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

TL;DR

This paper proves that solutions to the Helmholtz equation in 3D domains lack critical points for a finite set of frequencies, with explicit construction of these frequencies and conditions on boundary data.

Contribution

It establishes the absence of critical points for solutions to the Helmholtz equation in 3D, identifying a finite frequency set and providing explicit construction methods.

Findings

Critical points are absent for solutions at certain frequencies.

A finite set of frequencies ensures non-vanishing gradients in subdomains.

The set of admissible boundary data is generically dense.

Abstract

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $\Omega\subset\mathbb{R}^{3}$, given by \[ \left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_{\omega}^{g})-\omega qu_{\omega}^{g}=0\quad\text{in $\Omega$,}\\ u_{\omega}^{g}=g\quad\text{on $\partial\Omega$.} \end{array}\right. \] We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overline{\Omega}=\cup_{\omega\in K}\Omega_{\omega}$ such that $|\nabla u_{\omega}^{g}(x)|>0$ for every $\omega\in K$ 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 $\Omega$, the admissibility condition on $g$ is a generic property.