In this expository talk, I will first explain the usefulness of a log change of variable for studying certain PDE, and will then recall the connections with classical ODE theory for Riccati equations. These relationships are interesting since the second derivatives of solutions to Hamilton-Jacobi PDE formally solve a system of Riccati equations. This last insight leads to some conjectures for weak KAM theory, for which I will provide some heuristic calculations.

Lawrence Craig Evans got his PhD in 1975 at the University of California, Los Angeles, under the supervision of Mike Crandall. He held position at the Universities of Kentucky and Maryland, and is professor at the University of California in Berkeley since 1989. He gave fundamental contributions to the theory of Partial Differential Equations, in particular on the regularity of solutions to fully nonlinear elliptic equations, the theory of viscosity solutions, the homogenization of nonlinear PDEs, and optimal transportation. In 2004 he was awarded the Steele Prize for Seminal Contribution to Research of the American Mathematical Society; he gave an invited lecture at the International Congress of Mathematicians 1986 in Berkeley. Evans has written more than 100 journal papers and several influential books, including ``Partial Differential Equations'', currently regarded worldwide as the standard graduate-level textbook in the field. He has been the advisor of 34 PhD students and has trained a dozen of postdocs, many of whom now hold positions at major institutions in three continents.

More infos are available here:

https://math.berkeley.edu/~evans/

https://en.wikipedia.org/wiki/Lawrence_C._Evans