REACTION-DIFFUSION ELLIPTIC EQUATIONS AND MINIMAL SURFACES
Xavier Cabré
ICREA and UPC, Barcelona
We will present recent developments on solutions of
reaction-diffusion elliptic equations that are strongly related to some
classical results in the theory of minimal surfaces.
The connection between reaction-diffusion (or semilinear) elliptic
equations and minimal surfaces originates in semilinear models of phase transitions.
As the reaction term becomes stronger, interfaces between two states
tend to minimize their area.
A classical result in minimal surface theory is the flatness of minimal
graphs up to dimension 7. Its semilinear analogue is a
conjecture posed by E. De Giorgi in 1978
for which progress has been made only recently.
We will describe these developments. They establish rich relations among
different qualitative properties of solutions: their
stability, minimality, monotonicity in one variable, and their symmetry.
We will also discuss semilinear analogues (which concern saddle shaped solutions)
of the existence of singular minimal cones
(such as the Simons cone) in high dimensions.