The MichaelSimon Allard inequality
Speaker 
Dr. Gyula Csato, Universitat Politecnica de Catalunya, Spain


When 
Feb 05, 2019
from 04:00 PM to 05:00 PM 
Where  LH006 
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The MichaelSimonAllard inequality, proven first independently by Allard and MichaelSimon 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
\(\u\_{L^{p^{\ast}}(M)}\leq C\left(\int_M\nabla u^p+\int_Mu^pH^p\right)^{\frac{1}{p}}\quad\forall\, u\in C_c^{\infty}(M),\)
where
\(p^{\ast}=\frac{np}{np},\quad \text{ and $H$ is the mean curvature of $M$}\)
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.