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.
Best Answer
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.