Columbia Mathematics Department Colloquium


Convexity of solutions to nonlinear partial differential equations


 Pengfei Guan  

McGill University



In many natural settings, solutions of nonlinear partial

differential equations carry some special convex properties. For example,

the level-sets of equilibrium potential of a convex domain is convex. The

classical result of Brascamp and Lieb states that the first eigenfunction

of the Laplace equation in a convex domain is log–concave. There are two

types of methods which are quite different in treating the convexity. The

macroscopic convexity principle is based on the convex

hull of the solution and related to the notion of viscosity solutions. The

microscopic convexity principle is based on constant rank type theorem for

the Hessian matrix of the solution, which was observed by

Caffarelli-Friedman and Yau in 1980s. The advantage of the microscopic

convexity principle is that it is effective to treat nonlinear geometric

differential equations on general manifolds. In the talk, we will discuss

some recent development in this direction and discuss some related open problems.




  March 10th, Wednesday, 5:00-6:00 pm

Mathematics 520

Tea will be served at 4:30pm