I will start recalling the definitions and basic properties of viscosity solutions to fully nonlinear degenerate elliptic equations, in particular the comparison principles. The main goal of the course is discussing two properties of subsolutions: the Strong Maximum Principle (SMP), i.e., if a subsolution in an open connected set attains an interior maximum then it is constant, and the Liouville property, i.e., if a subsolution in the whole space is bounded form above then it is constant. They are standard results for classical solutions of linear elliptic PDEs, and many extensions are known, especially for divergence form equations. My goal is explaining how the viscosity methods allow to turn around the difficulties of non-smooth solutions, fully nonlinear equations, and their possible degeneracies.
Martino Bardi is a professor of Mathematical Analysis and the chairman of the PhD Program in Mathematical Sciences at the University of Padova, Italy. He graduated in Padova. He is coauthor with I. Capuzzo Dolcetta of the book “Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations”, reprinted by Modern Birkhauser Classics in 2008. He published over 80 scientific papers in the area of nonlinear PDEs and their applications, mainly to deterministic and stochastic optimal control and differential games. He received the Isaacs Award of the International Society for Dynamic Games in 2016. He was member of the Executive Board and Vice President of the International Society of Dynamic Games, member of the Scientific Committee of the Italian Mathematical Union (U.M.I), and he is currently in the Editorial Committee of NoDEA and in the Editorial Board of 4 other international journals.