Speaker: Juncheng Wei

Geomterization Program of Semilinear Elliptic PDEs

Abstract: Understanding the entire solutions of nonlinear elliptic equations

in $R^N$ such as $Δu + f(u) = 0$ is a basic problem in PDE research.
This is the context of various classical results in literature like the
Gidas-Ni-Nirenberg theorems on radial symmetry, Liouville type theorems,
or the achievements around De Giorgi’s conjecture. In those results, the
geometry of level sets of the solutions turns out to be a posteriori very
simple (planes or spheres). On the other hand, problems of this form do
have solutions with more interesting patterns, and the structure of their
solution sets has remained mostly a mystery. A major aspect of our
research program is to bring ideas from Differential Geometry into the analysis
and construction of entire solutions for two important equations: (1) the Allen-
Cahn equation and (2) the nonlinear Schrodinger equation. Though
simple-looking, they are typical representatives of two classes of semilinear
elliptic problems. The structure of entire solutions is quite rich. In this talk,
we shall establish an intricate correspondence between the study of entire
solutions of some scalar equations and the theories of minimal surfaces and
constant mean curvature surfaces (CMC).
Bio: Prof. Juncheng Wei at University of British Columbia is a Canada Research
Chair in Partial Differential Equations. He is a prolific researcher who has written
over 280 articles since 1995. His paper coauthored with M. Del Pino and
M. Kowalczyk (Annals of Mathematics 174 (2011)) resolved the De Giorgi’s
Conjecture in dimensions greater than 8. For this and other significant works,
he is invited to give a lecture at 2014 International Congress of Mathematician