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On the growth properties of interior transmission eigenfunctions near corners

Emilia L. K. Blåsten, Valter Pohjola

TL;DR

This work analyzes how L^2 interior transmission eigenfunctions behave near polygonal corners. It combines corner-singularity theory for the biharmonic equation with a buckling-type reformulation and an auxiliary eigenproblem to construct singular, localized ITEFs at non-convex corners, while proving vanishing at convex corners under weaker regularity. The main contributions include the first rigorous proof of corner-induced localization for ITEFs and a convex-corner vanishing result, clarifying the exotic nature of these eigenfunctions and their relation to corner scattering. These insights have implications for inverse scattering and imaging, illustrating that some ITEFs cannot arise from standard scattering solutions in non-convex geometries.

Abstract

We investigate the localization and vanishing of $L^2$ interior transmission eigenfunctions at corners. Past numerical computations suggest that these eigenfunctions localize at non-convex corners. This phenomenon has, however, not been proven theoretically. We show that localization does indeed occur for some eigenfunctions at a non-convex corner. We also investigate the vanishing of interior transmission eigenfunctions at a convex corner. We prove that these eigenfunctions vanish at convex corners with reduced smoothness assumptions compared to earlier results.

On the growth properties of interior transmission eigenfunctions near corners

TL;DR

This work analyzes how L^2 interior transmission eigenfunctions behave near polygonal corners. It combines corner-singularity theory for the biharmonic equation with a buckling-type reformulation and an auxiliary eigenproblem to construct singular, localized ITEFs at non-convex corners, while proving vanishing at convex corners under weaker regularity. The main contributions include the first rigorous proof of corner-induced localization for ITEFs and a convex-corner vanishing result, clarifying the exotic nature of these eigenfunctions and their relation to corner scattering. These insights have implications for inverse scattering and imaging, illustrating that some ITEFs cannot arise from standard scattering solutions in non-convex geometries.

Abstract

We investigate the localization and vanishing of interior transmission eigenfunctions at corners. Past numerical computations suggest that these eigenfunctions localize at non-convex corners. This phenomenon has, however, not been proven theoretically. We show that localization does indeed occur for some eigenfunctions at a non-convex corner. We also investigate the vanishing of interior transmission eigenfunctions at a convex corner. We prove that these eigenfunctions vanish at convex corners with reduced smoothness assumptions compared to earlier results.

Paper Structure

This paper contains 11 sections, 23 theorems, 145 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A condition for the appearance of a singular term
  4. The condition for a singular term on the bounded domain $D$
  5. Localization at a non-convex corner
  6. A buckling type problem
  7. An auxiliary eigenvalue problem
  8. Finding a singular ITEF
  9. Vanishing near convex corners
  10. Relation to corner scattering and open problems
  11. Open questions

Key Result

Theorem 1.1

Let $\omega \in (\omega_0, 2\pi)$. Then there exists a corner domain $D$ satisfying eq_def_D, with a corner point at $x_0=0$, and positive constants $n \neq 1$, and $k > 0$, such that the solutions $v$ and $w$ of eq_ite, are such that for some $\alpha \in (0,1)$ and where $C(\theta) \not \equiv 0$.

Figures (2)

  • Figure 1: An example domain $D$ with a non-convex corner of opening angle $\omega > \pi$. At this type of corner the ITEF can become unbounded.
  • Figure 2: An example domain $D$ with a convex corner of opening angle $\omega < \pi$. At this type of corner the ITEF vanishes. The localization of an ITEF is thus not possible for convex corners.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Lemma 3.1
  • ...and 30 more