The Poisson embedding approach to the Calder\'on problem

TL;DR

This paper introduces a novel Poisson embedding method for the anisotropic Calderón problem, providing new uniqueness results and a different proof approach for inverse problems on real analytic Riemannian manifolds.

Contribution

It develops the Poisson embedding technique to identify manifold points via boundary distributions, offering new uniqueness theorems and a simplified proof for the Calderón problem.

Findings

New Poisson embedding approach for inverse problems

Uniqueness results for quasilinear Calderón problems

Alternative proof of Calderón problem on real analytic manifolds

Abstract

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.