Boundary determination of electromagnetic parameters from local data

TL;DR

The paper addresses the inverse boundary problem for Maxwell equations with an embedded obstacle, aiming to determine the electromagnetic parameters $\mu$ and $\gamma$ at the boundary from local data. It extends and simplifies previous work by using only the local admittance map on a subset $\Gamma$ and local Cauchy data of the fundamental solution, without requiring full data or inversion of $\Lambda$. The authors prove that $\mu$ and $\gamma$ are uniquely determined to infinite order at $\Gamma$ from partial data, through a detailed singularity analysis of specially constructed Maxwell solutions and Green's representations, even in the presence of an unknown obstacle. The results advance partial-data boundary determination and yield a Maxwell-focused framework potentially extensible to other PDE systems.

Abstract

In this paper, we extend and simplify the methods in [13] to improve the results on uniqueness of the boundary determination for the Maxwell equation. In particular, we show that the electromagnetic parameters are uniquely determined to infinite order at the boundary from the local admittance map, disregarding the presence of an unknown obstacle, where actually only the local Cauchy data of the fundamental solution are used. The proof mainly relies on an elaborate singularity analysis on certain singular solutions to the Maxwell equation.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Theorem 2.7