# Boundary values of $f$, bounded linear operator

I have a question about Sobolev spaces

Let $U$ be a bounded Lipschitz domain of $\mathbb{R}^{d}$. $H^{1}(U)$ denotes the first order $L^2$-Sobolev space on $U$ with Neumann boundary condition.

It is well known that there exists a bounded linear operator \begin{equation*} T:H^{1}(U) \to L^{2}(\partial U;\mathcal{H}^{d-1}) \end{equation*} such that $Tf=f$ on $\partial U$ for all $f\in H^{1}(U)\cap C(\bar{U})$. Here $\mathcal{H}^{d-1}$ denotes the $d-1$ Hausdorff measure.

My question

Is there generalization of this theorem? Even if $U$ is not Lipschitz domain, trace operator $T$ exist?

If you know related results, please let me know.

## 1 Answer

Take a look into Maz'Yas book "Sobolev spaces", Chapter 1.4.7, Corollary 2.

There it is shown that if $\Omega$ is an extension domain (for $W^{1,p}$ and $L^p$ simultaneously) and $\mu$ is a measure supported in $\overline\Omega$ satisfying $$\sup_{x \in \mathbb{R}^n}\sup_{r \in (0,1)} r^{1-n}\mu(B(x,r)) < \infty,$$ then $$\|f\|_{L^p(\overline\Omega;\mu)} \leq C \|f\|_{W^{1,p}(\Omega)}$$ for all $f \in C^1(\Omega)$.

The restriction of the $(n-1)$-dimensional Hausdorff measure to $\partial\Omega$ satisfies this condition if $\mathcal{H}^{n-1}(\partial\Omega) < \infty$ (cf. for instance Chapter 2.3 in "Measure Theory and Fine Properties of Functions" by Evans and Gariepy).

• These references was very helpful for me. – sharpe Oct 31 '16 at 14:04
• @sharpe you might also find the following article interesting: Biegert, Markus. On traces of Sobolev functions on the boundary of extension domains. Proc. Amer. Math. Soc. 137 (2009), no. 12, 4169–4176. MR2538577 – anonymous Jan 14 '17 at 15:15