Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

Abstract

We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region \(\Omega\subset\mathbb{R}^3\) of cardiac tissue satisfies \[ -\nabla\!\cdot(\gamma\nabla u)+\alpha u^3=0 \quad \text{in }\Omega,\qquad \gamma\nabla u\cdot\nu=g \quad \text{on }\partial\Omega, \] where $\gamma$ is an anisotropic conductivity tensor and $\alpha$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partial\Omega}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subset\Omega$ where $\gamma$ is piecewise constant. We address the inverse problem of determining $\gamma$ from the Neumann-to-Dirichlet (NtD) map, assuming that $\alpha$ and $D$ are known. To our knowledge, uniqueness in the case of NtD data with anisotropic conductivities in this nonlinear setting has not been analyzed in previous work. Using a first-order linearization around a nontrivial pacing current, we prove uniqueness for $\gamma$.