The Morse property for functions of Kirchhoff-Routh path type

For a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} let \begin{document}$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $\end{document} be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary \begin{document}$ {\mathcal C}^2 $\end{document} function \begin{document}$ f:{\mathcal D}\to\mathbb{R} $\end{document} , defined on an open subset \begin{document}$ {\mathcal D}\subset\mathbb{R}^{nN} $\end{document} , and fixed coefficients \begin{document}$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $\end{document} we consider the function \begin{document}$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $\end{document} defined as \begin{document}$ 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). $\end{document} We prove that \begin{document}$ f_\Omega $\end{document} is a Morse function for most domains \begin{document}$ \Omega $\end{document} of class \begin{document}$ {\mathcal C}^{m+2,\alpha} $\end{document} , any \begin{document}$ m\ge0 $\end{document} , \begin{document}$ 0 . This applies in particular to the Robin function \begin{document}$ h:\Omega\to\mathbb{R} $\end{document} , \begin{document}$ h(x) = H_\Omega(x,x) $\end{document} , and to the Kirchhoff-Routh path function where \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} , \begin{document}$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $\end{document} , and \begin{document}$ 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|. $\end{document}