Some extensions of Weyl's lemma

Question

The Weyl's lemma said that: If $u$ is a continuous function on the open set $\Omega$ such that it satisfies $\Delta u=0$ in the distributive sense, that is

$$\int_\Omega u \Delta\phi = 0$$

for all $\phi \in C^2_0(\Omega)$, then $u$ is $C^2$ and $\Delta u=0$ classically. The proof that I found uses the mean value property of harmonic function and it seems that we can just assume $u\in L^1_{loc}(\Omega)$.

My question is, is there any generalization of the Weyl's lemma of the form: Let $u \in L^1_{loc}(\Omega)$ satisfies $a_{ij} u_{ij} = f$ in the distributive sense, that is

$$\int_\Omega u \big( a_{ij} \phi\big)_{ij} = \int_\Omega f \phi $$

for all test function $\phi$ (In here we assume that $(a_{ij})$ are uniformly elliptic and $C^\infty$). What can we say about the regularity of $u$ given the regularity of $f$? For example, if $f$ is $C^{0, \alpha}(\Omega)$, can we say $u \in C^{2, \alpha}(\Omega)$? How about $f\in L^p(\Omega)$?

Answer 1

In Weyl's lemma we do not need to assume that $u\in L_{\mathrm{loc}}^1(\Omega)$, but rather that $u$ is locally expressible as a linear combination of (finitely many) partial derivatives of locally integrable functions.

Weyl's lemma is generalized in several directions: For non-constant coefficients, for arbitrary degree elliptic operators (even pseudo-differential), for manifolds instead of euclidean spaces.

The best (but not readable by everybody) reference in L. Hörmander's first volume.

Indeed $f\in C^{k,a}$ implies that $u\in C^{k+2,a}$, for $k\ge 0$ and $a\in (0,1)$. But for $a=0$ IT DOES NOT HOLD!

Also, if $f\in W^{k,p}$, then $u\in W^{k+2,p}$, for all $k\ge 0$. (Restrictions related to the smoothness of the boundary apply.)