Conjugate harmonic functions in 3D with respect to a unitary gradient

Abstract: We propose to relax the classic Cauchy-Riemann equations for a mapping. We support the interest of such a proposal by looking at one specific situation in 3D, and proving the existence of pairs of harmonic conjugate functions with respect to a unitary gradient as the title of this contribution conveys. We further investigate the relationship between boundary conditions for such pairs, the importance of the unitary constraint, and the eventual link of these ideas to Calder\'on's problem in 3D.