17-28 June 2024

Regularity theory studies how the structure of a given PDE affects the nature of its solutions. Typically, one considers a weak notion of solutions and supposes that a merely continuous function solves a given partial differential equation. The fundamental question in regularity theory is the following: does the structure of the PDE imply the solution is more regular? A paramount example concerns the solutions to the Laplace equation. A merely continuous function that solves the Laplace equation is indeed analytic.

The course of Edgard A. Pimentel (University of Coimbra, LMS Distinguished Visiting Fellow at ICMU) focuses on regularity theory for viscosity solutions to fully nonlinear equations, both in the elliptic and the degenerate cases. We detail a set of methods and techniques known as approximation methods. This class of techniques was introduced by Luis Caffarelli in the late ’80s and recently has been advanced to cover various models. Our goal is to discuss the basics of the regularity theory for viscosity solutions (including the breakthroughs in the Krylov-Safonov and the Evans-Krylov theorems), put forward the seminal contribution due to Caffarelli, and present a panorama of recent developments.

Edgard A. Pimentel is an LMS Distinguished Visitng Fellow at ICMU. He completed his PhD in 2013 at the University of Lisbon (Portugal) and obtained his Habilitation in 2016 at the University of São Paulo (Brazil). An Associate Fellow of the International Centre for Theoretical Physics in Trieste, Professor Pimentel was elected an Affiliate Member of the Brazilian Academy of Sciences in 2019. With previous professorial positions in São Paulo and Rio de Janeiro, Edgard is currently an assistant professor and a researcher at the Centre for Mathematics (CMUC) at the University of Coimbra in Portugal.

The hosting Ukrainian scientist is Professor Volodymyr Mikhailets, the Head of the Laboratory of Partial Differential Equations at the Institute of Mathematics of the National Academy of Sciences of Ukraine.


Please register here to take part in the course


ICMU Working Space

Kyiv School of Economics

Mykoly Shpaka str. 3, Kyiv