Differential equations

Cluster

Mathematical Analysis

Description

The interest is focused on problems such as the existence and non-existence, multiplicity, well-posedness and qualitative properties of solutions to differential equations (to ordinary and partial derivatives) under different boundary or initial conditions. There is a special interest in those equations having direct applications in mathematical physics, differential geometry, and population dynamics.

The techniques employed range from the variational ones and those of the critical point theory to those generally described as topological methods (such as, to make just a few examples, the various notions of topological degree and the fixed-point theorems such as Poincaré-Birkhoff), and including all the tools supplied by functional analysis. The extension of known techniques and the implementation of new techniques foe the study of solutions of differential equations is, in itself, one of the interests of the research group.

As to differential equations to partial derivatives, elliptic and parabolic problems are taken into account that are associate with potential and hamiltonian types of systems, with particular attention being paid to non-compact problems and problems having a degenerate or singular character, pertaining to some systems that are relevant for applications. Other fields of interest concern the study of systems of conservation laws, both with regard to the general theory (existence, unicity of solutions, well-posedness), and to the applications, for instance gas kinetics, fluid dynamics or problems related with network traffic. The connections with the study of models describing the synchronisation behaviour of coupled oscillators also appear promising.

Within the field of ordinary differential equations the interest is about, just to make a few non-exhaustive examples, hamiltonian systems, their periodic solutions, and the study of non-local conserved quantities for variational and non-variational Lagrangian systems, in the spirit of Noether's theorem, with applications regarding problems in mechanics.

Research subjects

Variational methods and topological methods for boundary problems
Elliptic and parabolic problems: non-compact problems and degenerate or singular problems
Hyperbolic systems of conservation laws
Hamiltonian systems and periodic solutions
Non-local conserved quantities for lagrangian systems
Qualitative properties of solutions of PDE