Abstract

The mini-course is an introduction to the analysis of infinity−harmonic functions, a subject that grew mature in recent years in the field of nonlinear partial differential equations. The material covered ranges from the Lipschitz extension problem to questions of existence, uniqueness and regularity for infinity−harmonic functions. A rigorous and detailed analysis of the equivalence between being absolutely minimising Lipschitz, enjoying comparison with cones and solving the infinity–Laplace equation in the viscosity sense is the backbone of the course. A few regularity results (including the Harnack inequality and the local Lipschitz continuity) and an easy proof, due to Armstrong and Smart, of the celebrated uniqueness theorem of Jensen complete the course.

Brief Biography

José Miguel Urbano is a Professor of Mathematics at the University of Coimbra. 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 50 scientific papers in the area of Nonlinear Partial Differential Equations. He has supervised four PhD students and ten postdoctoral fellows and is an associate editor of the journal Nonlinear Analysis. 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).