Embedding for homogeneous Sobolev spaces

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. 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!

• 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 – 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.
• Many thanks! I have a followup question concerning the constant's dependence on $R$, in the edited version of the question above. – Siran Victor Li Feb 22 at 3:55