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 ℝ². Based on the delicate analysis about the distributions of connected components of the super-level sets {x ∈ Ω: u(x) > t} and sub-level sets {x ∈ 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 = ψ₁(x), u∣γ_E = ψ₂(x). When ψ₁(x) and ψ₂(x) have N₁ and N₂ local maximal points on γ_I and γ_E respectively, we deduce that $\sum\nolimits_{i = 1}^k {{m_i} \le {N_1} + {N_2}}$, where m₁,…,m_k are the respective multiplicities of interior critical points x₁,…,x_k of u. In addition, when ${\min _{{\gamma _E}}}{\psi _2}\left( x \right) \ge {\max _{{\gamma _I}}}{\psi _1}\left( x \right)$ and u has only N₁ and N₂ equal local maxima relative to $\overline \Omega$ on γ_I and γ_e respectively, we develop a new method to show that one of the following three results holds $\sum\nolimits_{i = 1}^k {{m_i} = {N_1} + {N_2}}$ or $\sum\nolimits_{i = 1}^k {{m_i} + 1 = {N_1} + {N_2}}$ or $\sum\nolimits_{i = 1}^k {{m_i} + 2 = {N_1} + {N_2}}$. 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.