Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following:

1. $S:=\lbrace v\in V:\|v\|_{L^\infty(\Omega)}\leq 1\rbrace$ is bounded in $V$;
2. $V$ has one degree of regularity, or maybe weaker, so that the divergence can be defined, for example, $\textrm{div}v\in L^2(\Omega)$;
3. The trace of $v$ can be suitbly defined on $\partial\Omega$.
4. $C_0^\infty(\Omega)$ is dense in $V$.

Thanks!

Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following:

1. $S:=\lbrace v\in V:\|v\|_{L^\infty(\Omega)}\leq 1\rbrace$ is bounded in $V$;
2. $V$ has one degree of regularity, or maybe weaker, so that the divergence can be defined, for example, $\textrm{div}v\in L^2(\Omega)$;
3. The trace of $v$ can be suitbly defined on $\partial\Omega$.
4. $C_0^\infty(\Omega)$ is dense in $V$.
5. $V$ is reflexive.

Thanks!