics, Volume2: Clifford Analysis[M]. Boston: Birkha¨user,2000.
[84] Bahmann M, Gu¨rlebeck K, Shapiro M. On a Modified Teodorescu Transform[J].Integral Transforms and Special Functions,2001,12(3):213–226.
[85] Zhang Z. Some Properties of Operators in Clifford Analysis[J]. Complex Variablesand Elliptic Equations,2007,52(6):455–473.
[86] Begehr H, Zhang Z, Ha V. Generalized Integral Representations in Clifford Anal-ysis[J]. Complex Variables and Elliptic Equations,2006,51:745–762.
[87] Spro¨ ig W. On Decompositions of Sobolev Spaces in Clifford Analysis[J]. Ad-vances in Applied Clifford Algebras,1995,5(2):167–185.
[88] Begehr H, Dubinskii J. Orthogonal Decompositions of Sobolev Spaces in CliffordAnalysis[J]. Annali di Matematica Pura ed Applicata,2002,181:55–71.
[89] Dubinski J A. On a Nonlinear Analytic Problem[J]. Doklady Mathematics,1998,48:370–375.
[90] Dubinskii J, Reissig M. Variational problems in Clifford analysis[J]. MathematicalMethods in the Applied Sciences,2002,25:1161–1176.
[91] Heinonen J, Kilpela¨inen T, Martio O. Nonlinear Porential Theory of DegenerateElliptic Equations[M]. Oxford: Oxford University Press,1993.
[92] Eleuteri M, Habermann J. Regularity Results for a Class of Obstacle Problemsunder Nonstandard Growth Conditions[J]. Journal of Mathematical Analysis andApplications,2008,344:1120–1142.
[93] Rodrigues J F, Sanchon M, Urbano J M. The Obstacle Problem for Nonlinear Ellip-tic Equations with Variable Growth and L1-data[J]. Monatshefte fur Mathematik,2008,154:303–322.
[94] Harjulehto P, Ha¨sto¨ P, Koskenoja M, et al. Obstacle Problem and SuperharmonicFunctions with Nonstandard Growth. Nonlinear Analysis[J]. Nonlinear Analysis,2007,67:3424–3440.
[95] Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities andTheir Applications[M]. New York: Academic Press,1980.
[96] Brezis H.泛函分析–理论和应用[M].清华大学出版社,2009.
[97] Ekeland I, Temam R. Convex Analysis and Variational Problems[M]. North-Holland: Amsterdam,1976.
[98] Eisen G. A Selection Lemma for Sequences of Measure Sets and Lower Semicon-tinuity of Multiple Integrals[J]. Mathematical Manuscripts,1979,27:73–79.
[99] Acerbi E, Fusco N. Semicontinuity Problems in the Calculus of Variations[J].Archive for Rational Mechanics and Analysis,1984,86:125–135.
[100] Liu F C. A Luzin Type Property of Sobolev Functions[J]. Indiana UniversityMathematics Journal,1977,26:645–651.
[101] Morrey C B. Multiple Integrals in the Calculus of Variations[M]. New Work:Springer,1966.
[102] Dacorogna B. Weak Continuity and Weak Lower Semicontinuity of NonlinearFunctionals[M]. New Work: Springer,1982.
[103] Tartar L. An Introduction to Sobolev Spaces and Interpolation Spaces[M]. Berlin:Springer-Verlag,2007.
[104] Yosida K. Functional Analysis[M]. Berlin: Springer-Verlag,1980.
[105] Amrouche C, Girault V. Decomposition of Vector Space and Application to theStokes Problem in Arbitrary Dimensional[J]. Czechoslovak Mathematicl Journal,1994,44:109–140.
[106] Finn R. On the Exterior Stationary Problem for the Navier-Stokes Equations andAssociated Pertubation Problems[J]. Archive for Rational Mechanics and Analysis,1965,19:363–406.
[107] Kim H. Existence and Regularity of Very Weak Solutions of the StationaryNavier-Stokes Equations[J]. Archive for Rational Mechanics and Analysis,2009,193:117–152.
