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COMPAT Report Summary

Project ID: 339958
Funded under: FP7-IDEAS-ERC
Country: Italy

Mid-Term Report Summary - COMPAT (Complex Patterns for Strongly Interacting Dynamical Systems)

This project focuses on nontrivial solutions of systems of differential equations characterized by strongly nonlinear interactions, with a special emphasis on the two complementary chapters of Complex dynamics for the classical N-body problem, featuring singular attractive interaction forces, and Pattern formation through spatial segregation, with mostly repulsive interactions.
During the first term of the project, we have pursued our aim of understanding the effect of such strong/singular nonlinearities on the emergence of non trivial self-organized structures in a methodologically unitary perspective, with a continuous exchange of strategies between the field of Hamiltonian systems in finite dimension and that of partial differential equations and systems in infinite dimension. The emerging patterns correspond to selected solutions of the differential system possessing special symmetries or shadowing particular shapes. In all the cases we consider, the configuration space is typically multi-dimensional or even infinite-dimensional, the equations are very far from being solved explicitly, and can not be seen in any extent as perturbations of simpler (e.g. integrable) systems.

The main results we have achieved during the first period can be summarized as follows:
• discovery of new entire solutions shadowing particular shapes and related with suitable spectral problems and/or optimal partition problems;
• unconventional classification of entire solutions and entire limiting profiles in terms of topological and/or analytical invariants (e.g. Maslov or Morse index, limiting Almgren’s frequency, degree of an algebraic growth rate, symmetry groups);
• full development of the asymptotic analysis of collision/ejection trajectories and of strongly competing systems in connection with the segregation phenomenon;
• full description of the geometry of of the nodal set of optimal partition problems involving higher eigenvalues and that of eigenfunctions of single or multipolar singular magnetic Schrödinger operators.

Besides the obvious field of Celestial Mechanics and dynamical astronomy, the main areas of application of our results arise from the field of dynamics of competing biological species and other relevant physical phenomena, among which the phase segregation in the study of multicomponent Bose-Einstein condensates. New connections have been foreseen with the field of differential game theory (Mean Field Games). We addressed our researches in interdisciplinary spirit, exploiting several mathematical theories: Calculus of Variations, Equivariant Topology, Morse and Critical Point theory, qualitative and regularity theory for elliptic, parabolic and hyperbolic PDE's and free boundary problems. When needed, we also intend to address the problems with the aid of numerical and computer assisted methods. The team has been reinforced in order to largely supply all the needed expertise. Our main results have been published in a series of papers in the major international mathematical journals. Full text access to all our papers is provided through the project website:

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