##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Pengfei Guan

#### McGill University

## Fully nonlinear equations, elementary symmetric functions and convexity property of solutions

##### Abstract:

We discuss convexity properties of solutions of fully nonlinear partial differential equations. The classical examples indicate that the level-sets of equilibrium potential in a convex domain is convex and the first eigenfunction of the Laplace equation in a convex domain is log-concave. Recently, there emerge two differente types of methods in the study of convexity of solutions of nonlinear equations. The macroscopic convexity principle is based on the convex hull of the solution. The microscopic convexity principle is based on constant rank type theorem for the Hessian matrix of the solution. The microscopic convexity principle is effective to treat nonlinear geometric differential equations on general manifolds. In the talk, we will explain how elementary symmetric functions can be used in a crucial way in this direction and what kind of "convexity" structural conditions are involved. The microscopic convexity principle shares close relationship with the Hamilton's maximum principle for general evolution equations. We will also discuss some related open problems.

Host: Ben Weinkove

### May 13, 2010

### 4:00 PM

### AP&M 6402

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