TY - JOUR

T1 - Vertex green's functions of a quarter-plane: Links between the functional equation, additive crossing and Lamé functions

AU - Assier, Raphael

AU - Shanin, A. V.

PY - 2021/5/15

Y1 - 2021/5/15

N2 - In our previous work (Assier & Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of plane-wave diffraction by a quarter-plane. 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 near-field are directly related to the eigenvalues of the Laplace-Beltrami operator. This is important since the correct near-field behaviour at the tip of the quarter-plane had so far never been obtained via a multivariable complex analysis approach.

AB - In our previous work (Assier & Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of plane-wave diffraction by a quarter-plane. 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 near-field are directly related to the eigenvalues of the Laplace-Beltrami operator. This is important since the correct near-field behaviour at the tip of the quarter-plane had so far never been obtained via a multivariable complex analysis approach.

U2 - 10.1093/qjmam/hbab004

DO - 10.1093/qjmam/hbab004

M3 - Article

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 1464-3855

ER -