TITLE: How exactly does the constant $C$ in the Sobolev inequality depend on the domain?
QUESTION [8 upvotes]: The Sobolev inequality theorem -as stated here- says

Let $U$ be a bounded open subset of $\mathbb{R}^N$, with a $C^1$ boundary. Assume $u \in W^{k,p}(U)$. If $k<n/p$ then $u \in L^q(U)$, where $$ \frac{1}{q}=\frac{1}{p}-\frac{k}{n}. $$ We have in addition the estimate $$ \Vert u \Vert_{L^q(U)} \leq C \Vert u \Vert_{W^{k,p}(U)} $$ the constant C depending only on $k,p,n$ and $U$.


My question is: How does $C$ depend on $U$ exactly? Does it depend only on the measure of $U$? Is it independent of translation of $U$?

The reason I'm asking this is that I am able to estimate solutions to some PDE as $$ \Vert u \Vert_{L^q(B(x,1))} \leq C \Vert u \Vert_{W^{k,p}(B(x,1))} $$ for some $q,k,p$ and any $x \in \mathbb{R}^N$. However, I need a uniform bound, i.e. a bound not depending on the center $x$ of the balls $B(x,1)$. So if the Sobolev inequality depended only on the measure of $U$, I'd be fine.

Here is a similar question, but I really don't understand the answer.

REPLY [3 votes]: In addition to the answer by user90090, it can be told that in Sobolev spaces by Adams, as well as in its version by Adams–Fournier, no appropriate details are to be found. The details of dependence of the best Sobolev constant on parameters of the cone condition,  usually referred to by experts in the field as well-known, belong to what is now called Mathematical folklore, i.e, something like check-it-youself-if-you-need (http://en.wikipedia.org/wiki/Mathematical_folklore). Cited from Mathematical folklore, the details look like this. For a domain $U\subset\mathbb{R}^n$ of diameter $D$, star-shaped wrt some ball $B\subset U$ of diameter $d$, the best Sobolev constant depends only on the ratio $d/D$, which is established by scaling argument.  For a bounded domain $U\subset\mathbb{R}^n$, represented by the Gagliardo lemma as a union of $N\geqslant 2$ domains $U_i\,$, each star-shaped wrt some ball $B_i\subset U_i\,$, the best Sobolev constant depends only on $N$ and the ratios $d_i/D_i\,$.