Singular and degenerate partial differential equations are unavoidable in the modelling of several phenomena, from phase transitions to flows in porous media or chemotaxis. They encompass a crucial issue in the analysis of pdes, namely wether we can still derive analytical estimates when the crucial algebraic assumption of ellipticity collapses. We provide a broad overview of qualitative versus quantitative regularity estimates for solutions of these equations, introducing the method of intrinsic scaling and deriving sharp estimates by means of geometric tangential analysis. We discuss, in particular, recent results concerning the Stefan problem, the parabolic p-Poisson equation and the porous medium equation.
José Miguel Urbano is a Professor of Mathematics at the University of Coimbra (Portugal). He studied in Coimbra, Ghent (Erasmus, 1992), Paris (École Polytechnique, 1995), Lisbon (PhD, 1999) and Chicago (postdoc, Northwestern University). He is the author of the book "The Method of Intrinsic Scaling" and of over 60 scientific publications in the area of Nonlinear Partial Differential Equations. He was a member of the Scientific Council for the Exact Sciences and Engineering of the Portuguese Science Foundation (FCT) and has served as evaluator of grants and research projects for the EU (Marie-Curie Fellowships), ERC (Starting Grants), the Academy of Finland, the Latvian Council of Science and FCT. He has supervised six PhD students and ten postdoctoral fellows and has taught short courses at IMPA (Rio de Janeiro, Brazil), the University of Florence (Italy), Aalto University (Finland), KAUST (Saudi Arabia) and Seoul National University (South Korea). He is a Corresponding Academician of the Sciences Class (Mathematics Section) of the Lisbon Academy of Sciences.