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

TL;DR

This paper introduces a relaxed version of the Cauchy-Riemann equations in 3D, establishing the existence of harmonic conjugate pairs with a unitary gradient and exploring their boundary conditions and connections to Calderón's problem.

Contribution

It proposes a novel relaxation of classic equations in 3D and proves the existence of harmonic conjugate functions under a unitary gradient constraint.

Findings

Existence of harmonic conjugate pairs in 3D with a unitary gradient

Analysis of boundary conditions for these pairs

Potential links to Calderón's inverse problem

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.