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.
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.
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.