The Morse property for functions of Kirchhoff-Routh path type
Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia · Discrete and Continuous Dynamical Systems - S · 2019
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 \begin{document}$ 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). $\end{document} 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 e x_k \; \text{for }\; j e k \} $, and \begin{document}$ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum\limits_{{j,k = 1}\atop{j e k}}^N\lambda_j\lambda_k\log|x_j-x_k|. $\end{document}