Boundary Value Problems with Mixed Dirichlet and Neumann Conditions for Three-Dimensional Degenerate Elliptic Equation
Ключевые слова:
Appell and Lauricella hypergeometric functions, three-dimensional degenerate elliptic equation, PDE-systems of hypergeometric type, fundamental solutions, mixed problems with Dirichlet and Neumann conditions, energy-integral methodАннотация
This article investigates two problems with mixed Dirichlet and Neumann conditions
for a three-dimensional degenerate elliptic equation. Fundamental solutions of the named equation are expressed through a triple Lauricella hypergeometric function and explicit solutions of
the mixed problems in the first octant are written out through a double Appell hypergeometric
function. The energy integral method is used to prove the uniqueness of the solutions to the
problems under consideration. In the course of proving the existence of the problem solution,
differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. The Gauss–Ostrogradsky formula
is used to express problem’s solutions in an explicit forms
