Iterated Neumann problem for the higher order Poisson equation

Abstract

Key wordsn u = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann-n problem for ∆ n u = f . The representation formula provides the tool to treat more general partial differential equations with leading term ∆ n u in reducing them into some singular integral equations.

Neumann problem for generalized n-Poisson equation

“…Using the higher-order Neumann functions and higher-order Cauchy-Pompeiu representations, Neumann problems for Poisson and n-Poisson equations are solved uniquely under some normalization and solvability conditions [7,5,8,9,6].…”

“…[5][6][7][8][9][10]. In this article, we discuss the Neumann problem for the "generalized n-Poisson equations".…”

Biharmonic Green function of a ring domain

“…Green functions for differential operators are an important tool as well for the existence theory of boundary value problems for partial differential equations, as for the representation of solutions of such problems (see [3], [4], [5], [6], [9], [12], [16]). Among the operators for which it is interesting to define Green functions is the biharmonic operator…”

“…In the case of the unit disk the explicit form for G 2 (z, ζ ) is given in [3], [6], [9], for G 2 (z, ζ ) it is presented in [2] (see also [3], [6], [9]). For a circular ring G 2 (z, ζ ) is obtained in [11].…”

Riemann boundary-value problem for doubly-periodic bianalytic functions

“…Recently, various boundary-value problems (BVP) of polyanalytic functions and other functions determined by the general partial differential equations have been widely investigated by Begehr, Schmersau, Hile, Vanegas, Kumar, Jinyuan Du, Yufeng Wang, Ying Wang, Zhihua Du and others (see, for example, ). The general partial differential equations include the inhomogeneous polyanalytic equation [5], the higher order Poisson equation [6], and polyharmonic equations [7,8].…”