The study of the structural properties of the set of points at which a solution u of a first order Hamilton-Jacobi equation fails to be differentiable—in short, the singular set of u—has been the subject of a long term project that started fifty years ago with a seminal paper by W. Fleming. Later on, such a topic was put into the right perspective by the introduction of viscosity solutions by M.Crandall and P.-L. Lions. All these years have registered enormous progress in the comprehension of the way how singularities propagate: a fine measure theoretical analysis of the singular set was developed, the dynamics of singularities was identified to be given by generalised characteristics, deep connections with weak KAM theory by A. Fathi were pointed out, and interesting topological applications were deduced. In this talk, I will revisit some of the milestones of this theory and discuss recent developments concerning uniqueness of singular characteristics.
Piermarco Cannarsa is a Full Professor in Mathematical Analysis of the Department of Mathematics at University of Rome “Tor Vergata”. He is also the President of the Italian Mathematical Union and Italian coordinator of the International Associated Laboratory (LIA) on‘Control, Optimization, Partial Differential Equations, and Scientific Computing’(COPDESC), issued by CNRS (France), INdAM (Italy) and the Max Planck Institutes (Germany).
Piermarco Cannarsa's research interests:
1. Hamilton-Jacobi equations, optimal control, and mean field games.
2. Nonsmooth analysis.
3. Analysis and control of evolution equations.
4. Partial differential equations of parabolic type with application to climate models.