Let $\Omega$ be a sufficiently nice domain in $\mathbb{R}^n$.
If $ 1 \leq p < n $ and $ p^* = \frac{np}{n-p} $ then there exists a constant $C_1$ such that for all $ u \in W^{1,p}(\Omega) $ we have $ (I)~~~~||u||_{L^{p^*}(\Omega)} \leq C_1||u||_{W^{1,p}(\Omega)}. $ If $ p > n $ then there exists a constant $C_2$ such that for all $ u \in W^{1,p}(\Omega) $ we have $ (II)~~~~||u||_{L^{\infty}(\Omega)} \leq C_2||u||_{W^{1,p}(\Omega)}. $
In general, the constants $C_i$ depend on the domain $\Omega$. Can someone point me to some references that discuss the dependence between the embedding constants and the domain?
I'm interested in conditions under which, given some family of domains $ \Omega_\alpha $, I can get Sobolev embedding inequalities as above with a constant that doesn't depend on $\alpha$. To be even more specific, I'm interested in the case $ n = 2 $ and when the domains are families of balls or annuli.
For example, if I consider inequality (II) and a family of balls, then an obvious sufficient condition is to have both an upper and a lower bound on the radii of the balls. One can't get away without a lower bound (consider the function $u \equiv 1$) but can get away without an upper bound by using a translation argument. What about inequality (I)? What can I use to answer such questions?
Since one way to prove the inequalities above is to use an extension operator, and then "steal" the inequality from $\mathbb{R}^n$, this question is related to dependence of the minimal norm of an extension operator $W^{1,p}(\Omega) \rightarrow W^{1,p}(\mathbb{R}^n)$ on the domain