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 $\sum_{i = 1}^k {{m_i}}\leq N_1+ N_2$, where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$. In addition, when $\min_{\gamma_E}\psi_2(x)\geq \max_{\gamma_I}\psi_1(x)$ 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.