[108] Galdi G P, Silvestre A L. The Steady Motion of a Navier-Stokes Liquid Around aRigid Body[J]. Archive for Rational Mechanics and Analysis,2007,184:371–400.
[109] Galdi G P, Kyed M. Steady-state Navier-Stokes Flows Past a Rotation Body: Ler-ay Solutions Are Physically Reasonable[J]. Archive for Rational Mechanics andAnalysis,2011,200:21–58.
[110] Wang G, Yang D. Decomposition of Vector-valued Divergence Free Sobolev Func-tions and Shape Optimization for Stationary Navier-Stokes Equations[J]. Commu-nications in Partial Differential Equations,2008,33:429–449.
[111] Ru zˇicˇka M. A Note on Steady Flow of Fluids with Shear Dependent Viscosity[J].Nonlinear Analysis,1997,30(5):3029–3039.
[112] Huber H. The Divergence Equation in Weighted-and Lp(·)-Spaces[J]. Mathematis-che Zeitschrift,2011,267:341–366.
[113] Arada N. A Note on the Regularity of Flows with Shear-dependent Viscosity[J].Nonlinear Analysis,2012,75(14):5401–5415.
[114] Diening L, Ru zˇicˇka M. Strong Solutions for Generalized Newtonian Fluids[J].Journal of Mathematica Fluid Mechanics,2005,7:413–450.
[115] Berselli L C, Diening L, Ru zˇicˇka M. Existence of Strong Solutions for Incom-pressible Fluid with Shear Dependent Viscosities[J]. Journal of Mathematica FluidMechanics,2010,12:101–132.
[116] Gu¨rlebeck K. Approximate Solution of the Stationary Navier-Stokes Equations[J].Mathematische Nachrichten,1990,145:297–308.
[117] Cerejeiras P, Ka¨hler U, Sommen F. Parabolic Dirac Operators and Navier-StokesEquations over Time-varying Domains[J]. Mathemetical Methods in the AppliedSciences,2005,28:1715–1724.
[118] Gu¨rlebeck K, Spro¨ ig W. Representation Theory for Classes of Initial Value prob-lems with Quaternionic Analysis[J]. Mathemetical Methods in the Applied Sci-ences,2002,25:1371–1382.
[119] Spro¨ ig W. Quaternionic Operator Methods in Fluid Dynamics[J]. Advances inApplied Clifford Algebras,2008,18:963–978.
[120] Spro¨ ig W, Gu¨rlebeck K. On the Treatment of Fluid Problems by Methods ofClifford Analysis[J]. Advances in Applied Clifford Algebras,1997,44:401–413.
[2] Sharapudinov I I. On the Topology of the Space Lp(t)([0,1])[J]. MathematicalNotes,1979,26:796–806.
[3] Zhikov V V. Averaging of Functionals of the Calculus Variations and Elastici-ty Theory[J]. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya,1986,877:675–710.
[4] Zhikov V V. On Lavrentiev’s Phenomenon[J]. Journal of Mathematical Physics,1995,3:249–269.
[5] Zhikov V V. Solvability of the Three-dimensional Thermistor Problem[J]. TrudyMatematicheskogo Instituta Imeni V. A. Steklova,2008,261:101–114.
[6] Kova′cˇik O, Ra′kosn′k J. On Spaces Lp(x)and Wk,p(x)[J]. Czechoslovak Mathemati-cal Journal,1991,41:592–618.
[7] Fan X, Shen J, Zhao D. Sobolev Embedding Theorems for Spaces Wk,p(x)()[J].Journal of Mathematical Analysis and Applications,2001,262:749–760.
[8] Fan X, Zhao D. On the Spaces Lp(x)() and Wm,p(x)()[J]. Journal of MathematicalAnalysis and Applications,2001,263:424–446.
[9] Fan X, Zhao Y, Zhao D. Compact Imbedding Theorems with Symmetry of Strauss-Lions Type for the Space W1,p(x)()[J]. Journal of Mathematical Analysis andApplications,2001,255:333–348.
