Let $D\subset\mathbb{R}^d$ be an open domain and let consider the open cylinder $D\times (0,T)\subset\mathbb{R}^{d+1}$ where $T\in (0,+\infty)$ arbitrary. Let $H^{1}(D\times (0,T))$ be the Sobolev space defined as usual. It is known that: if we assume that $D$ satisfies the cone condition, then we have the continuous embeding $$H^{1}(D\times (0,T))\mapsto L^{2}(D)\times \{\tau=t \}$$ for every $t\in (0,T)$ (see the book of Adams).

My question is: If $D$ is an unbounded domain with finite volume and it hasn't the cone condition,does the above embedding still true? and under which assypmtions? Thanks.

Let $D\subset\mathbb{R}^d$ be an open domain and let consider the open cylinder $D\times (0,T)\subset\mathbb{R}^{d+1}$ where $T\in (0,+\infty)$ arbitrary. Let $H^{1}(D\times (0,T))$ be the Sobolev space defined as usual. It is known that: if we assume that $D$ satisfies the cone condition, then we have the continuous embeding $$H^{1}(D\times (0,T))\mapsto L^{2}(D\times \{\tau=t \})$$ for every $t\in (0,T)$ (see the book of Adams).

My question is: If $D$ is an unbounded domain with finite volume and it hasn't the cone condition,does the above embedding still true? and under which assypmtions? Thanks.