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 $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distributions of connected components of the super-level sets $\{x\in Ω: u(x)>t\}$ and sub-level sets $\{x\in Ω: u(x)<t\}$ for some $t$, we obtain the geometric structure of interior critical point sets of $u$. Precisely, let $Ω$ be a multiply connected domain with the interior boundary $γ_I$ and the external boundary $γ_E$, where $u|_{γ_I}=ψ_1(x),~u|_{γ_E}=ψ_2(x)$. When $ψ_1(x)$ and $ψ_2(x)$ have $N_1$ and $N_2$ local maximal points on $γ_I$ and $γ_E$ 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 $\min_{γ_E}ψ_2(x)\geq \max_{γ_I}ψ_1(x)$ and $u$ has only $N_1$ and $N_2$ equal local maxima relative to $\overlineΩ$ on $γ_I$ and $γ_E$ 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.