TITLE: Trace inequality
QUESTION [6 upvotes]: Could you please give me a hint on how to prove the following inequality
$$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1}, \quad \forall u\in H^1.$$,

REPLY [2 votes]: In Brenner, Scott: "The Mathematical Theory of Finite Element Methods", 2nd edition, Springer-Verlag, New York (2002), the following theorem is stated (Theorem 1.6.6):
Suppose that $\Omega$ has a Lipschitz boundary, and that $p$ is a real number in the range $1\le p \le \infty$. Then there exists a constant, $C$, such that 
$$ \|v\|_{L^p(\partial\Omega)}\le C \|v\|_{L^p(\Omega)}^{1-1/p} \, \|v\|_{W^1_p(\Omega)}^{1/p} \quad \forall v \in W^1_p(\Omega).$$
A full proof is given only for the case of $\Omega$ being a unit disk in $\mathbb R^2$ and $p=2$. Hints for the proof of the theorem are given there.