2022/07/26 by Zicong Zhou, Zhou, Zicong, Guojun Liao +1 · 1 citation
Chemistry · Immunology and Microbiology · Mathematics · #49Q10 #49Q20 #65K10 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Carbohydrate Chemistry and Synthesis #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Macrophage Migration Inhibitory Factor
paper · pdf · doi:10.48550/arxiv.2207.13053
openalex publication_date 2022/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP seem to satisfy properties of a Lie group. Hence, it is conjectured that a unique diffeomorphism can be assured by its JD and curl (Uniqueness Conjecture). In this paper, (1) an observation based on VP is derived that a counter example to the Conjecture, if exists, should satisfy a particular property; (2) from the observation, an experimental strategy is formulated to numerically test whether a given diffeomorphism is a valid counter example to the conjecture; (3) a proof of an intermediate step to the conjecture is provided and referred to as the semi-general case, which argues that, given two diffeomorphisms, \pmbϕ and \pmbψ, if they are close to the identity map, \pmbid, then \pmbϕ is identical \pmbψ.