报告题目：On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains
时间：2017年12月26日上午10:30
地点：理工楼北楼601
主办：数学与信息学院
主讲：杨军
参加对象：数学系教师研究生和感兴趣的教师
报告人简介：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青年项目和面上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：Geometric and Functional Analysis、 Transactions of the American Mathematical Society、 Indiana University Mathematical Journal、Communications in Partial Differential Equations、SIAM Journal on Mathematical Analysis等。
报告简介：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 and Bin Xu.
