论文标题
关于通过规定的雅各布决定因素和卷曲确定的差异性唯一性问题的最新见解
Recent insights on the Uniqueness Problem of Diffeomorphisms determined by Prescribed Jacobian Determinant and Curl
论文作者
论文摘要
差异原理（VP）形成了规定的雅各布决定因素（JD）和卷曲的差异性。例子表明，（i）仅JD不能独立地确定没有卷发的差异性； （ii）VP的解决方案似乎满足了谎言群体的特性。因此，猜想可以通过其JD和Curl（唯一性猜想）来确保独特的差异性。在本文中，（1）得出了基于VP的观察结果，即存在对猜想的示例，如果存在，则应满足特定财产； （2）从观察结果中，制定了一种实验策略，以测试给定的差异是否是对猜想的有效典范； （3）提供了猜想的中间步骤的证明，并称为半通用案例，该案例认为，如果它们接近身份map，$ \ pmb {id} $，则$ \ pmb \ \ pmb $ $ \ pmbar $ \ pmbin。
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, $\pmb{id}$, then $\pmbϕ$ is identical $\pmbψ$.
