Numerical solution of the two-dimensional Calderon problem for domains close to a disk

For a compact Riemannian surface $(M,g)$ with non-empty boundary $Γ$, the Dirichlet-to-Neumann operator (DtN-map) $Λ_g:C^\infty(Γ)\to C^\infty(Γ)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_Γ$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_Γ=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $Λ_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}={(x,y)\mid x^2+y^2\le1}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain.

💡 Research Summary

The paper addresses the inverse Calderón problem in two dimensions: given the Dirichlet‑to‑Neumann (DtN) operator on the boundary of an unknown planar domain, recover the shape of the domain. After recalling the classical uniqueness result (Theorem 1.1), which states that a compact Riemannian surface is determined by its DtN map up to a boundary‑fixing diffeomorphism and a conformal factor equal to one on the boundary, the authors restrict attention to simply‑connected surfaces. By solving a scalar curvature equation they produce a conformal factor that flattens the metric, showing that any such surface can be isometrically immersed into the Euclidean plane. Consequently the Calderón problem reduces to recovering a simply‑connected planar domain Ω with smooth simple boundary Γ from its DtN map Λ_Ω.

The boundary is parametrised by arc‑length s∈