Eigenvalues of Laplace eq in the circle.

[krocho]

Does anybody know a web page or a book, or the general method to find the eigenvalues and the eigenfunctions of
laplacian u =lambda u inside the circle
u=0 in the boundary

thanks

 

[Astronuc]

If one is asking about

$$\nabla^2U + \lambda{U} = 0$$, this is the Helmholtz equation, which is normally solved in polar coordinates for a circle.

so U(r,θ) = R(r)Θ(θ), where R(r) is a Bessel's function.

http://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html

For Cartesian coordinates -
http://mathworld.wolfram.com/HelmholtzDifferentialEquationCartesianCoordinates.html

Laplace's equation

http://mathworld.wolfram.com/LaplacesEquation.html