Vertex Green's functions of a quarterplane. Links between the functional equation, additive crossing and Lamé functions
Abstract
In our previous work (Assier \& Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of planewave diffraction by a quarterplane. In particular, we showed that the unknown spectral function satisfies a condition of additive crossing about its branch set. In this paper, we study a very similar class of spectral problem, and show how the additive crossing can be exploited in order to express its solution in terms of Lamé functions. The solutions obtained can be thought of as tailored vertex Green's functions whose behaviours in the nearfield are directly related to the eigenvalues of the LaplaceBeltrami operator. This is important since the correct nearfield behaviour at the tip of the quarterplane had so far never been obtained via a multivariable complex analysis approach.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.08700
 arXiv:
 arXiv:2004.08700
 Bibcode:
 2020arXiv200408700A
 Keywords:

 Mathematics  Analysis of PDEs