On the continuous Sobolev embedding, smooth domain

I have a question about Sobolev spaces.

In the following, we assume $d \ge 2$. Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. Note that $D$ is not necessary bounded. $H^{1}(D)$ denotes first order $L^2$-Sobolev space on $D$ with Neumann boundary condition.

I am interested in when $H^{1}(D)$ is continuously embedded into $L^{2^{\ast}}(D)$. That is, there exists $C\ge0$ such that \begin{equation*} \left( \int_{D} |f|^{2^{\ast}}\,dx\right)^{1/2} \le C \left(\int_{D}|\nabla f|^{2}\,dx+\int_{D}|f|^{2}\,dx \right)\cdots(1) \end{equation*} Here $2^{\ast}=2d/(d-2)$ if $d\ge 3$, $2^{\ast}$ is any number in $(2,\infty)$ if $d=2$.

My question

In Ouhabaz's book, it is said that $(1)$ holds when $D$ has smooth boundary. But I couldn't find the definition of smooth boundary(I think there are many styles of definition of smooth boundary) and the proof of this claim in this book. When $D$ is bounded, there are many references, though.

If you know the details, please let me know.

• What do you mean when you say that $H^1(D)$ is the Sobolev space with Neumann boundary condition? – gerw Sep 19 '16 at 6:31
• Possible duplicate of On Sobolev embedding theorem – gerw Sep 19 '16 at 6:36