Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems

TL;DR

This paper introduces a frequency-based method for boundary control of PDE solutions to enforce local non-zero constraints, enhancing hybrid imaging techniques without relying on complex geometric optics solutions.

Contribution

It presents an alternative approach using multiple frequencies for boundary control in PDEs, independent of PDE coefficients, applicable to hybrid imaging.

Findings

Explicit boundary conditions and finite frequencies are constructed.

The method ensures solutions satisfy non-zero constraints inside the domain.

Applications to hybrid imaging modalities are demonstrated.

Abstract

We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on the solutions of the PDE do not vanish locally inside the domain. Suitable boundary conditions are classically determined by using complex geometric optics solutions. This work focuses on an alternative approach to this issue based on the use of multiple frequencies. Simple boundary conditions and a finite number of frequencies are explicitly constructed independently of the coefficients of the PDE so that the corresponding solutions satisfy the required constraints. This theory finds applications in several hybrid imaging modalities: some examples are discussed.