Proper domain for Laplacian

Question

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these eigenfunctions? The space should be probably $H^2(\mathbb{S}^2)$, but which boundary conditions are necessary, so that this is true?

Answer 1

The eigenfunctions are smooth, so you can eventually consider the space $C^\infty(\mathbb S^2)$. The manifold $\mathbb S^2$ has no boundary, so no boundary condition can be imposed.

The natural space to find these functions is $H^1(\mathbb S^2)$. These will be eigenfunctions in the weak sense (and, as it turns out, also in the strong sense). A convenient way to find eigenfunctions is to look for critical points of the Rayleight quotient $$ \frac{\int_{\mathbb S^2}|du|^2}{\int_{\mathbb S^2}|u|^2}. $$

The right domain depends on what you what to do with your operator. Note that $\Delta$ does not map $H^2$ to $H^2$ but only to $L^2$. If you are not satisfied with an operator $H^1\to H^{-1}$ or $H^2\to L^2$ but want something of the form $A\to A$, then there are not many choices left. Putting $A=C^\infty$ seems most natural to me.