On characterization of Dirichlet-to-Neumann map of Riemannian surface with boundary

Abstract

Let $(M,g)$ be a smooth compact orientable two-dimensional Riemannian manifold ({\it surface}) with a smooth metric tensor $g$ and smooth connected boundary $\Gamma$. Its {\it DN-map} ${\Lambda }_g:C^{\infty }(\Gamma )\to C^{\infty }(\Gamma )$ is associated with the (forward) elliptic problem ${\Delta }_{g}u=0\mathrm{in}M\setminus \Gamma ,u=f\mathrm{on}\Gamma$, and acts by ${\Lambda }_{g}f:={\partial }_{\nu }{u}^f\mathrm{on}\Gamma ,$ where ${\Delta }_g$ is the Beltrami-Laplace operator, $u=u^f(x)$ is the solution, $\nu$ is the outward normal to $\Gamma$. The corresponding {\it inverse problem} is to determine the surface $(M,g)$ from its DN-map ${\Lambda }_g$. We provide the necessary and sufficient conditions on an operator acting in $C^{\infty }(\Gamma )$ to be the DN-map of a surface. In contrast to the known conditions by G.Henkin and V.Michel in terms of multidimensional complex analysis, our ones are based on the connections of the inverse problem with commutative Banach algebras.

23 pages, 0 figures