Anisotropic Conductivities

In anisotropic tissue such as muscle, or in layered materials, the conductivity is a symmetric positive definite matrix field $\gamma (x)\in {\mathbb{R}}^{n\times n}$, and the equation is $\nabla \cdot (\gamma \nabla u)=0$.

Non-uniqueness. Let $\Phi :\bar{\Omega }\to \bar{\Omega }$ be a diffeomorphism with $\Phi {|}_{\partial \Omega }=\mathrm{id}$. The push-forward

$${\Phi }_{\ast }\gamma =\left(\frac{D\Phi \gamma D{\Phi }^{\top }}{|\det D\Phi |}\right)\circ {\Phi }^{-1}$$

has the same DtN map, ${\Lambda }_{{\Phi }_{\ast }\gamma }={\Lambda }_{\gamma }$. This change of variables in the Dirichlet energy was pointed out by Tartar (see Kohn–Vogelius 1984). Anisotropic conductivities can therefore only be recovered up to such diffeomorphisms. This is proven in 2D (Sylvester 1990; Astala, Lassas, Päivärinta 2005), and for real-analytic conductivities in $n\ge 3$ (Lee–Uhlmann 1989).

Relevance.

• The relaxed Kohn-Vogelius Functional produces anisotropic limits, which is one reason relaxation did not lead to a practical algorithm.
• The push-forward formula explains the symmetries of the forward map under rigid motions. In 2D, where conformal maps have $D\Phi D{\Phi }^{\top }=|\det D\Phi |I$, isotropic conductivities stay isotropic.

References

1. R. V. Kohn, M. Vogelius (1984). Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37(3), 289–298. doi:10.1002/cpa.3160370302
2. J. M. Lee, G. Uhlmann (1989). Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 42(8), 1097–1112. doi:10.1002/cpa.3160420804
3. J. Sylvester (1990). An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. 43(2), 201–232. doi:10.1002/cpa.3160430203
4. K. Astala, M. Lassas, L. Päivärinta (2005). Calderón’s Inverse Problem for Anisotropic Conductivity in the Plane. Comm. PDE 30(1–2), 207–224. doi:10.1081/PDE-200044485