Dirichlet problem for the multudimensional Helmholtz equation with one singular coefficient

T. G. Ergashev ; N. M. Safarbayeva

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 62 (2019), pp. 55-67

Résumé

In the study of boundary value problems for elliptic equations with singular coefficients, fundamental solutions play an important role, which is expressed by hypergeometric functions of one, two, or more variables depending on the number of the singularity. An interesting case is the Helmholtz equation with one or two singularities, and many authors solved various boundary value problems for a two-dimensional Helmholtz equation. However, relatively few works are devoted to the study of an equation with one singular coefficient, when the dimension of the equation exceeds three. The main obstacle in this direction is the lack of explicit fundamental solutions for the multidimensional Helmholtz equation with at least one singular coefficient. Fundamental solutions for the multidimensional Helmholtz equation with one singular coefficient in the half-space were found recently. In this paper, the Dirichlet problem for the abovementioned elliptic equation in a finite simply connected domain is studied. Using the properties of one of the fundamental solutions, the Green function was constructed. With the help of the function, the solution of the problem in a finite region bounded by the multidimensional hemisphere is found in an explicit form.

Détail

Keywords: multidimensional Helmholtz equation with one singular coefficient, Dirichlet problem, fundamental solution, Gauss–Ostrogradsky formula, Green function.