Numerical solution of the two-dimensional Calderón problem based on the Hilbert transform of a planar domain

Abstract

Let $(M,g)$ be a $C^{\infty}$ smooth compact connected Riemannian manifold with boundary $\partial M$. Consider the Dirichlet problem: $Δ_gu=0,\ u|_{\partial M}=f$ for $f\in C^{\infty}(\partial M)$. The Dirichlet-to-Neumann (DN) operator $Λ_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_{\partial M}$, where $ν$ is the unit outer normal to the boundary and $u$ is the unique solution to the Dirichlet problem. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderón problem is as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,Λ_g)$? In the two-dimensional case the surface $(M,g)$ is determined by the DN data uniquely up to conformal equivalence. Knowledge of the DN data is equivalent to knowledge of the Hilbert transform ${\mathcal H}_Ω:C^\infty(Γ)\to C^\infty(Γ)$ on the boundary curve $Γ=\partialΩ$ of a planar domain $Ω$. We study properties of the Hilbert transform. In particular, we obtain an integral formula for ${\mathcal H}_Ω$ for a simply connected $Ω$ which generalizes the classical integral formula for the Hilbert transform on the unit circle. This formula is the base of our algorithm for reconstructing a simply connected planar domain from the DN data. Several numerical reconstructions are presented.