The Morse property for functions of Kirchhoff-Routh path type

TL;DR

This paper proves that a class of functions related to Kirchhoff-Routh path problems are Morse functions for most smooth domains, which has implications for understanding the topology and critical points of these functions in fluid dynamics and potential theory.

Contribution

It establishes the Morse property for a broad class of functions associated with Kirchhoff-Routh path problems on generic smooth domains.

Findings

Most domains of class C^{m+2,α} make the functions Morse.

The result applies to the Robin function and Kirchhoff-Routh path functions.

Provides a generic Morse property for functions in fluid dynamics models.

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 \[ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). \] We prove that $f_\Omega$ is a Morse function for most domains $\Omega$ of class ${\mathcal C}^{m+2,\alpha}$, any $m\ge0$, $0<\alpha<1$. 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 \[ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum_{\genfrac{}{}{0pt}{}{j,k=1}{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|. \]