What is the regularity of the eigenfunctions for self-adjoint operators with non-smooth coefficients?

Question

There are well known results about elliptic operators, $L$, that guarantee that an operator of order $2k$ generates a basis for $H^{k}$ on smooth domains.( See Do eigenfunctions of elliptic operator form basis of $H^k(M)$?).

My question is regarding the regularity of this basis.

In particular,

Given $-Lu=\partial_{j}(a^{ij}\partial_{i}u)$ a self-adjoint compact operator in $L^{2}$ if $a^{ij}\in C^{k}$ what can we say about the regularity of the eigenbasis?

What can we say if we know that the regularity of $a^{ij}\in C^{\alpha,\beta}$ or $ H^{s}$ ?

Answer 1

The standard regularity theory for elliptic equations also handles the eigenvalue equation. See e.g. Theorem 8.13 of Gilbarg & Trudinger, which implies:

If $a^{ij} \in C^k$, $\phi \in H^{k+2}$ and $\partial \Omega \in C^{k+2}$ then any solution of $Lu = \lambda u$ in $\Omega$ with $u = \phi$ on the boundary has regularity $u\in H^{k+2}$.

The same will hold on a compact manifold. You can apply the Sobolev embedding theorem from here to get classical regularity, but you do lose $\rm dim/2$ derivatives.

Towards the end of chapter 8 you'll find the result $a^{ij} \in C^{\alpha} \implies u\in C^{1;\alpha}$. I'm not sure there are any higher Hölder estimates that work without assuming the classical derivatives exist.