[10] Fan X. Boundary Trace Embedding Theorems for Variable Exponent SobolevSpaces[J]. Journal of Mathematical Analysis and Applications,2008,339:1395–1412.
[11] Rajagopal K R, Ru zˇicˇka M. On the Modeling of Electrorheological Materials[J].Mechanics Research Communications,1996,23:401–407.
[12] Halsey T C. Electrorheological Fluids[J]. Science,1992,258:761–766.
[13] Ru zˇicˇka M. Electrorheological Fluids: Modeling and Mathematical Theory[M].Berlin: Springer-Verlag,2000.
[14] Acerbi G, E.and Mingione. Regularity Results for a Class of Functionals with Non-standard Growth[J]. Archive for Rational Mechanics and Analysis,2001,156:121–140.
[15] Acerbi G, E.and Mingione. Regularity Results for Stationary ElectrorheologicalFluids[J]. Archive for Rational Mechanics and Analysis,2002,164:213–259.
[16] Chen Y, Levine S, Rao M. Variable Exponent, Linear Growth Functionals in ImageRestoration[J]. SIAM Journal on Applied Mathematics,2006,66(4):1383–1406.
[17] Aboulaich R, Meskine R, Souissi D. New Diffusion Models in Image Process-ing[J]. Computers and Mathematics with Applications,2008,56:874–882.
[18] Diening L, Harjulehto P, Ha¨sto¨ P, et al. Lebesgue and Sobolev Spaces with VariableExponents[M]. New Work: Springer,2011.
[19] Diening L. Maximal Function on Generalized Lebesgue Spaces Lp(·)[J]. Mathe-matical Inequalities and Applications,2004,7:245–253.
[20] Alves C O, Souto M A. Existence of Solutions for a Class of Problems inRNIn-volving p(x)-Laplacian[J]. Progress in Nonlinear Differential Equations and TheirApplications,2005,66:17–32.
[21] Alves C O. Existence of Solution for a Degenerate p(x)-Laplacian Equation inRN[J]. Journal of Mathematical Analysis and Applications,2008,345:731–742.
[22] Miha ilescu M. Existence and Multiplicity of Solutions for an Elliptic Equation withp(x)-Growth Conditions[J]. Glasgow Mathematical Journal,2006,48:411–418.
[23] Miha ilescu M. Existence and Multiplicity of Solutions for a Neumann ProblemInvolving the p(x)-Laplace Operator[J]. Nonlinear Analysis,2007,67:1419–1425.
[24] Miha ilescu M. Elliptic Problems in Variable Exponent Spaces[J]. Bulletin of theAustralian Mathematical Society,2006,74(2):197–206.
[25] Miha ilescu M, Ra dulescu V. A Multiplicity Result for a Nonlinear DegenerateProblem Arising in the Theory of Electrorheological Fluids[J]. Proceedings of theRoyal Society A,2006,462:2625–2641.
[26] Miha ilescu M, Ra dulescu V. On a Nonhomogeneous Quasilinear Eigenvalue Prob-lem in Sobolev Spaces with Variable Exponent[J]. Proceedings of The AmericanMathematical Society,2007,135(9):2929–2937.
[27] Fan X, Han X. Existence and Multiplicity of Solutions for p(x)-Laplacian Equa-tions inRN[J]. Nonlinear Analysis,2004,59:173–188.
[28] Fan X, Zhang Q. Existence of Solutions for p(x)-Laplacian Dirichlet Problem[J].Nonlinear Analysis,2003,52:1843–1852.
[29] Fu Y, Dong Z, Yan Y. On the Existence of Weak Solutions for a Class of EllipticPartial Differential Systems[J]. Nonlinear Analysis,2002,48:961–977.
[30] Fu Y. The Existence of Solutions for Elliptic Systems with Nonuniform Growth[J].Studia Mathematica,2002,151(3):227–246.
