# General version of Weyl's lemma

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.

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $$a_{ij}$$. There is a huge literature on the subject. Just Google "regularity for uniformly elliptic operators" or check the Gilbarg - Trudinger book.
• For smooth coefficients the result is true. It follows by combining difference quotients methods and $W^{2,p}$ estimates if $u \in L^p_{loc}$ for some $p>1$. If $p=1$ one needs first an argument to show that it is in $L^p_{loc}$. Nov 10, 2022 at 17:54
• The argument is a variation of regularity results and I do not know any place where it is written in the form you ask for. However, if $p>1$, these arguments are explaind in Lemma 2.5 in uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/…. If you need $p=1$ Iet me know. Nov 12, 2022 at 14:12