主 题: De Giorgi Conjecture, Allen-Cahn Equation and Minimal Surfaces
报告人: Prof. Wei Jun-cheng (The Chinese University of Hong Kong)
时 间: 2009-03-17 下午 4:00 - 5:00
地 点: 资源大厦1328
A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\Delta u + (1-u^2) u = 0 \ \hbox{in} \ \R^N $ with
$\frac{\partial u}{\partial y_N} >0$ must be such that its level sets $\{u=\lambda\}$ are all hyperplanes, {\em \bf at least} for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist.
Based on a minimal graph $\Gamma$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\R^N$, $N\ge 9$, we prove that for any small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with $\frac{\partial u_\alpha}{\partial y_N} >>0$, which resembles $ \tanh \left ( \frac t{\sqrt{2}}\right ) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $\Gamma_\alpha := \alpha^{-1}\Gamma$.
This solution constitutes a counterexample to De Giorgi conjecture for $N\ge 9$. The methods allow us to establish a connection between minimal embedded surfaces in $R^3$ with finite topology and the Allen-Cahn equation with finite Morse index. (Joint work with M. del Pino
and M. Kowalczyk).
