Conformal mapping for cavity inverse problem: an explicit reconstruction formula

In this paper, we address a classical case of the Calder'on (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity $\omega$ (with boundary $\gamma$) contained in a domain $\mathrm{\Omega }$ (with boundary $\mathrm{\Gamma }$) from the knowledge of the Dirichlet-to-Neumann (DtN) map $\Lambda_\gamma: f \longmapsto \partial_n u^f|{\Gamma}$,where$u^f$isharmonicin$\Omega\setminus\overline{\omega}$,$u^f|{\Gamma}=f$and$u^f|{\gamma}=c^f$,$c^f$beingtheconstantsuchthat$\int{\gamma}\partial_n u^f,{\rm d}s=0$. We obtain an explicit formula for the complex coefficients $a_m$ arising in the expression of the Riemann map $z⟼a_{1}z+a_0+{\sum }_{m⩽-1}{a}_m{z}^m$ that conformally maps the exterior of the unit disk onto the exterior of $\omega$. This formula is derived by using two ingredients: a new factorization result of the DtN map and the so-called generalized P'olia-Szeg"o tensors (GPST) of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients $a_m$ with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method.