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Minimal solutions to nonlinear systems of PDEs

Description du projet

Solutions minimales pour la résolution d’équations aux dérivées partielles non linéaires

Financé par le programme Actions Marie Skłodowska-Curie, le projet MinSol-PDEs mènera une étude systématique des solutions minimales pour une grande classe d’équations aux dérivées partielles non linéaires. Une partie de la recherche sera orientée vers les problèmes de transition de phase décrits par l’équation d’Allen‑Cahn. L’idée principale est de réduire l’équation à un système hamiltonien afin de construire de nouvelles classes de solutions minimales et de comprendre les conditions impliquant la réduction des variables. Une autre partie de la recherche portera sur l’équation de Painlevé, qui joue un rôle crucial dans des domaines aussi divers que les matrices aléatoires, les systèmes intégrables et la supraconductivité. L’objectif principal est de classer et d’étudier les solutions minimales des systèmes de type Painlevé en basse dimension.

Objectif

The aim of this proposal is to provide a systematic study of minimal solutions for a large class of nonlinear systems of PDE. Namely we will construct minimal solutions with predefined characteristics and investigate their qualitative properties, addressing the fundamental challenges that appear in the case of systems and which cannot be tackled with tools from the scalar case.


The first part focuses on phase transition problems described by the Allen-Cahn system. This is a hot and difficult topic linking PDE with the theory of minimal surfaces. The main idea is to reduce the Allen-Cahn system to a Hamiltonian system in order to construct new classes of minimal solutions, and understand the conditions implying the reduction of variables (vector analog of the celebrated De Giorgi conjecture).

In the second part, our focus is on the Painlevé equation which plays a crucial role in areas as diverse as random matrices, integrable systems, and superconductivity. The objective is to classify and investigate the minimal solutions of Painlevé-type systems in low dimensions. These have direct applications in the study of vortices in liquid crystals and Bose-Einstein condensates. The proposed approach connects the Painlevé equation with a singular problem, easier to study.

The fellow has a strong research record on the Allen-Cahn system (a book + 6 papers), and has also worked on the Ginzburg-Landau model of liquid crystals. On the one hand, he will develop his own innovative approaches to the proposed problems, and transfer his expertise to the host. On the other hand, at BCAM and through a secondment, he will link his previous research on liquid crystals to other alternative models (for which the supervisor is a world-leading expert), and to the theory of Bose-Einstein condensates. He will also acquire new skills in simulation and computation. The achievement of this project will reinforce Fellow's reputation and support him in obtaining a strong academic position.

Régime de financement

MSCA-IF-EF-ST - Standard EF

Coordinateur

BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS
Contribution nette de l'UE
€ 160 932,48
Adresse
AL MAZARREDO 14
48009 Bilbao
Espagne

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Région
Noreste País Vasco Bizkaia
Type d’activité
Research Organisations
Liens
Coût total
€ 160 932,48