Eigenvalue problem for the Laplacian on the unit ball

Question

I want to find out what are the eigenvalues and eigenfunctions of the eigenvalue problem for the Laplacian on the unit ball in $\mathbb R^3$, with the Dirichlet boundary conditions.

Answer 1

Write the Laplacian on the unit ball in polar coordinates $(r,\xi)$, with $r\in (0,1],\xi\in S^{n-1}$, and separate variables. The basic form of the eigenfunctions will be $$ \big(\mbox{generalized Bessel function}\big)(r)\big(\mbox{spherical harmonic}\big)(\xi)$$ where the index of the Bessel function is determined by the Dirichlet condition. You then find the eigenvalues by examining the indices of the Bessel function and the spherical harmonic.

The general case of a ball in a constant-curvature space form is treated similarly, by taking geodesic polar coordinates and separating variables, and can be found in ch 2, section 5 of Chavel's book Eigenvalues in Riemannian Geometry.