Suppose we have a Lipschitz domain $\Omega \subset \mathbb{R}^2$. Let $u$ be a function in the Sobolev space $W^{1\,,\,p}(\Omega)$. Since $\Omega$ is Lipschitz, there is an extension operator $P: W^{1\,,\,p}(\Omega) \to W^{1\,,\,p}(\mathbb{R}^2)$ such that \begin{equation} P \, u \, |_{\Omega}=u \tag{1} \end{equation}

Q$1$. Can someone give me a reference for $(1)\,$?

Q$2$. How to extend $u$ in this case?


1 Answer 1


Let us say that a domain $\Omega \subset \mathbb{R}^n$ is an extension domain if there exists a bounded linear operator $\mathcal{E} \colon W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^n)$ such that $\mathcal{E}u(x) = u(x)$ for $x \in \Omega$.

It is proved in P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71–88, that every uniform domain is an extension domain. Since domains with Lipschitz boundary are uniform, the result follows.

In general, the construction of the extension is done via a local reduction (change of variables that reduced the boundary of $\Omega$ to a particular shape) and a patching by means of partitions of unity. You can find several examples in standard books about Sobolev spaces.


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