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 , given by ' {-div(a del u(omega)(g)) - omega qu(omega)(g) = 0 in Omega, u(omega)(g) = g on partial derivative Omega. We prove that for an admissible g there exists a finite set of frequencies K in a given interval and an open cover (Omega) over bar = boolean OR(omega is an element of K) Omega(omega) such that vertical bar del u(omega)(g)(x)vertical bar > 0for every omega is an element of K and x is an element of Omega(omega). The set K is explicitly constructed. If the spectrum of this problem is simple, which is true for a generic domain Omega, the admissibility condition on g is a generic property.