Suppose that a norm is defined on a subspace $V$ of a vector space based on the following inner product
$$\langle u, v\rangle_{V}=\int_V u v \quad d V .$$
Suppose that S is the boundary of V. I would like to write $\| u\|_S$ in terms of a norm in the space $V$. In other words, can we find any relation between $\| u\|_S$ and $\| U\|_V$, for some $U$ in $V$?
Any help is appreciated.