  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.
