Apresentações de matemática inspirada pela obra de Luis A Caffarelli, ganhador do Prêmio Abel 2023.
Quinta-feira, 25/05/2023
Departamento de Matemática, PUC-Rio
Rua Marquês de São Vicente, 225, Gávea
Rio de Janeiro, RJ - Brasil
Evento realizado em modo remoto.
Cartaz disponível aqui (em formato docx e formato pdf).
Programa
Hora: 14h00-14h50
Título: Global estimates for supersolutions and solutions of elliptic PDE
Palestrante: Boyan Sirakov
Palestra: Gravação disponível aqui
Resumo: The theory of elliptic PDE in nondivergence form and fully nonlinear equations of Isaacs type experienced tremendous growth in the 80s of the last century, thanks to the fundamental apriori and regularity estimates of N. Krylov, M. Safonov, and L. Caffarelli. In particular, they created a novel understanding of the connection between the maximum principle, its quantified forms (growth lemmas) and regularity. We review this already classical theory and Luis' unique contributions to its fundamentals. Then we proceed to describe some recent developments with particular emphasis on boundary and global regularity and estimates.
Hora: 15h00-15h50
Título: Approximation methods and consequences to regularity theory
Palestrante: Edgard Pimentel
Palestra: Gravação disponível aqui
Resumo: The regularity theory for fully nonlinear equations has benefitted tremendously from several, various ideas of Luis Caffarelli. In recent years, we have grown very fond of the following observation: given an equation whose regularity is unknown, one could approximate it by a different PDE with well-understood regularity properties. Ideally, one could transmit information from the latter to the former. The techniques stemming from this simple idea have played a major role in the analysis of PDE, with consequences for a wide range of problems. We discuss the method, its intuition and the way it has been implemented in the literature. Through a few examples, we highlight its power, as well as its limitations. The latter indicates further directions of research, which we briefly present.
Hora: 16h00-16h50
Título: Plateau problems and the Caffarelli-Nirenberg-Spruck method
Palestrante: Graham Smith
Palestra: Gravação disponível aqui
Resumo: In a remarkable series of papers Caffarelli-Nirenberg-Spruck (and also Kohn) developed a powerful technique for the treatment of boundary value problems for totally non-linear partial differential equations. This technique was adapted by Rosenberg-Spruck to the study of Plateau problems for hypersurfaces of constant extrinsic curvature in euclidean space, culminating in the remarkable existence results of Guan-Spruck and Trudinger-Wang. In this talk, we recall the simple ideas underlying this technique, we review its later developments, and we describe our own recent contributions to the field.