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

TL;DR

The paper addresses extending Cauchy–Riemann-type conjugacy to three dimensions by introducing harmonic conjugates with respect to a unit gradient $\nabla w$, defined via $\nabla u = \nabla v \wedge \nabla w$ and $\nabla v = \nabla w \wedge \nabla u$. It develops a variational framework, mirroring the classical 2D approach, to prove the existence of non-trivial pairs $(u,v)$ of harmonic functions for any unit gradient $\nabla w$ on a Lipschitz domain $\Omega$, and derives the associated boundary conditions that couple $u$ and $v$. The results are extended to non-unitary gradients and to weighted conductivity-type settings with $\gamma$, showing how these conjugate pairs satisfy modified equations and linking to Calderón’s inverse problem in 3D. The work highlights potential implications for 3D inverse conductivity, including possible non-uniqueness phenomena in the Dirichlet-to-Neumann map and a broader conductivity context via div-curl arguments. Overall, the paper provides a rigorous foundation for 3D harmonic conjugacy with respect to a unit (or weighted) gradient and situates it within inverse problems and variational methods.

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 Calderón's problem in 3D.

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 3.1
• Proposition 4.1
• Theorem 5.1