The classical **Weyl's lemma** say, suppose $u \in L^1_{loc}(\Omega)$ satisfies
$$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$
then $u$ is harmonic in $\Omega.$ What I want to ask is, whether the above claim holds in the case of general elliptic operator in divergence form.

For example, let an elliptic operator $Lu= -\sum_{i,j=1}^n\partial_{x_i}(a_{i,j}(x)\partial_{x_j}u)$, where the matrix $(a_{i,j})_{n\times n}$ is positive-definite and smooth. If $\int_{\Omega}u L \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$ I want to know whether $Lu=0$ in $\Omega$ implies $u$ is smooth in $\Omega$. Another question is, are there any results about Weyl's lemma for operators in **non-divergence form**?

PS: If there is any relevant literature, please let me know.