On the Dirichlet–Riquier problem for biharmonic equations

On Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

“…which coincides with (2) for n ≥ 3, because in this case, according to Remark 1, E 4 (x, ξ) = E 4 (x, ξ). If m = 3 and n ≥ 3, n = 4, then, again by Remark 1, we get E 6 (x, ξ) = E 6 (x, ξ) and hence, taking into account the previous equalities, (13) gives the equality…”

“…The Green's functions were found depending on the dimensionality n of the space and the order of the polyharmonicity m of the equation. Note that, in (13), the singularity of the Green's function is contained only in the first term. Of course, finding the explicit value of the integral from (27) for a specific function f (x) is not an easy task.…”

“…The papers [11,12] give an explicit form of the Green's functions for the biharmonic and 3-harmonic equations in the unit ball. As for the biharmonic equation, we can note the recent paper [13] devoted to the solvability conditions for some nonstandard problems in the ball. For the most general results on the generalized Neumann problem containing powers of normal derivatives in the boundary conditions, we note the paper [14].…”

Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

“…In connection with the biharmonic equation, we note the recent papers [11,12] devoted to the solvability conditions for some nonstandard problems in the ball for the biharmonic equation. As the most general results on the generalized Neumann problem containing powers of normal derivatives in boundary conditions, we note the paper [13].…”

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

“…В связи с бигармоническим уравнением отметим недавние работы [10,11], посвященные условиям разрешимости некоторых нестандартных задач в шаре для бигармонического уравнения. В качестве наиболее общих результатов по обобщённой задаче Неймана, содержащей степени нормальных производных в граничных условиях, отметим работу [12].…”