Trace inequality Could you please give me a hint on how to prove the following inequality
$$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1}, \quad \forall u\in H^1.$$,
 A: Typically, you get an estimate like
$$
\lVert u \rVert_{H^{s-1/2}(\Gamma)}
\le
C_s \, \lVert u \rVert_{H^{s}(\Omega)}
$$
for $s > 1/2$
by embedding theorems (maybe some regularity of the boundary is needed).
Here, $H^{s}(\Omega)$ is a fractional Sobolev space.
By interpolation theory, you get
$$
\lVert u \rVert_{H^{s}(\Omega)}
\le
C\,
\lVert u \rVert_{L^2(\Omega)}^{1-s}\,
\lVert u \rVert_{H^1(\Omega)}^{s}.
$$
Combining the estimates, we have
$$\lVert u \rVert_{L^2(\Gamma)} \le C \, \lVert u \rVert_{H^{s-1/2}(\Gamma)} \le C \, C_s \, \lVert u \rVert_{L^2(\Omega)}^{1-s}\,
\lVert u \rVert_{H^1(\Omega)}^{s}$$
for all $s > 1/2$.
Maybe, one can trace the dependence of $C_s$ as $s \to 1/2$ to conclude something.
A: 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.
