Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Beretta, Elena Francini, Elisa Pierotti, Dario Sincich, Eva |
| author_facet | Beretta, Elena Francini, Elisa Pierotti, Dario Sincich, Eva |
| contents | 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 $γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11987 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation Beretta, Elena Francini, Elisa Pierotti, Dario Sincich, Eva Analysis of PDEs 35R30, 35J25, 35J47 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 $γ$. |
| title | Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation |
| topic | Analysis of PDEs 35R30, 35J25, 35J47 |
| url | https://arxiv.org/abs/2602.11987 |