Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in ${\mathbb{R}}^3$

Abstract

Let $X$ be a smooth bordered surface in ${\Re }^3$ with smooth boundary and $\hat{\sigma }$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator ${\Lambda }_{\hat{\sigma }}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F:X\to Y$ which transforms $\hat{\sigma }$ into $\sigma$. As a corollary we obtain the following uniqueness result: if ${\sigma }_1,{\sigma }_2$ are two smooth anisotropic conductivities on $X$ with ${\Lambda }_{{\sigma }_1}={\Lambda }_{{\sigma }_2}$, then there exists a smooth diffeomorphism $\Phi :\bar{X}\to \bar{X}$ which transforms ${\sigma }_1$ into ${\sigma }_2$.

21 pages, no figures; added corrections in Theorem 5.1

References in corpus (1)

• On an inverse problem for anisotropic conductivity in the plane