Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$, given by $$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}(a\mathrm{\nabla }{u}_{\omega }^g)-\omega q{u}_{\omega }^g=0\text{in }\mathrm{\Omega }\text{,}\\ u_{\omega }^g=g\text{on }\mathrm{\partial }\mathrm{\Omega }\text{.}\end{array}$$ We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\bar{\mathrm{\Omega }}={\cup }_{\omega \in K}{\mathrm{\Omega }}_{\omega }$ such that $|\nabla u_{\omega}^{g}(x)|>0$forevery$\omega\in K$and$x\in\Omega_{\omega}$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $\mathrm{\Omega }$, the admissibility condition on $g$ is a generic property.