Let $\Omega \subseteq \mathbb R^d$ be a bounded Lipschitz domain. Then it is well known that the trace operator $$ \operatorname{tr} \colon W^{1, 1}(\Omega) \to L^1(\partial \Omega)$$ is well-defined and bounded. However, I was unable to find a citable reference for this fact. The references I found either focus on the $L^2$-setting, assume more regularity of the boundary of $\Omega$ (mostly $C^1$-boundary) or are kind of obscure. Another reference, is the seminal paper of Gagliardo for that fact, the drawback of this reference being that it is written in Italian. Nonetheless, I cannot imagine that there is not a well-cited book serving as a standard reference for PDEs or Sobolev spaces that contains the result in the above form. So it would be very nice if someone would know a reference including the result. Thanks in advance!


1 Answer 1


This is Theorem 15.10 in Giovanni Leoni's "A first course in Sobolev Spaces".

  • 4
    $\begingroup$ I would like to remark that it is Theorem 15.10 in the 1. Edition and Theorem 18.18 in the 2. Edition. $\endgroup$ Commented Nov 3, 2023 at 9:15

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .