Continuous Sobolev embedding

Question

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)^{2/2^{\ast}} \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 enter link description here, 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.

Answer 1

The abstract condition to have the Sobolev embeddings on domains in their known form as on Euclidean space is the existence of an extension operator for $D$, that is, a continuous linear mapping $E \colon W^{k,p}(D) \to W^{k,p}(\mathbb{R}^n)$ which serves as a right inverse for the restriction operator $R \colon W^{k,p}(\mathbb{R}^n) \to W^{k,p}(D)$, $u \mapsto u|_D$, and does so simultaneously for all $k \in \mathbb{N}_0$ (in your case, it would suffice for $k \in \{0,1\}$). This should be the actual requirement behind most smoothness assumptions in the various textbooks, with the actual requirement depending on how capable the extension operator introduced in the respective book is. There are authors with pose it as an abstract condition (a particular instance would be Maz'ya's "Sobolev spaces").

Such a linear extension operator allows you to transfer the embedding results from Euclidean space to the domain by the reasoning $$ \|u\|_{L^{p^*}(D)} \leq C\|Eu\|_{L^{p^*}(\mathbb{R}^n)} \leq C \|Eu\|_{W^{1,p}(\mathbb{R}^n)} \leq C\|u\|_{W^{1,p}(D)}. $$

The existence of these extension operators is an active (and rather fascinating) research field.

Notes: a) Compactness of the embeddings on Euclidean space also transfers, and b) if I am not missing something: in order to treat the borderline case $W^{1,n}(D)$ (so, $H^1(D)$ for space dimension $n = 2$ in your case), you need to tinker with the fact that the extension property of, say, $L^q(D)$ and $W^{1,q}(D)$ transfers to the interpolation spaces between them in order to get the whole range $(n,\infty)$ for the embedding as you stated.