# Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging

## TL;DR

This paper combines the Runge approximation and Whitney embedding theorem to enforce non-vanishing constraints on solutions of elliptic PDEs, with implications for hybrid imaging techniques.

## Contribution

It introduces a novel approach combining Runge approximation and Whitney embedding to ensure maximal rank Jacobians in solutions, applicable to hybrid imaging.

## Key findings

- The set of solutions with maximal Jacobian rank is open and dense in dimension d≥2.
- The method applies to less regular coefficients and other constraints.
- The approach is general and adaptable to various settings.

## Abstract

This paper addresses enforcing non-vanishing constraints for solutions to a second order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [Greene and Wu, Ann. Inst. Fourier (Grenoble), 25(1, vii):215-235, 1975]. The method is very general, and can be used in other settings.