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

TL;DR

This paper characterizes the Dirichlet-to-Neumann map for Riemannian surfaces with boundary, providing necessary and sufficient conditions based on Banach algebra connections, advancing inverse boundary value problem understanding.

Contribution

It introduces a new characterization of the DN-map for surfaces using Banach algebra methods, differing from previous complex analysis approaches.

Findings

Provides necessary and sufficient conditions for a boundary operator to be a DN-map.

Connects inverse problems on surfaces with commutative Banach algebra theory.

Offers a new perspective beyond classical complex analysis methods.

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_gu=0 \,\,\, {\rm in}\,\,M\setminus\Gamma,\,\,u=f \,\,\, {\rm on}\,\,\,\Gamma$, and acts by $ \Lambda_g f:=\partial_\nu u^f \,\,\, {\rm 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.