非线性离散Calderón问题唯一性研究
On the Uniqueness of a Nonlinear Discrete Calderón Problem
- Division of Science, New York University Abu Dhabi（纽约大学阿布扎比分校科学部）
- Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究方格上半线性椭圆方程的离散Calderón问题，通过角激发逐层重建、线性化DtN映射及有限非线性测量，建立了电导率恢复的三个唯一性结果。
AI中文摘要：
离散Calderón问题旨在从边界测量中恢复图边上的电导率，边界测量由离散Dirichlet-to-Neumann（DtN）映射编码。自Curtis和Morrow的开创性工作以来，该问题在方格上的线性情形已被深入研究。本文研究了方格上半线性二阶椭圆方程的离散Calderón问题的非线性类比。我们建立了电导率恢复的三个唯一性结果。首先，我们证明依赖于电导率的角激发允许逐层重建电导率。其次，我们研究非线性DtN映射在任意背景边界数据处的线性化，并证明电导率和背景势由一对非线性Cauchy数据和线性化DtN映射唯一确定。第三，我们表明线性化数据可以被有限多个非线性测量所替代，这些测量唯一确定电导率。
英文摘要：
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.
