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On a p-curl system arising in electromagnetism
1. | Department of Mathematics/CMAT, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal, Portugal |
2. | CMAF/Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa |
References:
[1] |
C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields, C. R. Math. Acad. Sci. Paris, 349 (2011), 529-534. |
[2] |
A. Bermúdez, R. Muñoz-Sola and F. Pena, A nonlinear partial differential system arising in thermoelectricity, European J. Appl. Math., 16 (2005), 683-712. |
[3] |
A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,'' Electromagnetism, Academic Press, Inc., San Diego, CA, 1998. |
[4] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 3, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. |
[5] |
A. Haraux, "Nonlinear Evolution Equations--Global Behavior of Solutions,'' Lecture Notes in Mathematics, 841, Springer-Verlag, Berlin-New York, 1981. |
[6] |
L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,'' Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
[7] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969. |
[8] |
F. Miranda, J.-F. Rodrigues and L. Santos, A class of stationary nonlinear Maxwell systems, Math. Models Methods Appl. Sci., 19 (2009), 1883-1905.
doi: 10.1142/S0218202509003966. |
[9] |
D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997), 131-192.
doi: 10.1007/BF02649132. |
[10] |
M. Mitrea, Boundary value problems for Dirac operators and Maxwell's equations in non-smooth domains, Math. Methods Appl. Sci., 25 (2002), 1355-1369.
doi: 10.1002/mma.375. |
[11] |
L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math., 7 (1996), 237-247. |
[12] |
L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition, Port. Math., 48 (1991), 441-468. |
[13] |
L. Santos, Variational problems with non-constant gradient constraints, Port. Math. (N.S.), 59 (2002), 205-248. |
[14] |
C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992. |
[15] |
J. Simon, Quelques propriétés de solutions d'équations et d'inéquations d'évolution paraboliques non linéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 585-609. |
[16] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. |
[17] |
S. Sobolev, "Applications of Functional Analysis in Mathematical Physics,'' Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. |
[18] |
W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143.
doi: 10.1002/mma.1670150206. |
[19] |
H.-M. Yin, On a nonlinear Maxwell's system in quasi-stationary electromagnetic fields, Math. Models Methods Appl. Sci., 14 (2004), 1521-1539.
doi: 10.1142/S0218202504003787. |
[20] |
H.-M. Yin, B. Li, and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Contin. Dyn. Syst., 8 (2002), 781-794.
doi: 10.3934/dcds.2002.8.781. |
[21] |
S. Zheng, "Nonlinear Evolution Equations,'' Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
show all references
References:
[1] |
C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields, C. R. Math. Acad. Sci. Paris, 349 (2011), 529-534. |
[2] |
A. Bermúdez, R. Muñoz-Sola and F. Pena, A nonlinear partial differential system arising in thermoelectricity, European J. Appl. Math., 16 (2005), 683-712. |
[3] |
A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,'' Electromagnetism, Academic Press, Inc., San Diego, CA, 1998. |
[4] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 3, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. |
[5] |
A. Haraux, "Nonlinear Evolution Equations--Global Behavior of Solutions,'' Lecture Notes in Mathematics, 841, Springer-Verlag, Berlin-New York, 1981. |
[6] |
L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,'' Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
[7] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969. |
[8] |
F. Miranda, J.-F. Rodrigues and L. Santos, A class of stationary nonlinear Maxwell systems, Math. Models Methods Appl. Sci., 19 (2009), 1883-1905.
doi: 10.1142/S0218202509003966. |
[9] |
D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997), 131-192.
doi: 10.1007/BF02649132. |
[10] |
M. Mitrea, Boundary value problems for Dirac operators and Maxwell's equations in non-smooth domains, Math. Methods Appl. Sci., 25 (2002), 1355-1369.
doi: 10.1002/mma.375. |
[11] |
L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math., 7 (1996), 237-247. |
[12] |
L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition, Port. Math., 48 (1991), 441-468. |
[13] |
L. Santos, Variational problems with non-constant gradient constraints, Port. Math. (N.S.), 59 (2002), 205-248. |
[14] |
C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992. |
[15] |
J. Simon, Quelques propriétés de solutions d'équations et d'inéquations d'évolution paraboliques non linéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 585-609. |
[16] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. |
[17] |
S. Sobolev, "Applications of Functional Analysis in Mathematical Physics,'' Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. |
[18] |
W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143.
doi: 10.1002/mma.1670150206. |
[19] |
H.-M. Yin, On a nonlinear Maxwell's system in quasi-stationary electromagnetic fields, Math. Models Methods Appl. Sci., 14 (2004), 1521-1539.
doi: 10.1142/S0218202504003787. |
[20] |
H.-M. Yin, B. Li, and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Contin. Dyn. Syst., 8 (2002), 781-794.
doi: 10.3934/dcds.2002.8.781. |
[21] |
S. Zheng, "Nonlinear Evolution Equations,'' Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
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