The Morse property for functions of Kirchhoff-Routh path type

Abstract

For a bounded domain $\Omega \subset {\mathbb{R}}^n$ let $H_{\Omega }:\Omega \times \Omega \to \mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal{C}}^2$ function $f:\mathcal{D}\to \mathbb{R}$, defined on an open subset $\mathcal{D}\subset {\mathbb{R}}^{nN}$, and fixed coefficients ${\lambda }_1,\dots ,{\lambda }_N\in \mathbb{R}\setminus \{0\}$ we consider the function $f_{\Omega }:\mathcal{D}\cap {\Omega }^N\to \mathbb{R}$ defined as \[ f_Ω(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N λ_jλ_k H_Ω(x_j,x_k). \] We prove that $f_{\Omega }$ is a Morse function for most domains $\Omega$ of class ${\mathcal{C}}^{m+2,\alpha }$, any $m\ge 0$, $0<\alpha <1$. This applies in particular to the Robin function $h:\Omega \to \mathbb{R}$, $h(x)=H_{\Omega }(x,x)$, and to the Kirchhoff-Routh path function where $\Omega \subset {\mathbb{R}}^2$, ${\mathcal D}=\{x\in\mathbb{R}^{2N}: \text{$x_j\ne x_k$for$j\ne k$}\}$, and \[ f(x_1,\dots,x_N) = - \frac{1}{2π}\sum_{\genfrac{}{}{0pt}{}{j,k=1}{j\ne k}}^Nλ_jλ_k\log|x_j-x_k|. \]

14 pages

References in corpus (1)

• Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Cited by in corpus (4)

• The N-vortex Problem on a Riemann Sphere
• Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two
• Existence of stationary vortex sheets for the 2D Euler equation
• The number of positive solutions to the Brezis-Nirenberg problem