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!



A suitable Sobolev-type space
I think that such a space will not exists. In particular your requirements (1) and (2) contradicts in the following sense: (1) says basically, that the norm in $V$ is weaker than the $L^\infty(\Omega)$-norm, whereas (2) requires that you can control the divergence by the norm in $V$. Imho, this is not possible.


