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# The Michael-Simon Allard inequality

Dr. Gyula Csato, Universitat Politecnica de Catalunya, Spain
 Speaker Dr. Gyula Csato, Universitat Politecnica de Catalunya, Spain Feb 05, 2019 from 04:00 PM to 05:00 PM LH006 vCal iCal

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.

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