Research group in partial differential equations and applications

Research Areas

The aims of the present Research Group are the study of several topics in the field of Partial Differential Equations (PDE’s) and their applications, also combined with the study of other mathematical tools to deal with some specific models. A particular emphasis is put into models that come from Physics, Mathematical Biology, and technological applications. 

We study the steady and time dependent states for several problems, mainly formulated in terms of PDE’s. We do this using analytical and numerical techniques, taking also care of the modeling process.

The particular topics in which we focus are

  • Regularity theory for linear, semilinear, and fully nonlinear elliptic and parabolic equations. We include here not only classical local PDEs, but also nonlocal or integro-differential equations with fractional diffusion. Of special interest are: nonlocal minimal surfaces and surfaces with constant nonlocal mean curvature, as well as local and nonlocal free boundary problems.
  • Qualitative properties of solutions to reaction-diffusion equations. We focus on: the regularity of stable solutions to local reaction-diffusion elliptic equations; the conjecture of de Giorgi on the Allen-Cahn equation and its connections with minimal surfaces (also in the nonlocal case); front propagation for nonlinear parabolic equations under fractional diffusion, also in heterogeneous media.

  • To study the relations between nonlocal fractional operators and Conformal Geometry. Here, fractional curvatures and the fractional Yamabe problem play a central role. Other geometric problems of our interest are: isoperimetric and Sobolev inequalities, transformations with a given Jacobian, linearization for evolution equations in infinite dimensions, and boundary problems on graphs.

  •  The analysis of several problems in mathematical physics and mathematical modelling: viscoelastic systems of wave equations, vortices in Ginzburg-Landau models, continuum chromatography, quantum cosmology, and the Dirac equation, age-estructured population approach for modeling of urban burglaries.

  • Delay equations formulation of structured population dynamics and relation with the standard partial differential formulation

  • The study of different models of cell growth and microbial interaction, using a semilinear formulation to determine existence and stability of equilibria, as well as possible periodic solutions.

  • Work on the measures-space formulation of the so-called selection-mutation equations and study the asymptotic behaviour of the time-solutions for small mutation rate.

  • The study the emergence of periodic behaviour in a model of price formation among a population of buyers and sellers

  • The study of dynamical processes defined on networks. We will focus on the introduction of behavioural responses in epidemic modelling, and on the propagation of multiple failures in communications networks.

  • Derivation and analysis of metapopulation models for epidemics with density-dependent migration and contact rates. Application of metapopulation models for the study of infectious diseases in Tropical areas.

  • To study models for multilayer networks and to develop algorithms for their generation and overlapping control.