# American Institute of Mathematical Sciences

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

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.
Citation: Marc Briane. Isotropic realizability of electric fields around critical points. Discrete and 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), 229-256. [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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##### 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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