Article Contents
Article Contents

# The Morse property for functions of Kirchhoff-Routh path type

• * Corresponding author: Thomas Bartsch

Dedicated to Norman Dancer, with friendship and esteem

The first author is supported by funds "Agreement between Sapienza University of Roma and University of Giessen"

• 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

$f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum\limits_{j,k = 1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k).$

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 $\Omega\subset\mathbb{R}^2$, ${\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \}$, and

$f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum\limits_{{j,k = 1}\atop{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|.$

Mathematics Subject Classification: Primary: 35J08; Secondary: 35J25, 35Q31, 76B47.

 Citation:

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