About the project

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

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 and Hamiltonian 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 PDE’s and free boundary problems, Morse and Critical Point theory, equivariant topology, geometric measure theory. We intend to address the following strongly interconnected themes:

A. Pattern formation in reaction-diffusion systems, phase separation and optimal partition problems , arising in multispecies and multiagent models. Includes the analysis of the interfaces between the different phases in the presence non local diffusions and interactions.

B. Shape optimization, free boundary problems , including the dynamics of nodal sets and free boundaries in evolutionary problems as well as spectral domain optimization issues.

C. Complex solutions in Celestial Mechanics and Hamiltonian PDEs , where we seek solutions with prescribed behavior to the N-body problem and study their stability properties, with the final intent of detecting the occurrence of controlled chaos. The same paradigms will be applied to the search of entire solutions of several classes of PDEs.

D. Complex domains and their effects on solutions to linear and nonlinear equations. We focus on the effects of global geometrical and topological properties, as well as on more local facts, like unique continuation and boundary effects.

E. Geometric Variational problems and concentration phenomena , as they appear in Prescribed curvature problems, Conformal Geometry, Mathematical Physics and in the study Partial Differential Equations and systems when critical nonlinearities occur or, for some limit values of a parameter, special solutions exhibiting a singular limiting behaviour appear.