General version of Weyl's lemma

Question

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.

Answer 1

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.

Answer 2

The result is true for elliptic PDE with smooth coefficients of any order!

Theorem. Let $a_\alpha:\Omega \to \mathbb{R}$ be smooth functions on $\Omega \subset \mathbb{R}^n$ and suppose that the differential operator $$P:=\sum_{|\alpha|\leq N}a_\alpha(x) \partial^\alpha_x $$ is elliptic, i.e. the principal symbol $$\sigma_N(P)(x,\xi):=\sum_{|\alpha|=N}a_\alpha(x)(i\xi)^{\alpha}\neq 0\qquad x\in \Omega,\xi\in \mathbb{R}^n,\xi\neq 0$$ Then, one has $$u\in \mathcal{D}'(\Omega),Pu\in C^\infty(\Omega) \Rightarrow u\in C^\infty(\Omega)$$

Here $\mathcal{D}'(\Omega)$ is the space of all distributions on $\Omega$. In particular, this theorem implies that $Pu =0$ on $\Omega$ means that $u$ is smooth.

This is proved by first constructing Elliptic parametrix. For references, see any book on microlocal analysis, for example P. Hintz, An introduction to microlocal analysis.