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

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 \[ \left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_ω^{g})-ωqu_ω^{g}=0\quad\text{in $\Omega$,}\\ u_ω^{g}=g\quad\text{on $\partial \Omega$.} \end{array}\right. \] 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.

14 pages

References in corpus (2)

• Optimal shape and location of sensors for parabolic equations with random initial data
• On local non-zero constraints in PDE with analytic coefficients

Cited by in corpus (3)

• Critical Points for Elliptic Equations with Prescribed Boundary Conditions
• Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging
• Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems