Linear water-wave problem in a pond with a shallow beach

Abstract

We consider the linear water-wave problem in a bounded water-basin with a shallow beach. In spite of the boundedness of the domain, the spectrum of the problem may have a continuous component, if the beach of the basin has a cuspidal form. Following the approach and methods of an earlier paper by S.A.Nazarov and the second named author, we improve the results of the citation by determining the spectrum in an open borderline case under weaker geometric assumptions.

Keywords:

• linear water-wave problem
• Steklov condition
• continuous spectrum
• shallow beach

AMS Subject Classifications::

• Primary 35J20
• Secondary 35P99
• 76B15

Acknowledgements

The authors want to thank Professor Sergei Nazarov for many discussions on the subject of this article, and the referees for many valuable remarks which were helpful to improve the presentation of the results.

This research was supported by the Academy of Finland project ‘Functional analysis and applications’.