走向现代数学学术报告 - 代国伟教授（No. 665）

报告题目：Sign-changing solution for an overdetermined elliptic problem on unbounded

报告人：代国伟 教授（大连理工大学）

报告时间：2023年11月17日，9:00

腾讯会议ID：958807677

报告摘要：We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that\begin{equation}-\Delta u=\lambda u\,\, \text{in}\,\,\Omega,\,\,u=0,\,\,\partial_\nuu=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber\end{equation}admits a sign-changing solution. The domains bifurcate from the straight cylinder $B_1\times\mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.

报告人简介：代国伟，大连理工大学教授、博士生导师，主持完成国家自然科学基金3项，在研面上基金1项。获省自然科学奖二等奖1次，高校科技进步奖一等奖3次。研究方向是分歧理论及应用，建立了连通分支逼近法，研究了特征值问题的谱理论，以及非线性椭圆方程解集的全局结构，在《Indiana Univ. Math. J.》、《J. Funct. Anal.》、《Calc. Var. Partial Differential Equations》、《J. Differential Equations》、《Nonlinearity》等学术期刊上发表学术论文100余篇，引用1000余次。在科学出版社出版学术专著1部。