[31] Fu Y. Weak Solution for Obstacle Problem with Variable Growth[J]. NonlinearAnalysis,2004,59:371–383.
[32] Chabrowski J, Fu Y. Existence of Solutions for p(x)-Laplacian Problems on aBounded Domain[J]. Journal of Mathematical Analysis and Applications,2005,306:604–618. Erratum in: Journal of Mathematical Analysis and Applications,2006,323:1483.
[33] Fu Y. Existence of Solutions for p(x)-Laplacian Problem on an Unbounded Do-main[J]. Topological Methods in Nonlinear Analysis,2007,30:235–249.
[34] Fu Y. The Principle of Concentration Compactness in Lp(x)Spaces and its Appli-cation[J]. Nonlinear Analysis,2009,71:1876–1892.
[35] Fu Y, Zhang X. A Multiplicity Result for p(x)-Laplacian Problem inRN[J]. Non-linear Analysis,2009,70:2261–2269.
[36] Fu Y, Zhang X. Multiple Solutions for a Class of p(x)-Laplacian Equations inRNInvolving the Critical Exponent[J]. Proceedings of the Royal Society A,2010,466:1667–1686.
[37] Lukkari T. Elliptic Equations with Nonstandard Growth Involving Measures[J].Hiroshima Mathematical Journal,2008,38:155–176.
[38] Fan X. Global C1,αRegularity for Variable Exponent Elliptic Equations in Diver-gence Form[J]. Journal Differential Equations,2007,235(2):397–417.
[39] Miha ilescu M. On a Class of Nonlinear Problems Involving a p(x)-Laplace TypeOperator[J]. Czechoslovak Mathematical Journal,2008,58(133):155–172.
[40] Diening L, Lengeler D, Ru zˇicˇka M. The Stokes and Poission Problem in VariableExponent Spaces. Complex Varibles and Elliptic Equations[J]. Complex Variblesand Elliptic Equations,2011,56:789–811.
[41] Miha ilescu M, Pucci P, Ra dulescu V. Eigenvalue Problems for Anisotropic Quasi-linear Elliptic Equations with Variable Exponent[J]. The Journal of MathematicalAnalysis and Applications,2008,340:687–698.
[42] Harjulehto P, Ha¨sto¨ P, V. Le U, et al. Nuortio. Overview of Differential Equationswith Non-standard Growth[J]. Nonlinear Analysis,2010,72:4551–4574.
[43]付永强,张夏.变指数函数空间在偏微分方程上的应用[M].科学出版社,2011.
[44] Clifford W K. Applications of Grassmann’s Extensive Algebra[J]. The AmericanJournal of Mathematics,1878,1:350–358.
[45] Dirac P A M. The quantum Theory of the Electron[J]. Proceedings of the RoyalSociety A,1928,117:610–624.
[46] Dalanghe R. Clifford Analysis: History and Perspective[J]. Computational Meth-ods and Function Theory,2001,1:107–153.
[47] Brackx F, Delanghe R, Sommen F. Clifford Analysis[M]. London: Pitman Re-search Notes in Mathematics,1982.
[48] Gu¨rlebeck K, Spro¨ ig W. Quaternionic and Clifford Calculus for Physicists andEngineers[M]. Chichester: John Wiley&Sons,1997.
[49] Gu¨rlebeck K, Spro¨ ig W. Quaternionic Analysis and Elliptic Boundary[M]. Berlin:Birha¨user,1990.
[50] Gu¨rlebeck K, Spro¨ ig W. Fluid Flow Problems with Quaternionic Analysis–AnAlternative Conception[J]. Geometric Algebra Computing,2010,2010:345–380.
[51] Gu¨rlebeck K, Spro¨ ig W. An Introduction to the Mathematical Theory of theNavier-Stokes Equations[M]. New York: Springer,2011.
[52] Teman R. Navier-Stokes Equations: Theory and Numerical Analysis[M]. Amster-dam: North–Holland Publishing Company,1977.
