Neumann problem
Suppose Ω is a region of ℝn and ∂Ω is the boundary of Ω. Further suppose f is a function f:∂Ω→ℂ, 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:Ω∪∂Ω→ℂ such that
∂ϕ∂n | = | 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
∫∂Ω∂ϕ∂n𝑑σ=∫Ω∇⋅(∇ϕ)𝑑τ=∫Ω∇2ϕdτ |
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 |