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

TL;DR

This work addresses recovering anisotropic conductivity $\gamma$ from the Neumann-to-Dirichlet map for the stationary semilinear elliptic equation $-\nabla\cdot(\gamma\nabla u)+\alpha u^3=0$ in a bounded domain $\Omega$, modeling pacing-guided cardiac tissue. The authors employ a first-order linearization around a nontrivial boundary input $g_0$ to link the nonlinear NtD map $\mathcal{N}_{NL}^{\gamma,\alpha}$ to a linear NtD map for the operator $L=-\nabla\cdot(\gamma\nabla\cdot)+3\alpha u_0^2$, where $u_0=u(\cdot,g_0)$. Under the assumption that $D$ is known and $\alpha$ is known, and with nonflat portions on the boundaries, they prove that the nonlinear NtD data uniquely determine $\gamma$ across $\Omega$, by combining boundary determination, Alessandrini-type identities, and unique continuation to recover $\gamma$ both outside and inside $D$. This constitutes the first uniqueness result for anisotropic conductivities from NtD data in a nonlinear setting, with implications for interpreting pacing-induced boundary measurements in EIT-based cardiac applications; future work includes stability analysis, reconstruction algorithms, and potential simultaneous recovery of $\gamma$ and $\alpha$.

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 \(Ω\subset\mathbb{R}^3\) of cardiac tissue satisfies \[ -\nabla\!\cdot(γ\nabla u)+αu^3=0 \quad \text{in }Ω,\qquad γ\nabla u\cdotν=g \quad \text{on }\partialΩ, \] where $γ$ is an anisotropic conductivity tensor and $α$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partialΩ}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subsetΩ$ where $γ$ is piecewise constant. We address the inverse problem of determining $γ$ from the Neumann-to-Dirichlet (NtD) map, assuming that $α$ 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 $γ$.

Theorems & Definitions (36)

• Definition 2.1
• Proposition 2.1
• proof
• Definition 2.2: Neumann-to-Dirichlet map
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Remark 2.5