Elastic scattering problems by penetrable obstacles with embedded objects

TL;DR

The paper analyzes 3-D elastic scattering by penetrable obstacles containing embedded objects, establishing well-posedness of the direct problem through boundary-integral methods and deriving Lippmann–Schwinger-type formulations for density contrasts. It then addresses the inverse problem at a fixed frequency by introducing interior transmission and modified interior transmission problems to prove uniqueness: the obstacle boundary is uniquely determined by the far-field pattern under suitable contrast conditions. The results extend uniqueness theory to penetrable elastic inclusions with embedded cores and provide a rigorous operator-theoretic framework for the associated singular integral systems, with implications for nondestructive testing and geophysical imaging. Overall, the work advances the mathematical understanding of elastic inverse scattering in complex penetrable media.

Abstract

This paper considers 3-D elastic scattering problems by penetrable obstacles with embedded objects. The well-posedness of transmission problem is proved by employing integral equation method. Then the Inverse Problems , which is to recover the obstacle by the far-field pattern measurement, is considered. It is shown that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency.

Figures (2)

• Figure 1: Scattering by inhomogeneous medium with embedded objects.
• Figure 2: Two different scattering obstacles.

Theorems & Definitions (25)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Proposition 3.4
• Theorem 3.5
• Remark 3.6
• proof