Inverse problem by Cauchy data on arbitrary subboundary for system of elliptic equations

TL;DR

This paper addresses an inverse problem for elliptic systems, showing that identical Cauchy data on any subboundary implies the coefficient matrices satisfy certain PDEs, leading to uniqueness results.

Contribution

It establishes the uniqueness of coefficient matrices in elliptic systems based on partial Cauchy data on arbitrary subboundaries, under certain conditions.

Findings

Same Cauchy data implies coefficient matrices satisfy PDEs

Uniqueness of coefficient matrices given partial boundary data

Results applicable to arbitrary subboundary data

Abstract

We consider an inverse problem of determining coefficient matrices in an $N$-system of second-order elliptic equations in a bounded two dimensional domain by a set of Cauchy data on arbitrary subboundary. The main result of the article is as follows: If two systems of elliptic operators generate the same set of partial Cauchy data on an arbitrary subboundary, then the coefficient matrices of the first-order and zero-order terms satisfy the prescribed system of first-order partial differential equations. The main result implies the uniqueness of any two coefficient matrices provided that the one remaining matrix among the three coefficient matrices is known.