Green function representation in the Dirichlet problem for polyharmonic equations in a ball

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

“…The similar results in the class of nonuniform biharmonic and triharmonic functions in a sector were obtained in [11,12]. The Green's function of the Dirichlet problem for a polyharmonic equation in the multidimensional ball was constructed in the explicit form in [13][14][15][16][17]. We also note that works [18][19][20] are devoted to the construction of the Green's function of the Robin problem in the explicit form in a circle.…”

About solvability of boundary value problems for the nonhomogeneous polyharmonic equation in a ball

“…with homogeneous boundary conditions In this case the rank of the matrix A 0 D A.k 1 ; k 2 ; : : : ; k m / is m. It is known that problem (3.1)-(3.2) is uniquely solvable for any right-hand side, and the solution is represented by the Green function [7,14,15]. As a result we constructed the Green function in explicit form.…”

On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation

“…Let us first consider the case when ϕ1(x) ≡ ϕ2(x) ≡ 0. It is known (see, for example, [9,10,11]), that if g1(x) is sufficiently smooth function, then the solution of problem (3.1) exists, it is unique and has the form…”

“…Then the necessary and sufficient solvability condition for the Neumann boundary value problem(1.5),(1.6) has the formyjg(y)dy = ∂Ω yj [ψ2(y) − ψ1(y)] dSy, j = 1, 2, ..., nIf a solution exists, then it is unique up to a first order polynomial and can be represented asu(x) = c0 + − s)s −2 v(sx)ds,where cj, j = 0, 1, ..., n, are arbitrary constants, v(x) is the solution of Dirichlet problem (3.1) with the functions g1(x) = r ∂ ∂r + 4 r ∂ ∂r + 3 g(x), ϕ1(x) = ψ1(x), ϕ2(x) = ψ2(x) + 2ψ1(x) and with the additional conditions v(0) = 0, ∂v(0) ∂xj = 0, j = 1, 2, ..., n. In the section two we study properties of some integro-differential operators, which we then use throughout the paper. In section 3 we investigate the Dirichlet problem for biharmonic equation, making use of the explicit form of the Green function found in[9,10,11]. Then in the following section 4, reducing the Neumann problem (1.5),(1.6) to the considered Dirichlet problem, we give the necessary and sufficient solvability conditions for the Neumann problem for the biharmonic equation.…”