理学院数理讲坛（2017年第14讲）
报告题目：On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains
报告人：杨军 教授
单 位：华中师范大学
报告时间：2017年5月5日（星期五）10:00-10:50
报告地点：龙赛理科楼北楼 116
报告摘要：
We consider the problem
$$
\epsilon^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad\mbox{in}\quad\Omega,
\quad\frac{\partial u}{\partial \nu}\,=\,0\quad\mbox{on}\quad\partial \Omega,
$$
where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies {\it stationary and non-degeneracy conditions} with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac{1}{2}$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni (p.327, Indiana Univ. Math. J. 53 (2004), no. 2). This is a joint work with Suting Wei, Fang Xiao and Bin Xu.
报告人简介：
杨军，华中师范大学教授，博士生导师，2007年获得香港中文大学数学哲学博士学位，访问过多个国际著名数学研究中心，主持国家自然科学基金面上基金和青年项目等多个国家课题。主要研究方向是非线性偏微分方程和非线性分析，在多个国际高水平学术期刊上发表论文，如：Geom. Funct. Anal.、 TAMS、 Indiana University Math. J.、Comm. PDE、SIAM J. Math. Anal.等。