Sobolev Embedding Theorem

Question

In the Sobolev Embedding Theorem, what does it mean for the constant to depend on the domain $\Omega$? I know the constant depends on the dimension of the domain, but when you say $C$ depends on $\Omega$, how is it deferent from the dependence of $C$ on the dimension?

Answer 1

This depends on the specific formulation of the Sobolev embedding theorem. In general for these kinds of things, you just have to go through the proof and carefully keep track of the dependence of constants.

The Sobolev embedding for $W^{1,p}(\mathbb R^n)$ can be established with respect to a constant $C(n,p).$ Usually one extends this to the case $W^{1,p}(U)$ with $U \subset \mathbb R^n$ a bounded domain with $C^1$ boundary by using the extension operator, $$ E : W^{1,p}(U) \rightarrow W^{1,p}(\mathbb R^n). $$ This gives a chain of inequalities (e.g. if $p < n$), $$ \lVert u \rVert_{W^{1,p}(U)} \leq \lVert Eu \rVert_{W^{1,p}(\mathbb R^n)} \leq C_1 \lVert Eu \rVert_{L^{p^*}(\mathbb R^n)} \leq C_2 \lVert u \rVert_{L^{p^*}(U)}. $$ We know $C_1$ depends on $n$ and $p.$ The $C_2$ dependence comes from the extension operator, which is a bit harder.

When we prove the extension theorem, we locally flatten the boundary and use a linear-type extension. The dependence on $\Omega$ comes from this flattening bit, in particular the diffeomorphisms $U \cap B_r(x_0) \rightarrow B_1^+$ and their inverses. A similar argument applies for the $p > n$ case.

In particular, this does not depend on the volume $\Omega.$