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Title: Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity (English)
Author: Wellander, Niklas
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940
Volume: 47
Issue: 3
Year: 2002
Pages: 255-283
Summary lang: English
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Category: math
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Summary: The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations. (English)
Keyword: nonlinear PDEs
Keyword: Maxwell’s equations
Keyword: nonlinear conductivity
Keyword: homogenization
Keyword: existence of solution
Keyword: unique solution
Keyword: two-scale convergence
Keyword: corrector results
Keyword: heterogeneous materials
Keyword: compactness result
Keyword: non-periodic medium
MSC: 35B27
MSC: 35Q60
MSC: 74Q10
MSC: 74Q15
MSC: 78A25
idZBL: Zbl 1090.35504
idMR: MR1900514
DOI: 10.1023/A:1021797505024
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Date available: 2009-09-22T18:10:11Z
Last updated: 2015-05-19
Stable URL: http://hdl.handle.net/10338.dmlcz/133894
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Related article: http://dml.cz/handle/10338.dmlcz/133893
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Reference: [1] G.  Allaire: Homogenization and two-scale convergence.SIAM J.  Math. Anal. 23 (1992), 1482–1518. Zbl 0770.35005, MR 1185639, 10.1137/0523084
Reference: [2] M. Artola, M. Cessenat: Diffraction d’une onde électromagnetique par un obstacle borné à permittivité et perméabilité élevées.C. R.  Acad. Sci. Paris, Sér.  I  Math. 314 (1992), 349–354. MR 1153713
Reference: [3] A.  Bensoussan, J. L. Lions and G.  Papanicolaou: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications.North-Holland Publishing Company, Amsterdam-New York-Oxford, 1978. MR 0503330
Reference: [4] B. Birnir, N. Wellander: Homogenized Maxwell’s equations; a model for ceramic varistors.Submitted.
Reference: [5] E. Coddington, N. Levinson: Theory of Ordinary Differential Equations.McGraw-Hill, New York, 1955. MR 0069338
Reference: [6] G. Duvaut, J. L. Lions: Inequalities in Mechanics and Physics.Springer Verlag, Berlin-Heidelberg-New York, 1976. MR 0521262
Reference: [7] L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, 1992. MR 1158660
Reference: [8] A. Holmbom: The concept of parabolic two-scale convergence, a new compactness result and its application to homogenization of evolution partial differential equations.Research report 1994-18, Mid-Sweden University Östersund.
Reference: [9] A. Holmbom: Some modes of convergence and their application to homogenization and optimal composites design.Ph.D. thesis, Luleå University of Technology, 1996.
Reference: [10] P. A. Markowich, F.  Poupaud: The Maxwell equation in a periodic medium: Homogenization of the energy density.Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 301–324. MR 1433425
Reference: [11] C.-W. Nan, D. R. Clarke: Effect of variations in grain size and grain boundary barrier heights on the current-voltage characteristics of ZnO varistors.J. Am. Ceram. Soc. 79 (1996), 3185–3192. 10.1111/j.1151-2916.1996.tb08094.x
Reference: [12] A. Negro: Some problems of homogenization in quasistationary Maxwell equations.In: Applications of Multiple Scaling in Mechanics, Proc. Int. Conf., Ecole Normale Superieure, Paris 1986, Rech. Math. Appl. 4, Masson, Paris, 1987, pp. 246–258. Zbl 0644.73077, MR 0901998
Reference: [13] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization.SIAM J.  Math. Anal. 20 (1989), 608–623. Zbl 0688.35007, MR 0990867, 10.1137/0520043
Reference: [14] E.  Sanchez-Palencia: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127.Springer-Verlag, Berlin-Heidelberg-New-York, 1980. MR 0578345
Reference: [15] A. Vojta, Q. Wen and D. R.  Clarke: Influence of microstructural disorder on the current transport behavior of varistor ceramics.Comput. Mater. Sci. 6 (1996), 51–62. 10.1016/0927-0256(96)00011-0
Reference: [16] A. Vojta, D. R. Clarke: Microstructural origin of current localization and “puncture” failure in varistor ceramics.J. Appl. Phys. 81 (1997), 1–9.
Reference: [17] N. Wellander: Homogenization of the Maxwell equations: Case  I. Linear theory.Appl. Math. 46 (2001), 29–51. Zbl 1058.78004, MR 1808428, 10.1023/A:1013727504393
Reference: [18] N. Wellander: Homogenization of some linear and nonlinear partial differential equations.Ph.D.  thesis, Luleå University of Technology, 1998.
Reference: [19] E.  Zeidler: Nonlinear Functional Analysis and its Applications, Volumes IIA and IIB.Springer-Verlag, Berlin, 1990.
Reference: [20] V. V. Zhikov, S. M. Kozlov and O. A.  Oleinik: Homogenization of Differential Operators and Integral Functionals.Springer-Verlag, Leyden, 1994. MR 1329546
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