Nonradiating sources and transmission eigenfunctions vanish at corners and edges

TL;DR

This paper proves that acoustic sources and transmission eigenfunctions must vanish at corners and edges, using a novel complex geometrical optics approach, advancing understanding in inverse source problems.

Contribution

It introduces a new method employing complex geometrical optics solutions with logarithmic branches to analyze sources at corners and edges in inverse problems.

Findings

Nonradiating sources vanish at boundary corners and edges.

Transmission eigenfunctions also vanish at corners and edges.

The method handles non-convex geometries effectively.

Abstract

We consider the inverse source problem of a fixed wavenumber: study properties of an acoustic source based on a single far- or near-field measurement. We show that nonradiating sources having a convex or non-convex corner or edge on their boundary must vanish there. The same holds true for smooth enough transmission eigenfunctions. The proof is based on an energy identity from the enclosure method and the construction of a new type of planar complex geometrical optics solution whose logarithm is a branch of the square root. The latter allows us to deal with non-convex corners and edges.