Holmgren problem for multudimensional elliptic equation with two singular coefficients

T. G. Ergashev ; N. J. Komilova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 63 (2020), pp. 47-59

Résumé

Fundamental solutions of the two-dimensional elliptic equation were known in the first half of the last century and they were successfully used in solving the basic boundary value problems and constructing the theory of potential for this equation. Relatively few papers have been devoted to the study of boundary value problems for multidimensional (greater than two-dimensional) elliptic equations with singular coefficients. For example, main boundary value problems for twodimensional and three-dimensional elliptic equations with two singular coefficients in finite and infinite domains have been studied by many authors; however, the study of the Holmgren problem was limited to the two-dimensional case. This work is devoted to finding a unique solution to the Holmgren problem for a multidimensional elliptic equation with two singular coefficients in a quarter of a ball. Using the “abc” method, the uniqueness for the solution of the Holmgren problem is proved. Applying the method of Green's function, we are able to find the solution of the problem in an explicit form. Moreover, the decomposition formula, formula of differentiation, and some adjacent relations for Appell's hypergeometric functions were used in order to find the explicit solution for the formulated problem.

Détail

Keywords: multidimensional elliptic equation with two singular coefficients, Holmgren problem, fundamental solution, Gauss–Ostrogradsky formula, Green function.