[53] Teman R. Navier-Stokes Equations and Nonlinear Functional Analysis[M]. Soci-ety for Industrial and Applied Mathematics,1995.
[54] Spro¨ ig W. Fluid Flow Equations with Variable Viscosity in Quternionic Setting[J].Advances in Applied Clifford Algebras,2007,17:259–272.
[55] Ka¨hler U. On a Direct Decomposition in the Space Lp()[J]. Zeitschrift fu¨r Anal-ysis und ihre Anwendungen,1999,4:839–848.
[56] Ka¨hler U. Elliptic Boundary Value Problems in Bounded and Unbounded Do-main[M]. Harlow: Addison Wesley,1998.
[57] Ka¨hler U. Clifford Analysis and the Navier-Stokes Equations over UnboundedDomains[J]. Advances in Applied Clifford Algebras,2001,11:305–318.
[58] Gu¨rlebeck K, Ka¨hler U, Ryan J. Clifford Analysis over Unbounded Domain[J].Advances in Applied Mathematics,1997,19:216–239.
[59] Cerejeiras P, Ka¨hler U. Elliptic Boundary Value Problems of Fluid Dynamics overUnbounded Domains[J]. Mathemetical Methods in the Applied Sciences,2000,23:81–101.
[60] Borchers W, Miyakawa T. On Stability of Exterior Stationary Navier-StokesFlows[J]. Acta Mathematica,1995,174:311–382.
[61] Novotny A, Padula M. Note on Decay of Solutions of Steady Navier-Stokes Equa-tions in3-D Exterior Domains[J]. Differential Integral Equations,1995,8:1833–1842.
[62] Kozono H, Sohr H. New A-priori-estimates for the Stokes Equations in ExteriorDomains[J]. Indiana University Mathematics Journal,1991,40:1–27.
[63] Kozono H, Sohr H. On a New Class of Generalized of Solutions of the StokesEquations in Exterior Domains[J]. Annali della Scuola Normale Superiore di Pisa,1992,19:155–181.
[64] Varnhorn W. The Stokes Equations[M]. Berlin: Akademie–Verlag,1994.
[65] Nolder C A. A-Harmonic Equations and the Dirac Operator[J]. Journal of Inequal-ity and Applications,2010,2010:1–9.
[66] Nolder C A. Nonlinear A-Dirac Equations[J]. Advances in Applied Clifford Alge-bras,2011,21:429–440.
[67] Nolder C A, Ryan J. p-Dirac Operators[J]. Advances in Applied Clifford Algebras,2009,19:391–402.
[68] Nolder C A. Conjungate Harmonic Functions and Clifford Algebras[J]. Journal ofMathematical Analysis and Applications,2005,302:137–142.
[69] Abreu-Blaya R, Bory-Reyes J, Delanghe R, et al. Duality for Harmonic DifferentialForms Via Clifford Analysis[J]. Advances in Applied Clifford Algebras,2007,17:589–610.
[70] Brackx F, Delanghe R, Sommen F. Diffrential Forms and/or Multi-vector Functionsin Hermitean Clifford Analysis[J]. Cubo,2011,13:85–117.
[71] Gilbert J, Murray M A M. Clifford Algebra and Dirac Oprators in HarmonicAnalysis[M]. Oxford: Oxford University Press,1993.
[72] Delanghe R, Sommen F, Soucek V. Clifford Algebra and Spinor-valued Func-tion[M]. Dordrecht: Kluwer Academic Publishers,1992.
[73] Gu¨rlebeck K, Spro¨ ig W. Holomorphic Functions in the Plane and n-dimensionalSpace[M]. Boston: Birkha¨user,2008.
[74] Huang S, Qiao Y, Wen G. Real and Complex Clifford Analysis[M]. New Work:Springer,2006.
[75] Du J, Zhang Z. A Cauchy’s Integral Formula for Functions with Values in a Uni-versal Clifford Algebra and its Applications[J]. Complex Variables,2002,47:915–928.
