MinSol-PDEs · Minimal solutions to nonlinear systems of PDEs
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2019-12-01 → 2022-04-01
- EU contribution
- €160,932
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
Lines connect the coordinator with its partners.
Results in brief
Minimal solutions to nonlinear systems of PDEs
Many physical phenomena are ruled by the principle of least action, according to which the temporal evolution of a physical system tends to minimize its "energy". For instance, "the catenoid", the curve that a hanging chain or cable assumes under its own weight, minimizes its potential energy. In my project I studied minimization problems from the mathematical viewpoint. On the one hand, I developed the mathematical theory of these minimal solutions which are also related to the geometry of minimal surfaces. The problem is to classify and describe the solutions (or the surfaces) whose local deformations result in an increase of their energy (or area). On the other hand, my research has a direct application in the Physics of liquid crystals. The objective is to model the orientation of the molecules in a liquid crystal film illuminated by a laser light, and understand the structure of the vortices that are created. Manipulating light vortices has technological applications in the areas of quantum computing, telecommunications and astronomy (improvement of astronomical images, detection of exoplanets).
Data: CORDIS, © European Union
Project objective
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.
Original text from CORDIS.
Participants
- BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS · BilbaoCoordinatorSpain
Links
Data: CORDIS, © European Union