We are interested in the effect of the nonlinearities on the emergence of non trivial patterns in several differential models arising in physics and other sciences. Such self organized structures correspond to selected solutions of the differential problem, possessing some special symmetries or shadowing particular shapes. We wish to understand the main analytical mechanisms involved in this process in terms of the common variational structure of the problems . A feature of this project rests indeed in the interchange of attack strategies between different specific applications in the fields of partial differential equations and systems. There is a remarkable unity in methodology across the different parts of the project. On the other hand, all the proposed issues must be addressed in interdisciplinary spirit and require expertise in several fields of mathematics: variational and topological methods, qualitative and regularity theory for partial differential equations and free boundary problems, Morse and critical point theory. We intend to address the following strongly interconnected themes:
(a) Pattern formation in reaction-diffusion systems and phase arising in multispecies and multiagent models.
(b) The effect of the geometry of the domain on the shape of the solutions to linear and nonlinear.
(c) Singular perturbation problems and concentration phenomena, as they appear in the study of partial differential equations and systems when, for some limiting values of a parameter, special solutions exhibiting a singular limiting behaviour appear.
This proposal aims at approaching all these different issues with the same basic methodology which relies on their common perturbative and/or variational structure. This requires expertise in different fields of mathematical analysis, Riemannian geometry and partial differential equations.
The project is dedicated to different problems inspired by models from Theoretical Physics, Materials Science, Biology and Social Sciences (e.g. the optimal partition problems describe the dynamics of populations of competing species or cells). Thus, the project will have an impact on all these fields through the analysis of the mathematical aspects of these models. The focus of the project is on the theoretical aspects of the problems and the models described. The methods and tools that we will use and develop come from the field of Mathematical Analysis and Riemannian Geometry, more specifically, the analysis of nonlinear (partial and ordinary) differential equations and systems, and one of our main objectives is to contribute to the evolution of this field. Still, we believe that the methods we aim to develop and the results we aim to obtain go beyond the specific problems described in the present project and will have an impact on a larger scale. In particular, some of the phenomena we study (for instance, spatial segregation, concentration phenomena, pattern formation, overdetermined problems) are common for many different problems from the fields of Analysis of PDEs, Calculus of Variations, Geometric Analysis and Mathematical Physics.
The project gathers together a team of Italian mathematicians recognized on an international level. The team is composed of specialists with competences in several different fields of Analysis and Geometry. The different tasks of the project are dedicated to problems coming from all these fields but which require similar analytical tools and methodology. Thus, the present project will contribute to establish new connections between these fields and to stimulate new discussions and collaborations. We expect that this traversal project will have a positive impact to the senior researchers, as well as to the Master and PhD students, and postdocs working in the field. This project will give us the possibility to organize synergic scientific events and to set up collaborations with scientists from different fields and disciplines. The numerous collaborations of the members of the team and the relevance of the research themes will highlight the role of La Sapienza in the European and the international research.
The overall objectives will therefore be mainly the following: ¿ to provide effective responses to a series of open problems in this research area that are of current interest both from the methodological point of view and from that of applications;
1) to disseminate the results through the organization of scientific events (workshops and schools) on an annual basis;
2) to invest in the training of talented young people for which we will dedicate a significant part of the requested funding;
3) to participate in the international scientific debate and competition through the participation in international conferences;
4) to increase the already existing international collaborations, establish new ones and contribute to the exchange of information and expertise, this increasing the effective collaboration within the group itself;
5) to provide to participants in this project the opportunity to carry out their research in the present context of research funds shortage and acquire the resources necessary to cover their basic academic needs.