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.