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Isotropic realizability of electric fields around critical points
1. | Institut de Recherche Mathématique de Rennes & INSA de Rennes, 20 avenue des Buttes de Cöesmes, CS 70839, 35708 Rennes Cedex 7, France |
References:
[1] |
G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 229-256. |
[2] |
G. Alessandrini & R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.
doi: 10.1137/S0036141093249080. |
[3] |
G. Alessandrini & V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Rational Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[4] |
D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtejn and V. Z. Grines, Dynamical Systems. I, translated from the Russian, D. V. Anosov and V. I. Arnold (eds), Encyclopaedia Math. Sci. 1, Springer-Verlag, Berlin, 1988, pp. 233.
doi: 10.1007/978-3-642-61551-1. |
[5] |
V. I. Arnold, Ordinary Differential Equations, translated from the third Russian edition by R. Cooke, Springer Textbook, Springer-Verlag, Berlin, 1992, pp. 334. |
[6] |
P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization, Indiana Univ. Math. J., 50 (2001), 747-757.
doi: 10.1512/iumj.2001.50.1832. |
[7] |
M. Briane, G. W. Milton & A. Treibergs, Which electric fields are realizable in conducting materials? to appear in Math. Mod. Num. Anal., arXiv: 1301.1613. |
[8] |
D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001, pp. 517. |
[9] |
P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations (I), Amer. J. Math., 75 (1953), 449-476.
doi: 10.2307/2372496. |
[10] |
M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, second edition, Pure and Applied Mathematics (Amsterdam) 60, Elsevier/Academic Press, Amsterdam, 2004, pp. 417. |
[11] |
J. Milnor, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies 51, Princeton University Press, Princeton, New Jersey, 1963, pp. 153. |
[12] |
F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions, Lecture Notes in Mathematics 1445, Springer-Verlag, Berlin, 1990, pp. 123. |
show all references
References:
[1] |
G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 229-256. |
[2] |
G. Alessandrini & R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.
doi: 10.1137/S0036141093249080. |
[3] |
G. Alessandrini & V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Rational Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[4] |
D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtejn and V. Z. Grines, Dynamical Systems. I, translated from the Russian, D. V. Anosov and V. I. Arnold (eds), Encyclopaedia Math. Sci. 1, Springer-Verlag, Berlin, 1988, pp. 233.
doi: 10.1007/978-3-642-61551-1. |
[5] |
V. I. Arnold, Ordinary Differential Equations, translated from the third Russian edition by R. Cooke, Springer Textbook, Springer-Verlag, Berlin, 1992, pp. 334. |
[6] |
P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization, Indiana Univ. Math. J., 50 (2001), 747-757.
doi: 10.1512/iumj.2001.50.1832. |
[7] |
M. Briane, G. W. Milton & A. Treibergs, Which electric fields are realizable in conducting materials? to appear in Math. Mod. Num. Anal., arXiv: 1301.1613. |
[8] |
D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001, pp. 517. |
[9] |
P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations (I), Amer. J. Math., 75 (1953), 449-476.
doi: 10.2307/2372496. |
[10] |
M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, second edition, Pure and Applied Mathematics (Amsterdam) 60, Elsevier/Academic Press, Amsterdam, 2004, pp. 417. |
[11] |
J. Milnor, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies 51, Princeton University Press, Princeton, New Jersey, 1963, pp. 153. |
[12] |
F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions, Lecture Notes in Mathematics 1445, Springer-Verlag, Berlin, 1990, pp. 123. |
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