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arXiv · 2609.15175

On the Uniqueness of a Nonlinear Discrete Calderón Problem

Elena Beretta·Maolin Deng·Alberto Gandolfi·Bangti Jin
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Abstract

The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.

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Elena Beretta, Maolin Deng, Alberto Gandolfi, Bangti Jin. 2026-09-14. On the Uniqueness of a Nonlinear Discrete Calderón Problem. https://arxiv.org/abs/2609.15175

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