Isotropic realizability of electric fields around critical points

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March 2014, 19(2): 353-372. doi: 10.3934/dcdsb.2014.19.353

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

Received June 2013 Revised September 2013 Published February 2014

Abstract
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In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $u$ is solution to the equation div$\left(\sigma\nabla u\right)=0$ for some isotropic conductivity $\sigma>0$. The case of a smooth function $u$ without critical point was investigated in [7] thanks to a dynamical system approach which yields a global isotropic realizability result in $\mathbb{R}^d$. The presence of a critical point $x^*$ needs a specific treatment according to the behavior of the gradient flow in the neighborhood of $x^*$. The case where the hessian matrix $\nabla^2 u(x^*)$ is invertible with both positive and negative eigenvalues is the most favorable: the anisotropic realizability is a consequence of Morse's lemma, while the Hadamard-Perron theorem leads us to a characterization of the isotropic realizability around $x^*$ through some boundedness condition involving the laplacian of $u$ along the gradient flow. When the matrix $\nabla^2 u(x^*)$ has $d$ positive eigenvalues or $d$ negative eigenvalues, we get a strong maximum principle under the same boundedness condition. However, when the matrix $\nabla^2 u(x^*)$ is not invertible, the derivation of the isotropic realizability is much more intricate: the Hartman-Wintner theorem gives necessary conditions for the isotropic realizability in dimension two, while the dynamical system approach provides a criterion of non realizability in any dimension. The two methods are illustrated by a two-dimensional and a three-dimensional example.

Keywords: critical point., gradient system, isotropic conductivity, Electric field.

Mathematics Subject Classification: 35B27, 78A30, 37C1.

Citation: Marc Briane. Isotropic realizability of electric fields around critical points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353

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.
[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. doi: 10.1137/S0036141093249080.
[3]	G. Alessandrini & V. Nesi, Univalent $\sigma$-harmonic mappings,, Arch. Rational Mech. Anal., 158* (2001), 155. 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, 1 (1988). doi: 10.1007/978-3-642-61551-1.
[5]	V. I. Arnold, Ordinary Differential Equations,, translated from the third Russian edition by R. Cooke, (1992).
[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. 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., (1301).
[8]	D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of the 1998 edition, (1998).
[9]	P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations (I),, Amer. J. Math., 75 (1953), 449. doi: 10.2307/2372496.
[10]	M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos,, second edition, 60 (2004).
[11]	J. Milnor, Morse Theory,, based on lecture notes by M. Spivak and R. Wells, 51 (1963).
[12]	F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions,, Lecture Notes in Mathematics 1445, 1445 (1990).

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.
[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. doi: 10.1137/S0036141093249080.
[3]	G. Alessandrini & V. Nesi, Univalent $\sigma$-harmonic mappings,, Arch. Rational Mech. Anal., 158* (2001), 155. 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, 1 (1988). doi: 10.1007/978-3-642-61551-1.
[5]	V. I. Arnold, Ordinary Differential Equations,, translated from the third Russian edition by R. Cooke, (1992).
[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. 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., (1301).
[8]	D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of the 1998 edition, (1998).
[9]	P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations (I),, Amer. J. Math., 75 (1953), 449. doi: 10.2307/2372496.
[10]	M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos,, second edition, 60 (2004).
[11]	J. Milnor, Morse Theory,, based on lecture notes by M. Spivak and R. Wells, 51 (1963).
[12]	F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions,, Lecture Notes in Mathematics 1445, 1445 (1990).

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