Brief description

Linear and non-linear PDEs of local type were key in many settings during the 20th century. For example, the heat equation, which involves a Laplacian. This equation is linked to Brownian Motion, a continuous stochastic process. In the past decades, this idea has evolved in two directions. Brownian-like models lead to spatial exponential decay of the density, which does not capture many practical problems (e.g., in finance), and more general model with power-type decay could be introduced, for example through Lévy flights. These are related to some non-local operators like the fractional Laplacian. We will study theoretical and numerical questions related to this operator: well-posedness, homogenisation, rearrangement, approximation schemes, … Another interesting topic is Stochastic Differential Games, such as tug-of-war games. Some choices lead to the p-Laplace equation, and one of our aims is to study examples leading to non-local linear and non-linear problem. We will also study a different type of non-local problem. Considering more than one particle interacting a distance (e.g., gravitational forces, chemical interaction, …) leads to non-local PDEs. A particularly interesting set of these problems is the family known as Aggregation-Diffusion problems, in which some members of the team have been working over the last years.

 

Researchers

• David Gómez Castro. PhD Teaching Assistant (Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM (PI)
• Ángel René Arroyo García. Assistant Professor (Profesor Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM
• Jesús Ildefonso Díaz Díaz. Professor Emeritus (Catedrático Emérito), Member of the Royal Academy of Sciences (Miembro de la RAC)
• Juan Carlos Felipe Navarro. Assistant Professor (Profesor Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM

 

External Collaborators

• Gregorio Díaz Díaz (Retired Prof of Applied Mathematics, UCM)
• Jesús Hernández (Retired Prof of Mathematics, UCM)

 

Publications

• L. Boccardo, J. I. Díaz, D. Gómez-Castro. Failure of the Hopf-Oleinik Lemma for a linear elliptic problem with singular convection of non-negative divergence. Electronic Journal of Differential Equations, Vol. 2024. 2024. DOI: 10.58997/ejde.2024.13.
• J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy. Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size. Electric Journal of Differential Equations, Vol. 2024. 2024. DOI: 10.58997/ejde.2024.03.
• J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy. Aperiodical Isoperimetric Planar Homogenization with Critical Diameter: Universal Non-local Strange Term for a Dynamical Unilateral Boundary Condition. Doklady Mathematics, 109, 12–19. 2024. DOI: 10.1134/S1064562424701734.
• J. I. Díaz, J. Hernández. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions. Advances in Nonlinear Analysis, 13 (1),. 2024. DOI: 10.1515/anona-2023-0128.
• P. Bégout, J. I. Díaz. Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains. Journal of Mathematical Analysis and Applications, 538 (1). 2024. DOI: 10.1016/j.jmaa.2024.128329.
• J. Ildefonso Díaz, A. V. Podolskiy, T. A. Shaposhnikova. Unexpected regionally negative solutions of the homogenization of Poisson equation with dynamic unilateral boundary conditions: critical symmetric particles. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 118, 9. 2024. https://doi.org/10.1007/s13398-023-01503-w
• P. Bégout and J. I. Díaz, Finite time extinction for critically damped Schrodinger equation with a sublinear nonlinearity. Advances in Differential Equations, Volume 28, Numbers 3-4 (2023), 311-340. https://hal.science/hal-03805319v1

 

News

• 27 November 2023. Four IMI members and four members of its Scientific Committee are in the lists of the most highly cited scientists worldwide, published by Stanford University.  The selection is based on the top 100,000 scientists by c-score or a percentile rank of 2% or above in the sub-field. The four IMI members mentioned are Juan Luis Vázquez Suárez, Julián López-Gómez, Jesús Ildefonso Díaz Díaz and Francisco Javier Montero de Juan. Also appearing in the same lists are IMI Scientific Committee members Paul Rabinowitz, Simon Donaldson, Herbert Amann and Dikran Dikranjan.