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ANOPTSETCON

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Analysis of optimal sets and optimal constants old questions and new results

The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versio... ver más

31/07/2015

FAU

540K€

Presupuesto del proyecto: 540K€

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FRIEDRICHALEXANDERUNIVERSITAET ERLANGENNUERNB... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2015-07-31 No tenemos la información de la convocatoria

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3 Participantes

FAU

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UNIFI

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Líder del proyecto

FRIEDRICHALEXANDERUNIVERSITAET ERLANGENNUERNB... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 540K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versions of the) isoperimetric inequality and the Sobolev-like inequalities. Much is known about equality cases and best constants, but there are still many questions which seem quite natural but yet have no answer. For instance, it is not known, even in the 2-dimensional space, the answer of a question by Brezis: which set, among those with a given volume, has the biggest Sobolev-Poincaré constant for p=1? This is a very natural problem, and it appears reasonable that the optimal set should be the ball, but this has never been proved. The interest in problems like this relies not only in the extreme simplicity of the questions and in their classical flavour, but also in the new ideas and techniques which are needed to provide the answers. The main techniques that we aim to use are fine arguments of symmetrization, geometric constructions and tools from mass transportation (which is well known to be deeply connected with functional inequalities). These are the basic tools that we already used to reach, in last years, many results in a specific direction, namely the search of sharp quantitative inequalities. Our first result, together with Fusco and Maggi, showed what follows. Everybody knows that the set which minimizes the perimeter with given volume is the ball. But is it true that a set which almost minimizes the perimeter must be close to a ball? The question had been posed in the 1920's and many partial result appeared in the years. In our paper (Ann. of Math., 2007) we proved the sharp result. Many other results of this kind were obtained in last two years.

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