Skip to main content
European Commission logo
español español
CORDIS - Resultados de investigaciones de la UE
CORDIS

Minimal solutions to nonlinear systems of PDEs

Descripción del proyecto

Soluciones mínimas para resolver ecuaciones diferenciales parciales no lineales

En el proyecto MinSol-PDEs, financiado por las Acciones Marie Skłodowska-Curie, se llevará a cabo un estudio sistemático de soluciones mínimas para una gran clase de ecuaciones diferenciales parciales no lineales. Parte de la investigación se orientará a los problemas de transición de fase descritos por la ecuación de Allen-Cahn. La idea principal es reducir la ecuación a un sistema hamiltoniano para construir nuevas clases de soluciones mínimas y comprender las condiciones que implican la reducción de las variables. Otra parte de la investigación se centrará en la ecuación de Painlevé, que desempeña un papel crucial en áreas tan diversas como las matrices aleatorias, los sistemas integrables y la superconductividad. El objetivo principal es clasificar e investigar las soluciones mínimas de los sistemas de tipo Painlevé en dimensiones bajas.

Objetivo

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.

Régimen de financiación

MSCA-IF-EF-ST - Standard EF

Coordinador

BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS
Aportación neta de la UEn
€ 160 932,48
Dirección
AL MAZARREDO 14
48009 Bilbao
España

Ver en el mapa

Región
Noreste País Vasco Bizkaia
Tipo de actividad
Research Organisations
Enlaces
Coste total
€ 160 932,48