A proof of the C^p'-regularity conjecture in the plane

Overview

Abstract

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of degenerate elliptic type, which yields a precise control on the growth rate of solutions near their set of critical points. We then apply this new tool to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C^p'=C^{1,1/p-1}, an optimal regularity result. This is joint work with Eduardo Teixeira (University of Central Florida, USA) and Damião Araújo (Universidade Federal da Paraíba, Brazil). 

Brief Biography

José Miguel Urbano is a Professor of Mathematics at the University of Coimbra (Portugal). He holds a PhD from the University of Lisbon and did a postdoc at Northwestern University in Chicago. He is the author of the book "The Method of Intrinsic Scaling" and of over 60 scientific papers in the area of Nonlinear Partial Differential Equations. He has supervised four PhD students and ten postdoctoral fellows. 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 taught short courses at IMPA (Rio de Janeiro, Brazil), the University of Florence (Italy), Aalto University (Finland), the Federal University of Ceará (Fortaleza, Brazil), 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.