Preamble
I am considering a sequence of bounded domains, $\Omega_\epsilon \subset \mathbb{R}^N$, $\epsilon>0$, with Lipschitz continuous boundaries $\partial \Omega_\epsilon$. As the trace operator is compact this implies that there exists a constant $C(\Omega_\epsilon)$ such that $$ \|{\rm T} u\|_{H^{1/2}(\partial \Omega_\epsilon)}\leqslant C(\Omega_\epsilon)\|u\|_{H^1(\Omega_\epsilon)}.$$
I wish to understand how $C$ explicitly depends on $\Omega_\epsilon$ and therefore what conditions must $\Omega_\epsilon$ satsify such that $C(\Omega_\epsilon)$ is bounded as $\epsilon\rightarrow 0$.
Background
I have read "Elliptic Problems in Nonsmooth Domains" by Pierre Grisvard (2011), lemma 1.5.1.9 and theorem 1.5.1.10. Which bounded $L^p(\partial \Omega)$ by a constant muliplied by the $W^{1,p}(\Omega)$ norm. I am looking for a sharper bound than this, as I am only considering $H^1$ functions with $H^{1/2}$ smooth traces.