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Partial Differential Equations: reaction-diffusion, integro-differential, and geometric problems

Total activity: 12
Type of activity
Competitive project
Funding entity
Funding entity code
95.711,00 €
Start date
End date
This project is concerned with the analysis of partial differential equations and their applications. Our main areas of interest are: reactiondiffusion
equations, fractional diffusion and integro-differential equations, PDEs in some geometric problems (especially in conformal
geometry and with fractional diffusion), PDEs in some problems of mathematical physics and mathematical modelling.
Our group has been pioneer in a very active line of research worldwide started ten years ago: nonlinear elliptic and parabolic equations
with fractional diffusion. It deals with integro-differential operators (such as the fractional Laplacian) which are, therefore, nonlocal. Among
our main results are: the exponential rate of invasion of the KPP fronts with fractional diffusion, the discovery of the Pohozaev identity for
the fractional Laplacian, and the regularity theory for fully nonlinear integro-differential equations.
Our main goals within the field of reaction-diffusion equations, integro-differential operators, and PDEs in geometric problems are:
1) The regularity theory for elliptic and parabolic linear, semilinear, and fully nonlinear integro-differential equations. We include here the
regularity for general stable operators and the recent nonlocal Monge-Ampère equations. We will also continue to develop the regularity
theory for stable solutions of local elliptic reaction-diffusion equations.
2) The propagation of fronts for nonlinear parabolic equations under fractional diffusion, also in heterogeneous media.
3) To study the relationship between nonlocal fractional operators and conformal geometry. Here, fractional curvatures and the fractional
Yamabe problem play a central role.
4) Local and non-local isoperimetric inequalities. Emphasis will be placed on perimeters with weights or densities, as well as the recent
notion of nonlocal minimal surfaces. Other geometric problems of our interest are: transformations with prescribed Jacobian, linearization
of evolution equations in infinite dimensions, and boundary value problems on graphs.
Our goals in PDEs of mathematical physics and mathematical modeling are:
5) Study of viscoelastic systems of waves equations, continuum chromatography, quantum cosmology, and the Dirac equation.
Adm. Estat
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Resoluton year
Funcding program
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia
Funding subprogram
Subprograma Estatal de Generación de Conocimiento
Funding call
Excelencia: Proyectos I+D
Grant institution
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco


Scientific and technological production

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