Geodesic
Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin--Neumann boundary conditions
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 121-134

Voir la notice de l'article provenant de la source Math-Net.Ru

This work studies eigenfunction problem of the Laplace operator in the angular domain with the Robin-type boundary condition on the upper side of the angle and the Neumann-type boundary condition on the bottom side of the angle. From the physical point of view, such eigenfunctions describe waves over sloping beach. Negative values of the spectral parameter were considered. We obtained the eigenfunction of the essential spectrum and studied a special case of eigenfunction, which are elementary functions. The Sommerfeld integral representation of an eigenfunction of the negative part of the essential spectrum of the Laplace operator was obtained. Moreover, we calculated it's asymptotic far away from the angle's vertex. It is bounded on the top side of the angle and vanishes exponentially in the angle's interior with its bottom side. So, the eigenfunction of essential spectrum behaves like a surface wave. 
@article{ZNSL_2022_516_a5,
     author = {M. A. Lyalinov and N. S. Fedorov},
     title = {Eigenfunctions of the essential spectrum of the {Laplace} operator in an angle with the {Robin--Neumann} boundary conditions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {121--134},
     publisher = {mathdoc},
     volume = {516},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/}
}
TY  - JOUR
AU  - M. A. Lyalinov
AU  - N. S. Fedorov
TI  - Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin--Neumann boundary conditions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2022
SP  - 121
EP  - 134
VL  - 516
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/
LA  - ru
ID  - ZNSL_2022_516_a5
ER  - 
%0 Journal Article
%A M. A. Lyalinov
%A N. S. Fedorov
%T Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin--Neumann boundary conditions
%J Zapiski Nauchnykh Seminarov POMI
%D 2022
%P 121-134
%V 516
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/
%G ru
%F ZNSL_2022_516_a5
M. A. Lyalinov; N. S. Fedorov. Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin--Neumann boundary conditions. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 121-134. http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/