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.


1 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.

  • $\begingroup$ So when a_{i,j} is smooth, does the claim hold? In this case I think the regularity is not a problem, but I am still not sure whether the Weyl lemma holds. $\endgroup$
    – W.J.
    Nov 10, 2022 at 13:17
  • 2
    $\begingroup$ 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 $L^p_{loc}$. $\endgroup$ Nov 10, 2022 at 17:54
  • $\begingroup$ Yes, for smooth coefficients your statement is true. $\endgroup$ Nov 11, 2022 at 14:45
  • $\begingroup$ @Giorgio Metafune Thanks a lot! So can you recommend me some books or papers about this method that you said? I am a little confused about difference quotients methods and the problem of p. $\endgroup$
    – W.J.
    Nov 12, 2022 at 9:58
  • $\begingroup$ 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. $\endgroup$ Nov 12, 2022 at 14:12

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