A note on the critical points of the localization landscape

Let $Omegasubsetmathbb{C}$ be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function $v=v_Omega$ that solves the elliptic problem $Delta v = -2$ in $Omega,$ with boundary values $v=0$ on $partialOmega.$ This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of $v$ when $Omega$ belongs to a special class of domains in the plane, namely, domains for which the boundary $partialOmega$ is contained in ${z:|z|^2 = f(z) + overline{f(z)}},$ where $f(z)$ is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when $Omega$ is a quadrature domain, and conclude the note by stating some open problems and conjectures.