A1 Journal article (refereed)
Free boundary methods and non-scattering phenomena (2021)

Salo, M., & Shahgholian, H. (2021). Free boundary methods and non-scattering phenomena. Research in the Mathematical Sciences, 8(4), Article 58. https://doi.org/10.1007/s40687-021-00294-z

JYU authors or editors

Salo, Mikko

Publication details

All authors or editors: Salo, Mikko; Shahgholian, Henrik

Journal or series: Research in the Mathematical Sciences

ISSN: 2522-0144

eISSN: 2197-9847

Publication year: 2021

Publication date: 05/10/2021

Volume: 8

Issue number: 4

Article number: 58

Publisher: Springer Science and Business Media LLC

Publication country: Netherlands

Publication language: English

DOI: https://doi.org/10.1007/s40687-021-00294-z

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/78289

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2106.15154

Abstract

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

Keywords: inverse problems; partial differential equations

Fields of science: