# Estimates for the Sobolev inequality

How to prove the Sobolev estimate:

If $\Omega$ is a bounded open subset of $\mathbb R^N$, then for any $q>1$ $$\|u\|_{L^{q}(\Omega)} \leq C|\Omega|^{1/q} q^{1- 1/N}\| \nabla u \|_{L^{N} (\Omega)} ; \forall u\in W^{1, N}_{0} (\Omega),$$
where the constant $C>0$ depends on $N$ only.

• You have both $n$ and $N$. are they the same? Commented Jun 25, 2018 at 5:35
• Yes, they are the same. Thanks for pointing me out. Commented Jun 25, 2018 at 5:59
• I believe the power of q is correct. But the constant should not depend on u. Commented Jun 25, 2018 at 21:32
• Check if my editing is correct. Commented Jun 26, 2018 at 3:50
• This follows from Moser--Trudinger inequality. Commented Jun 26, 2018 at 6:56

## 1 Answer

If $$p = \frac{qN}{q+N},$$ then $$q = \frac{pN}{N-p}$$ Therefore, since, $u$ is compactly supported in $\Omega$, by the sharp Sobolev inequality of Aubin and Talenti (see Talenti's paper) and the Holder inequality, $$\|u\|_{L^q(\Omega)} \le C(N,p)\|\nabla u\|_{L^p(\Omega)} \le C(N,p)|\Omega|^{\frac{1}{p}-\frac{1}{N}}\|\nabla u\|_{L^N(\Omega)} = C(N,p)|\Omega|^{\frac{1}{q}}\|\nabla u\|_{L^N(\Omega)},$$ where $$C(N,p) = \pi^{-\frac{1}{2}}N^{-1/p}\left(\frac{p-1}{N-p}\right)^{1-1/p} \left\{\frac{\Gamma(1+N/2)\Gamma(N)}{\Gamma(N/p)\Gamma(1+N-N/p)}\right\}^{1/N}$$ It is now straightforward to check that if $q \rightarrow \infty$, then $p\rightarrow N$ and $C(N,p) \sim q^{1-1/N}$.