CRM: Centro De Giorgi
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A Pisan workshop in Geometric Analysis

From a higher order isoperimetric inequality for non-convex domains to an exotic functional inequality

speaker: Federico Glaudo (ETH Zurich)

abstract: One of the Alexandrov-Fenchel inequalities states that, among convex domains $K\subseteq\mathbb Rn$ with perimeter equal to the perimeter of the ball, the quantity $\int{\partial K} H$ is minimized by the ball ($H$ denotes the mean curvature of the boundary). This is a higher-order isoperimetric inequality: instead of comparing volume and perimeter, we are comparing perimeter and integral of the mean curvature. The validity of the inequality is open for mean-convex (i.e., $H\ge 0$) domains.

We will consider the inequality without the assumption of mean-convexity, but replacing $H$ with its absolute value $
$ and restricting our study to domains $K$ which are $C1$-perturbations of the ball. Under these assumptions, we will explain why the desired inequality is "morally" equivalent to the following functional inequality: given $u:\mathbb S{n-1}\to\R$ with $
{C1}$ sufficiently small, it holds $\int (\Delta u)
\nabla u
2 \le \frac{n-2}{n-1}(\sup\Delta u)\int
\nabla u
2$. Thanks to this insight, one can prove the inequality in some special cases.

Wed 13 Oct, 16:30 - 18:00, Aula Dini
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