Critical Points of Solutions to Exterior Boundary Problems

Abstract

In this article, we mainly study the critical points of solutions to the Laplace equation with Dirichlet boundary conditions in an exterior do-main in ℝ². Based on the fine analysis about the structures of connected components of the super-level sets $\{x\in {ℝ}^2\backslash \mathrm{\Omega }:u(x)>t\}$ and sub-level sets $\{x\in {ℝ}^2\backslash \mathrm{\Omega }:u(x)<t\}$ for some t, we get the geometric distributions of interior critical point sets of solutions. Exactly, when Ω is a smooth bounded simply connected domain, ${u|}_{\mathrm{\partial }\mathrm{\Omega }}=\psi (x)$, ${lim}_{|x|\to \mathrm{\infty }}u(x)=-\mathrm{\infty }$ and $\psi (x)$ has K local maximal points on ∂Ω, we deduce that ${\sum }_{i=1}^l{m}_i\le K$, where m₁, ..., m_l are the multiplicities of interior critical points x₁, ..., x_l of solution u respectively. In addition, when $\psi (x)$ has only K global maximal points and K equal local minima relative to ${ℝ}^2\backslash \mathrm{\Omega }$ on ∂Ω, we have that ${\sum }_{i=1}^l{m}_i=K$. Moreover, when Ω is a domain consisting of l disjoint smooth bounded simply connected domains, we deduce that ${\sum }_{x_i\in \mathrm{\Omega }}{m}_i+\frac{1}{2}{\sum }_{x_j\in \mathrm{\partial }\mathrm{\Omega }}{m}_j=l-1$, and the critical points are contained in the convex hull of the l simply connected domains.

Keywords

Critical points / level set / multiplicity / location / exterior boundary problem

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