CORDIS
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CORDIS

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

Projektinformationen

ID Finanzhilfevereinbarung: 258685

Status

Abgeschlossenes Projekt

  • Startdatum

    1 August 2010

  • Enddatum

    31 Juli 2015

Finanziert unter:

FP7-IDEAS-ERC

  • Gesamtbudget:

    € 540 000

  • EU-Beitrag

    € 540 000

Veranstaltet durch:

FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN-NUERNBERG

Deutschland

Ziel

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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Hauptforscher

Aldo Pratelli (Prof.)

Gastgebende Einrichtung

FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN-NUERNBERG

Adresse

Schlossplatz 4
91054 Erlangen

Deutschland

Aktivitätstyp

Higher or Secondary Education Establishments

EU-Beitrag

€ 392 044

Hauptforscher

Aldo Pratelli (Prof.)

Kontakt Verwaltung

Kathrin Linz-Dinchel (Ms.)

Begünstigte (3)

FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN-NUERNBERG

Deutschland

EU-Beitrag

€ 392 044

UNIVERSITA DEGLI STUDI DI FIRENZE

Italien

EU-Beitrag

€ 15 077

UNIVERSITA DEGLI STUDI DI PAVIA

Italien

EU-Beitrag

€ 132 879

Projektinformationen

ID Finanzhilfevereinbarung: 258685

Status

Abgeschlossenes Projekt

  • Startdatum

    1 August 2010

  • Enddatum

    31 Juli 2015

Finanziert unter:

FP7-IDEAS-ERC

  • Gesamtbudget:

    € 540 000

  • EU-Beitrag

    € 540 000

Veranstaltet durch:

FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN-NUERNBERG

Deutschland