The Michael-Simon Allard inequality

The Michael-Simon-Allard inequality, proven first independently by Allard and Michael-Simon is a generalization of the Sobolev (respectively isoperimetric) inequality to manifolds. It states the following: There is a universal constant \(C\) depending only on the dimension \(n\) and the exponent \(p,\) such that for any \(n\)-dimensional submanifold \(M\subset\mathbb{R}^{n+1}\) one has

\(\|u\|_{L^{p^{\ast}}(M)}\leq C\left(\int_M|\nabla u|^p+\int_M|u|^p|H|^p\right)^{\frac{1}{p}}\quad\forall\, u\in C_c^{\infty}(M),\)

where

\(p^{\ast}=\frac{np}{n-p},\quad \text{ and $H$ is the mean curvature of $M$}\)

There is still no other proof known, except in dimension \(2.\) Neither is the best constant known, not even for \(p=1\) (analogue of isoperimetric inequality). I will give an overview on this inequality: some ideas of the proof, connection to minimal surfaces and an application to mean curvature flow. If time permits I will talk about a first result on this type of inequality in the nonlocal setting by Cabr\'e and Cozzi.