[76] Edmunds D, Ra′kosn′k J. Sobolev Embedding with Variable Exponent[J]. StudiaMathematica,2000,143:267–293.
[77] Edmunds D, Ra′kosn′k J. Sobolev Embedding with Variable Exponent, II[J]. Math-ematische Nachrichten,2002,246:53–67.
[78] Hudzik H. The Problems of Separability, Duality, Reflexivity and of Comparisonfor Generalized Orlicz-Sobolev Spaces Wkm()[J]. Commentationes Mathematicae(Prace Matematyczne),1979,21:315–324.
[79] Doran C, Lasenby A. Geometric Algebra for Physicists[M]. Cambridge: Cam-bridge University Press,2003.
[80] Dubinskii J, Reissig M. Variational Problems in Clifford Analysis[J]. MathematicalMethods in the Applied Sciences,2002,25:1161–1176.
[81] Kravchenko V, Oviedo H. On a Quaternionic Reformulation of Maxwell’s Equa-tions[J]. Zeitschrift fu¨r Analysis and ihre Anwendungen,2003,22:569–589.
[82] Ablamowicz R. Clifford Algebras and Their Applications in Mathematical Physics,Volume1: Algebra and Physics[M]. Boston: Birkha¨user,2000.
[83] Ryan J, Spro¨ ig W. Clifford Algebras and Their Applications in MathematicalPhysics, Volume2: Clifford Analysis[M]. Boston: Birkha¨user,2000.
[84] Bahmann M, Gu¨rlebeck K, Shapiro M. On a Modified Teodorescu Transform[J].Integral Transforms and Special Functions,2001,12(3):213–226.
[85] Zhang Z. Some Properties of Operators in Clifford Analysis[J]. Complex Variablesand Elliptic Equations,2007,52(6):455–473.
[86] Begehr H, Zhang Z, Ha V. Generalized Integral Representations in Clifford Anal-ysis[J]. Complex Variables and Elliptic Equations,2006,51:745–762.
[87] Spro¨ ig W. On Decompositions of Sobolev Spaces in Clifford Analysis[J]. Ad-vances in Applied Clifford Algebras,1995,5(2):167–185.
[88] Begehr H, Dubinskii J. Orthogonal Decompositions of Sobolev Spaces in CliffordAnalysis[J]. Annali di Matematica Pura ed Applicata,2002,181:55–71.
[89] Dubinski J A. On a Nonlinear Analytic Problem[J]. Doklady Mathematics,1998,48:370–375.
[90] Dubinskii J, Reissig M. Variational problems in Clifford analysis[J]. MathematicalMethods in the Applied Sciences,2002,25:1161–1176.
[91] Heinonen J, Kilpela¨inen T, Martio O. Nonlinear Porential Theory of DegenerateElliptic Equations[M]. Oxford: Oxford University Press,1993.
[92] Eleuteri M, Habermann J. Regularity Results for a Class of Obstacle Problemsunder Nonstandard Growth Conditions[J]. Journal of Mathematical Analysis andApplications,2008,344:1120–1142.
[93] Rodrigues J F, Sanchon M, Urbano J M. The Obstacle Problem for Nonlinear Ellip-tic Equations with Variable Growth and L1-data[J]. Monatshefte fur Mathematik,2008,154:303–322.
[94] Harjulehto P, Ha¨sto¨ P, Koskenoja M, et al. Obstacle Problem and SuperharmonicFunctions with Nonstandard Growth. Nonlinear Analysis[J]. Nonlinear Analysis,2007,67:3424–3440.
[95] Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities andTheir Applications[M]. New York: Academic Press,1980.
[96] Brezis H.泛函分析–理论和应用[M].清华大学出版社,2009.
[97] Ekeland I, Temam R. Convex Analysis and Variational Problems[M]. North-Holland: Amsterdam,1976.
[98] Eisen G. A Selection Lemma for Sequences of Measure Sets and Lower Semicon-tinuity of Multiple Integrals[J]. Mathematical Manuscripts,1979,27:73–79.
