Iterated Neumann problem for the higher order Poisson equation

Abstract

Abstract Rewriting the higher order Poisson equation Δ n u = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher‐order Neumann (Neumann‐ n ) problems for Δ n 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 Δ n u = f . The representation formula provides the tool to treat more general partial differential equations with leading term Δ n u in reducing them into some singular integral equations. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)