The constant in the Sobolev trace theorem inequality The trace theorem for nice enough domains states that there is a operator $T:H^1(\Omega) \to L^2(\partial \Omega)$ such that
$$|Tu|_{L^2(\partial \Omega)} \leq C|u|_{H^1}.$$
My question, is there an expression for the constant $C$? I want to see exactly how it depends on the domain $\Omega.$ This is because I want to see how the constant varies (eg. continuously) if I vary the domain.
 A: Such an estimate can be found in Grisvard "Elliptic problems in nonsmooth domains", Theorem 1.5.1.10.
It basically says
$$
\delta \|u\|_{L^2(\partial\Omega)}^2 \le \|\mu\|_{C^1(\bar\Omega)}
\left(\epsilon^{1/2}\|\nabla u\|_{L^2(\Omega)}^2
+ (1+\epsilon^{-1/2}) \|u\|_{L^2(\Omega)}^2 
\right)
$$
for all  $\epsilon\in(0,1)$, $u\in H^1(\Omega)$. The vector field $\mu$ has to be chosen to be $C^1(\bar\Omega,\mathbb R^n)$ such that
$$
\mu \cdot \nu \ge \delta 
$$ on $\partial \Omega$ with $\nu$ the outer normal vector.
This could give you an estimate of the constant for a fixed domain at least. It should help to prove continuity with respect to domain variations as well.
A: This is an old question, but since I stumbled upon it when wondering the same thing, I'll just add two points:
(1) The trace theorem is actually sharper; you can bound even the $H^{1/2}$ norm (not just the $L^2$ norm) of the trace on $\partial \Omega$ in terms of the $H^1$ norm of the function on $\Omega$. 
(2) I found this reference and also this which characterize and approximate the value of the constant in the Sobolev trace theorem.  I haven't read either one carefully, but they appear interesting and on point.
