Isotropic realizability of electric fields around critical points

Abstract

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.