# Neumann Problem

## Neumann problem

[′nȯi‚män ‚präb·ləm]
(mathematics)
The determination of a harmonic function within a finite region of three-dimensional space enclosed by a closed surface when the normal derivatives of the function on the surface are specified.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Neumann Problem

(also the second boundary value problem of potential theory), a boundary value problem posed for second-order partial differential equations. In the simplest cases, particularly for the Laplace equation, the Neumann problem consists in finding in some region a solution of the equation having a given normal derivative on the boundary of the region. The problem was first systematically studied in 1877 by C. Neumann.

References in periodicals archive ?
Neumann functions are analogous to Green's functions for Dirichlet problems, so they are often also called Green's function for the Neumann problem or Green's function of the second kind.
In this paper, we study the bifurcation and the exact multiplicity of solutions for the Neumann problem
We point out the fact that if n = 1 and [M.sub.1] (t) = 1 for all t [member of] [R.sup.+], then ([N.sub.[lambda],[mu]]) becomes the nonhomogeneous Neumann problem
His topics include the Kerzman-Stein operator and kernel,the Ahlfors map of a multiply connected domain, the classical Neumann problem, arc length quadrature domains, and Green's function and the Bergman kernel.
The authors of  used the bifurcation techniques to obtain multiple positive solutions for the Neumann problem. We only mention several among many results based on applications of a Guo-Krasnosel'skii fixed point theorem and fixed point index computations.
Variational Forms of the Dirichlet and Neumann Problems. We recall that we are considering the shape optimization problem (7) where [u.sub.D] solves the pure Dirichlet problem (4) and [u.sub.N] solves the Neumann problem (6).
To the best of our knowledge, there is no analogous study for the Neumann problem. Certain resonant Neumann problems, were studied by Iannacci-Nkashama , , Kuo , Mawhin-WardWillem , Rabinowitz .
They cover the Cauchy problem, the Dirichlet problem, the Neumann problem, the Neumann problem for a nonlocal nonlinear diffusion equation, nonlocal p-Laplacian evolution problems, the nonlocal total variation flow, and nonlocal models for sandpiles.
Neumann Problem. Let f [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
By analogy with the Green function for the Dirichlet problem for a domain [OMEGA], we consider the Green type function for the Neumann problem for [OMEGA] (also known as Neumann's function, or Green's function for the Neumann problem or Green's function of the second kind).
The principal focus will be on what happens to the eigenstructure of the Neumann problem ([sigma] = 0) as [sigma] proceeds along rays emanating from the origin toward the point at infinity in the complex plane.
The interior Neumann problem (3.4) is equivalent to the following exterior problem: find a such that

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