[99] Acerbi E, Fusco N. Semicontinuity Problems in the Calculus of Variations[J].Archive for Rational Mechanics and Analysis,1984,86:125–135.
[100] Liu F C. A Luzin Type Property of Sobolev Functions[J]. Indiana UniversityMathematics Journal,1977,26:645–651.
[101] Morrey C B. Multiple Integrals in the Calculus of Variations[M]. New Work:Springer,1966.
[102] Dacorogna B. Weak Continuity and Weak Lower Semicontinuity of NonlinearFunctionals[M]. New Work: Springer,1982.
[103] Tartar L. An Introduction to Sobolev Spaces and Interpolation Spaces[M]. Berlin:Springer-Verlag,2007.
[104] Yosida K. Functional Analysis[M]. Berlin: Springer-Verlag,1980.
[105] Amrouche C, Girault V. Decomposition of Vector Space and Application to theStokes Problem in Arbitrary Dimensional[J]. Czechoslovak Mathematicl Journal,1994,44:109–140.
[106] Finn R. On the Exterior Stationary Problem for the Navier-Stokes Equations andAssociated Pertubation Problems[J]. Archive for Rational Mechanics and Analysis,1965,19:363–406.
[107] Kim H. Existence and Regularity of Very Weak Solutions of the StationaryNavier-Stokes Equations[J]. Archive for Rational Mechanics and Analysis,2009,193:117–152.
[108] Galdi G P, Silvestre A L. The Steady Motion of a Navier-Stokes Liquid Around aRigid Body[J]. Archive for Rational Mechanics and Analysis,2007,184:371–400.
[109] Galdi G P, Kyed M. Steady-state Navier-Stokes Flows Past a Rotation Body: Ler-ay Solutions Are Physically Reasonable[J]. Archive for Rational Mechanics andAnalysis,2011,200:21–58.
[110] Wang G, Yang D. Decomposition of Vector-valued Divergence Free Sobolev Func-tions and Shape Optimization for Stationary Navier-Stokes Equations[J]. Commu-nications in Partial Differential Equations,2008,33:429–449.
[111] Ru zˇicˇka M. A Note on Steady Flow of Fluids with Shear Dependent Viscosity[J].Nonlinear Analysis,1997,30(5):3029–3039.
[112] Huber H. The Divergence Equation in Weighted-and Lp(·)-Spaces[J]. Mathematis-che Zeitschrift,2011,267:341–366.
[113] Arada N. A Note on the Regularity of Flows with Shear-dependent Viscosity[J].Nonlinear Analysis,2012,75(14):5401–5415.
[114] Diening L, Ru zˇicˇka M. Strong Solutions for Generalized Newtonian Fluids[J].Journal of Mathematica Fluid Mechanics,2005,7:413–450.
[115] Berselli L C, Diening L, Ru zˇicˇka M. Existence of Strong Solutions for Incom-pressible Fluid with Shear Dependent Viscosities[J]. Journal of Mathematica FluidMechanics,2010,12:101–132.
[116] Gu¨rlebeck K. Approximate Solution of the Stationary Navier-Stokes Equations[J].Mathematische Nachrichten,1990,145:297–308.
[117] Cerejeiras P, Ka¨hler U, Sommen F. Parabolic Dirac Operators and Navier-StokesEquations over Time-varying Domains[J]. Mathemetical Methods in the AppliedSciences,2005,28:1715–1724.
[118] Gu¨rlebeck K, Spro¨ ig W. Representation Theory for Classes of Initial Value prob-lems with Quaternionic Analysis[J]. Mathemetical Methods in the Applied Sci-ences,2002,25:1371–1382.
[119] Spro¨ ig W. Quaternionic Operator Methods in Fluid Dynamics[J]. Advances inApplied Clifford Algebras,2008,18:963–978.
[120] Spro¨ ig W, Gu¨rlebeck K. On the Treatment of Fluid Problems by Methods ofClifford Analysis[J]. Advances in Applied Clifford Algebras,1997,44:401–413.