Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

Abstract

In this paper, we mainly study the critical points and critical zero points of solutions $u$ to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain $\Omega$ in ${\mathbb{R}}^2$. Based on the fine analysis about the distributions of connected components of the super-level sets $\{x\in \Omega :u(x)>t\}$ and sub-level sets $\{x\in \Omega :u(x)<t\}$ for some $t$, we obtain the geometric structure of interior critical point sets of $u$. Precisely, let $\Omega$ be a multiply connected domain with the interior boundary ${\gamma }_I$ and the external boundary ${\gamma }_E$, where $u{|}_{{\gamma }_I}={\psi }_1(x),u{|}_{{\gamma }_E}={\psi }_2(x)$. When ${\psi }_1(x)$ and ${\psi }_2(x)$ have $N_1$ and $N_2$ local maximal points on ${\gamma }_I$ and ${\gamma }_E$ respectively, we deduce that ${\sum }_{i=1}^k{m}_i\le N_1+N_2$, where $m_1,\cdots ,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots ,x_k$ of $u$. In addition, when ${\min }_{{\gamma }_E}{\psi }_2(x)\ge {\max }_{{\gamma }_I}{\psi }_1(x)$ and $u$ has only $N_1$ and $N_2$ equal local maxima relative to $\overline{\Omega }$ on ${\gamma }_I$ and ${\gamma }_E$ respectively, we develop a new method to show that one of the following three results holds ${\sum }_{i=1}^k{m}_i=N_1+N_2$ or ${\sum }_{i=1}^k{m}_i+1=N_1+N_2$ or ${\sum }_{i=1}^k{m}_i+2=N_1+N_2$. Moreover, we investigate the geometric structure of interior critical zero points of $u.$ We obtain that the sum of multiplicities of the interior critical zero points of $u$ is less than or equal to the half of the number of its isolated zero points on the boundaries.

23 pages, 10 figures. arXiv admin note: text overlap with arXiv:1712.08458