On local non-zero constraints in PDE with analytic coefficients

Abstract

We consider the Helmholtz equation with real analytic coefficients on a bounded domain $Ω\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $ω$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_ω^{i}$ of the form $ζ(u_ω^{1},\ldots,u_ω^{d+1},\nabla u_ω^{1},\ldots,\nabla u_ω^{d+1})\neq0$, where $ζ$ is analytic, which is satisfied in $Ω$ when $ω=0$. We show that for any $Ω^{\prime}\SubsetΩ$ and almost any $d+1$ frequencies $ω_{k}$ in $[a,b]$, there exist $d+1$ subdomains $Ω_{k}$ such that $Ω^{\prime}\subset\cup_{k}Ω_{k}$ and $ζ(u_{ω_{k}}^{1},\ldots,u_{ω_{k}}^{d+1},\nabla u_{ω_{k}}^{1},\ldots,\nabla u_{ω_{k}}^{d+1})\neq0$ in $Ω_{k}$. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems.

9 pages, 1 figure