Analysis of subwavelength resonances in high contrast elastic media by a variational method

TL;DR

This work analyzes subwavelength resonances in high-contrast elastic media by recasting elastic scattering in a bounded domain $D$ through a Dirichlet-to-Neumann map and a specially crafted auxiliary sesquilinear form. Resonances are characterized by the vanishing determinant $\det(\boldsymbol{\mathscr{I}}-\boldsymbol{\mathscr{A}}(\omega,\delta))=0$, enabling explicit asymptotic expansions of resonant frequencies via Gohberg–Sigal theory and Puiseux series; leading terms are governed by eigenvalues of a real symmetric matrix $\boldsymbol{\mathscr{Q}}$ and the density contrast parameter $\tau$. The interior field exhibits resonant enhancement linked to the imaginary parts of $\omega$, while the exterior field is described by well-separated transversal and longitudinal far-field patterns $\boldsymbol{u}_{s,\infty}$ and $\boldsymbol{u}_{p,\infty}$. The results provide a flexible, variational framework that avoids layer-potential spectral analysis and yields comprehensive interior and exterior asymptotics for subwavelength elastic resonances in three dimensions.

Abstract

In this paper, we present a mathematical study of wave scattering by a hard elastic obstacle embedded in a soft elastic body in three dimensions. Our contributions are threefold. First, we characterize subwavelength resonances using the Dirichlet-to-Neumann map and an auxiliary variational form, showing that these resonances occur when the determinant of a specific matrix vanishes. Second, employing Gohberg-Sigal theory and Puiseux series expansions for multi-valued functions, we derive the asymptotic expansions of subwavelength resonant frequencies in the low-frequency regime through this explicit characterization. Finally, we provide a representation of the scattered field in the interior domain, where the enhancement coefficients are governed by the imaginary parts of the resonant frequencies. Additionally, we establish the transversal and longitudinal far-field patterns for the scattered field in the exterior domain.

Theorems & Definitions (40)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Proposition 2.8
• proof