The Calder\'on problem for quasilinear elliptic equations

TL;DR

This paper proves the uniqueness of the conductivity in a quasilinear Calderón inverse problem, where the conductivity depends nonlinearly on the potential and its gradient, using complex analysis and linearization techniques.

Contribution

It establishes the first uniqueness result for a quasilinear Calderón problem with nonlinear dependence on the potential and its gradient.

Findings

Uniqueness of the conductivity under certain structural assumptions.

Use of complex-valued test functions and linearization of the DN map.

Applicability to conductivities with small analytic continuation.

Abstract

In this paper we show uniqueness of the conductivity for the quasilinear Calder\'on's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induces a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.