Preamble
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with Lipschitz continuous boundary $\partial \Omega$. Trace theorems imply that there exists a constant $C(N,\Omega, p)$ such that $$ \|{\rm T} u\|_{L^p(\partial \Omega)}\leqslant C\|u\|_{W^{1,p}(\Omega)}.$$
I wish to know how the explicit dependence between the constant $C$ and $\partial \Omega$. Is there an explicit formula for this constant? Can it be bounded from above by another constant that is dependent only on $N$ and $p$?
Background
I have been researching trace theorems of Sobolev functions in relation to homogenisation of the solutions to PDE's. I am considering a sequence of domains $\Omega_\epsilon \subset \Omega$, for $\epsilon>0$, which converge to $\Omega$ as $\epsilon \rightarrow 0$. The theorem that I am working on requires me to consider the limit supremum of $C(N,\Omega_\epsilon, 2)$ as $\epsilon\rightarrow 0$ and $\Omega_\epsilon \rightarrow \Omega$.
I have read Evans (Partial Differential Equations), Jindrich Necas (Direct Methods in the Theory of Elliptic Equations) and Emmanuele DiBenedetto (Real Analysis), yet these books typically consider the hyperplane or just do not say what the constant is, just claiming it to be dependent on $\partial \Omega$,$N$ and $p$.
Answer
Thank you to all that commented. The answer is given in "Elliptic Problems in Nonsmooth Domains" by Pierre Grisvard (2011), lemma 1.5.1.9 and theorem 1.5.1.10. I will state both results for future readers.
Lemma 1.5.1.9
If $\Omega$ is a bounded domain with a Lipschitz continuous boundary, then there exists a constant $\delta>0$ and vector field $\underline{\mu}\in C^1(\overline{\Omega})$ such that $\underline{\mu}\cdot \underline{n} \geqslant \delta$ on $\partial \Omega$, for $\underline{n}:\partial \Omega \rightarrow \mathbb{S}^{N-1}$ the outward pointing normal.
Theorem 1.5.1.10
The following inequality $$\delta \|{\rm T} u\|^p_{L^p(\partial \Omega)} \leqslant \|\underline{\mu}\|_{C^1(\overline{\Omega})}\left(\left(a^{1-1/p}\right)\|\nabla u\|^p_{L^p\left(\Omega,\mathbb{R}^N\right)}+ \left(1+ (p-1) a^{-1/p}\right)\|u\|^p_{L^p(\Omega)}\right) $$ holds for all $a\in (0,1)$ and $u \in W^{1,p}(\Omega)$.