Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions

TL;DR

This paper analyzes the geometric structure of interior critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions, establishing relationships between critical point multiplicities and boundary maxima.

Contribution

It introduces a new method to relate interior critical points' multiplicities to boundary maximum points for solutions in various domain types.

Findings

For simply connected domains, sum of critical point multiplicities plus one equals the number of boundary maxima.

In annular domains, the sum of critical point multiplicities is less than or equal to the number of boundary maxima.

The paper provides geometric insights into the distribution of critical points based on boundary conditions.

Abstract

In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $\Omega$ in $\mathbb{R}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in \Omega: u(x)>t\}$ for any $\mathop {\min}_{\partial\Omega}u(x)<t<\mathop {\max}_{\partial\Omega}u(x)$, we obtain the geometric structure of interior critical points of $u$. Precisely, when $\Omega$ is simply connected, we develop a new method to prove $\Sigma_{i = 1}^k {{m_i}}+1=N$, where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$ and $N$ is the number of global maximal points of $u$ on $\partial\Omega$. When $\Omega$ is an annular domain with the interior boundary $\gamma_I$ and the external boundary $\gamma_E$, where $u|_{\gamma_I}=H,~u|_{\gamma_E}=\psi(x)$ and $\psi(x)$ has $N$ local (global) maximal points on $\gamma_E$. For the case $\psi(x)\geq H$ or $\psi(x)\leq H$ or $\mathop {\min}\limits_{\gamma_E}\psi(x)<H<\mathop {\max}\limits_{\gamma_E}\psi(x)$, we show that $\Sigma_{i = 1}^k {{m_i}} \le N$ (either $\Sigma_{i = 1}^k {{m_i}}=N$ or $\Sigma_{i = 1}^k {{m_i}}+1=N$).