Neumann problem

Suppose Ω is a region of n and Ω is the boundary of Ω. Further suppose f is a function f:Ω, and suppose n corresponds to taking a derivative in a direction normal to the boundary Ω at any point. Then the Neumann problem is to find a function ϕ:ΩΩ such that

ϕn = f,on Ω,
2ϕ = 0,in Ω.

Here 2 represents the Laplacian operator and the second condition is that ϕ be a harmonic function on Ω. The condition for the existence of a solution ϕ of the Neumann problem is that integral of the normal derivative of the function ϕ, calculated over the entire boundary Ω, vanish. This follows from the identic equation


and from the fact that 2ϕ=0.

Title Neumann problem
Canonical name NeumannProblem
Date of creation 2013-03-22 15:19:59
Last modified on 2013-03-22 15:19:59
Owner dczammit (9747)
Last modified by dczammit (9747)
Numerical id 10
Author dczammit (9747)
Entry type Definition
Classification msc 31B15
Classification msc 31B05
Classification msc 31A05
Related topic HarmonicFunction