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
Best Answer
By a scaling argument, one does not need upper bound on the radii in (I) as well. The lower bound is needed because of the inhomogeneous term in the Sobolev norm in the right hand side. If you use the Sobolev seminorm $\|Du\|_{L^p}$ instead of the full norm you will have a scale independent inequality.