How to prove that $\|u\|_\infty\leq C_p\|u'\|_{L^p},\ \forall\ u\in W^{1,p},\ u(0)=u(T)$ with $\int_0^T u=0$ and $C_p=\frac{1}{2}\left[\frac{T(p-1)}{2p-1}\right]^{\frac{p-1}{p}}$?

It is easy to show that $\|u\|_\infty\leq T^{\frac{p-1}{p}}\|u'\|_{L^p}$, but $C_p$ is a sharp constant.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.