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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.

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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.

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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.

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