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Stable interfaces: phase transitions, minimal surfaces, and free boundaries

Description du projet

Voir les relations stables sous un nouveau jour avec l’aide des mathématiques avancées

Lorsque Joseph Louis Lagrange a dérivé en 1762 l’équation des surfaces minimales, il ne pouvait pas soupçonner les liens profonds de celle-ci avec la théorie des transitions de phase, qui ne sera développée que deux siècles plus tard. Nous sommes tous habitués à certaines transitions de phase, comme la fonte de la glace dans l’eau. Mais il en existe des dizaines qui sont essentielles à l’activité quotidienne et à l’innovation humaine: les métaux dans un alliage, la supraconductivité, les limites de décision dans la finance, les cristaux liquides, la combustion, la conception optimale des isolateurs, et tant d’autres. Malgré la nature et l’importance omniprésentes des transitions de phase, notre capacité à analyser mathématiquement leurs comportements stables est étonnamment limitée. Le projet StableIF, financé par l’UE, combine des avancées récentes avec les outils classiques de la théorie des surfaces minimales afin de développer l’analyse mathématique qui permettra d’améliorer notre compréhension des transitions de phase stables.

Objectif

One of the main drivers of development for the theory of nonlinear elliptic PDE during the second half of the XX century has been the mathematical analysis of physical models for “inter- faces”. Depending on the specific model, these “interfaces” are called minimal surfaces, phase transitions, free boundaries, etc. These models are very important in applications and, due to their strong geometric content and the interdisciplinary methods required for their study, also from a “pure mathematics” perspective. One of the simplest semilinear PDE exhibiting an interface is the classical Allen-Cahn equation. Originally proposed as a model for metal alloys, it gained mathematical notoriety due to its deep connection with the minimal surface equation and many other important PDE. It is very related to the Cahn-Hiliard equation (phase separation in binary fluids), to the Peierls-Nabarro equation (crystal dislocations), and to the Ginzburg-Landau theory (phase transitions, super-conductivity). In addition, it has similarities with other important models such as Bernoulli’s free boundary problem (flame propagation and shape optimization) or the Eriksen-Leslie system (liquid crystals). In the last four decades, outstanding works led to a very deep understanding of the structure of (absolute) energy minimizers for most of the previous models. Still, up to very few exceptions, almost nothing is known today on the structure of stable solutions —i.e. (roughly speaking) minimizers with respect to sufficiently small perturbations. Since stable solutions are “the ones observable in Nature”, their understanding is a fundamental question. Even though it is a very challenging mathematical problem, all the new analysis tools developed in the last decades plus some recent progress give us now an excellent opportunity to address it. In three words, the very ambitious goal of this ERC project is to “understand stable interfaces”.

Régime de financement

ERC-STG - Starting Grant

Institution d’accueil

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Contribution nette de l'UE
€ 1 348 125,00
Adresse
Raemistrasse 101
8092 Zuerich
Suisse

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Région
Schweiz/Suisse/Svizzera Zürich Zürich
Type d’activité
Higher or Secondary Education Establishments
Liens
Coût total
€ 1 348 125,00

Bénéficiaires (2)