Let $B_R$ be a ball of radius $R$ in $\mathbb{R}^d$. Under what conditions does the Sobolev embedding $\dot{W}^{1,p}_0(B_R) \hookrightarrow L^{p^*}(B_R)$ holds, where $p^* = \frac{np}{n-p}$?

Here $\dot{W}^{1,p}_0(B_R)$ is the homogeneous Sobolev space, and $1<p<\infty$. Thanks!

In the answer, the following version of the Poincare inequality is stated: \begin{equation} \int_\Omega |u|^p\,dx \leq C(\Omega,p,n)\int_\Omega |\nabla u|^p\,dx, \end{equation} if $\Omega$ is a bounded domain in $\mathbb{R}^n$. It manifests that $\dot{W}^{1,p}(\Omega) = W^{1,p}(\Omega)$, of which I am convinced.

Now, in the case $\Omega = B_R$, I am interested in getting explicit dependence of the constant with respect to $R$. For example, is it possible to get an estimate of the form \begin{equation} \bigg(\int_{B_R} |u|^{p^*}\,dx\bigg)^{1/p^*} \leq C(p,n) R^\beta\bigg(\int_{B_R} |\nabla u|^p\,dx\bigg)^{1/p}, \end{equation} where $u \in W^{1,p}(B_R)$ with $p \in ]1,n[$ and $\beta \in \mathbb{R}$ is some scaling index?

(I know that, roughly speaking, for Poincare inequality over $B_R$ we will get an $R$ in the constant, while the Sobolev inequality corresponding to $W^{1,p}(B_R) \hookrightarrow L^{p^*}(B_R)$ should be independent of $R$. But, in the case without the average term, I wonder if we can still get something neat?)

Thank you!

  • $\begingroup$ Assuming bounded domain (not necessarily a ball), $W^{1,p}$ is compactly embedded in $L^{q}$ for $1\leq p < d$ and $1 \leq q < p^{*}$ with $d$ being the dimension of the space. (Rellich-Kondrachov Compactness Theorem) You can read L.C. Evans Partial Differential Equations Book for the details and other useful Sobolev Embeddings $\endgroup$ – Evan William Chandra Feb 21 at 6:20

If $\Omega$ is a bounded domain, then we have the following Poincaré inequality: There is a constant $C>0$ such that $$ \int_{\Omega}|u|^p\leq C\int_{\Omega}|\nabla u|^p $$ for all $u\in C_c^\infty(\Omega)$. In other words, $\|\nabla\cdot\|_p$ and $\|\cdot\|_p+\|\nabla\cdot\|_p$ are equivalent norms on $C_c^\infty(\Omega)$ and thus $W^{1,p}_0(\Omega)=\dot{W}^{1,p}_0(\Omega)$ with equivalent norms. Hence you can apply the classical embedding theorems for Sobolev spaces that Evan William Chandra mentioned in the comments to get the embedding you want.

  • $\begingroup$ Many thanks! I have a followup question concerning the constant's dependence on $R$, in the edited version of the question above. $\endgroup$ – Siran Victor Li Feb 22 at 3:55
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    $\begingroup$ We have a "one question at a time" policy here at stackexchange. If you have a follow-up question, better post a new one instead of modifying an existing one. $\endgroup$ – MaoWao Feb 22 at 10:16

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