**We are the CAPDE Millenium Nucleus Center for the Analysis of Partial Differential Equations** (from spanish *Núcleo Milenio Centro para el Análisis de Ecuaciones en Derivadas Parciales*), a joint research group in Mathematics composed by researchers from several Chilean universities and institutions.

### Main institutions in CAPDE collaboration

### Objectives of the nucleus

- To obtain substantial progress in the research program proposed in the 3-year period of the Nucleus.
- To train young researchers in the topics of the program: recruitment of new students and postdocs, and the involvement of new young, excellent researchers in PDEs that will integrate the Chilean system in the next two years , such as Claudio Muñoz (Paris, dispersive equations), Darío Valdebenito (Minneapolis, nonlinear diffusions), Gonzalo Dávila (UBC, Vancouver, fully nonlinear elliptic problems).
- Joint organization of thematic conferences and schools, and the involvement of renowned experts in a program of research visits that includes the realization of short courses of their expertise.
- In the long term, to position this group as a highly prestigious international center for nonlinear analysis.

### Lines of Research

### Singularity formation in nonlinear parabolic problems

This part of the project deals with the analysis of singularities in evolution equations of parabolic type. This area sits within a field of great scope stretching from fundamental questions in fluid dynamics, geometry and modeling of biological pattern formation. At the heart of each of the topics above lies an evolution PDE of parabolic type, with key research challenges that are remarkably similar. Each of these equations could be perhaps a law of physics, or an equation modeling an industrial or biological process. Smooth solutions to evolution geometric PDE have been extremely successful in applications to pure and applied problems. The most famous application in recent years has been the resolution of the Poincaré conjecture, which was named by the journal `Science’ as the scientific `Breakthrough of the year, 2006,’ but is considered by many to be the greatest achievement of mathematics in the past 100 years.

### Analysis of variational models in mechanics by singular perturbations

In 1998, Francfort and Marigo proposed a variational model for fracture mechanics, in analogy to the well-known model by Mumford & Shah (1989) in computational image segmentation. In this model a fracture is conceived as the discontinuity set of the displacement function of the material. The propagation of the fracture is associated to the minimization of a functional among configurations where these discontinuities are allowed. For the theoretical model of cavitation and fracture for ductile materials and incompressible polymers, Henao and collaborators have formulated and analyzed a relaxed energy functional in which the test functions are smooth with large gradients on a thick interface, interpreted as the “damage zone”. This approximation theory, in the spirit of that by Ambrosio & Tortorelli (1990) for the Mumford & Shah functional, is crucial for the numerical simulation of fractures. The relaxed energy involved has strong analogies with the Allen-Cahn and Ginzburg-Landau functionals in the theories of phase transitions and superconductivity, which have been broadly treated in the PDE and Calculus of Variations literature. On the other hand, the relaxed functional in fracture mechanics involves heavy technical challenges, such as lack of convexity in the gradient terms and its vector-valued character.

### Fractional nonlinear elliptic equations

Singular integrals and nonlocal (especially fractional) operators are a classical topic in harmonic analysis and operator theory and they are now becoming impressively fashionable because of their connection with many real-world phenomena. Our main purpose in this part of the project is to develop and apply singular perturbation techniques to questions involving fractional elliptic operators.