This subproject concerns the analysis of Partial Differential Equations and their applications. Our main topics of interest are: reactiondiffusion and integro-differential equations, as well as PDEs in some geometric problems. Our group has been pioneer in a very active line of research started worldwide about twelve years ago: nonlinear elliptic and parabolic equations with fractional diffusion associated to integro-differential operators. Among our most relevant results (some published in the two most cited 2014 articles in all areas of Mathematics) are: the regularity theory for nonlinear integro-differential equations (including free boundary problems) and the discovery of the Pohozaev identity for the fractional Laplacian. Our main goals are: 1) Regularity theory for nonlinear elliptic and parabolic integro-differential equations, including the free boundaries in obstacle problems for general integro-differential operators. 2) We will continue developing the results on boundedness of stable solutions to local and nonlocal reaction-diffusion elliptic equations. 3) Study of nonlocal minimal surfaces and surfaces with constant nonlocal mean curvature. We will optimize the gradient estimate for nonlocal minimal graphs, an open problem recently solved by the group (but not yet announced). We will develop the results already found on the classification of nonlocal minimal cones (the main open problem in this area, of major implications also for the next goal). The study of periodic solutions to fractional problems will also be pursued (such as some periodic nonlocal minimal surfaces already found by the group). 4) Classification of stable solutions to the classical and the fractional Allen-Cahn equation, pushing our results towards the total resolution of the conjecture of De Giorgi on 1D symmetry of solutions in both cases. 5) Other geometric problems: Sobolev and isoperimetric inequalities for classical or fractional perimeters, maps with a given Jacobian, linearization for evolution equations in infinite dimensions, and boundary value problems on graphs. 6) Analysis of several problems of physics and mathematical modeling: viscoelastic systems of wave equations, vortices in the Ginzburg- Landau models, continuous chromatography, quantum cosmology, and the Dirac equation.
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia