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 Ω⊂ ℝ3, given by (Formula presented.). We prove that for an admissibleg there exists a finite set of frequencies K in a given interval and an open cover Ω¯ = ∪ ω∈KΩω such that |∇uωg(x)|>0 for every ω∈ K and x∈ Ωω. The set K is explicitly constructed. If the spectrum of this problem is simple, which is true for a generic domain Ω , the admissibility condition on g is a generic property.