Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

Abstract

The Green’s function of the Riquier–Neumann problem for the polyharmonic equation in the unit ball is constructed. Using the obtained Green’s function, an integral representation of the solution to the Riquier–Neumann problem in the unit ball is found.

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

“…In ref. [17], these results are extended to the polyharmonic equation. The Green's function is also used to study nonlocal equations.…”

“…In ref. [17], a function E 2m (x, ξ) was considered, which is related to the function E 2m (x, ξ) by the following equality…”

“…In Lemma 2 [17], it is proved that the function E 2m (x, ξ) is m-harmonic in x ∈ S for x = ξ. Therefore, for n ∈ N m , the function E 2m (x, ξ) is also m-harmonic in x ∈ S for x = ξ.…”

“…In ref. [17], an elementary solution E 2m (x, ξ) of the polyharmonic equation is used, which is related to the function E 2m (x, ξ) by the equality (9). This function E 2m (x, ξ) has the property…”

“…In the present paper, another representation of the Green's function G 2m (x, ξ) is given, which coincides, in particular, with Formulas (2) and (4). We use the notion of a fundamental solution instead of the previously used notion of an elementary solution E 2m (x, ξ) for the Riquier-Neumann problem [17].…”