Mini Course Part 1 of 4. The course is a very short introduction to regularity for linear elliptic pdes of second order. We start with equations with regular coefficients and the difference quotient method of Nirenberg. We then treat the case of coefficients that are merely measurable and bounded, putting forward the basics of De Giorgi-Nash-Moser theory. If time permits, we present some characterizations of Hölder spaces which are very useful in regularity theory.
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).