## Columbia Mathematics Department
Colloquium

*Convexity of solutions to nonlinear partial differential
equations*

### by

### Pengfei Guan

### McGill
University

Abstract:

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**