中科院数学与系统科学研究院

数学研究所

学术报告

偏微分方程研讨班

报告人：代国伟  教授（大连理工大学）

题  目：Sign-changing solution for an overdetermined elliptic problem on unbounded domain

时  间：2023.09.21（星期四）16:00-17:00

地  点：思源楼S813

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_\nu u=\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.