论文标题
生成的Jacobian方程的规律性和唯一性结果
Regularity and uniqueness results for generated Jacobian equations
论文作者
论文摘要
这是关于生成的雅各布方程的博士学位论文；我们的目的是双重的。首先，我们提供了这些方程式的介绍，同时，整理了一些散布在整个文献中的结果。另一个目标是在这些方程式上介绍作者自己的结果。这些结果都涉及生成的雅各布方程的解决方案，通常与第二个边界值问题配对。在最佳假设下，我们证明了严格的凸度和$ c^1 $可区分性结果，并且在较高维度的情况下，同样的结果和一些额外的假设。我们还考虑了第二个边界价值问题的唯一性结果，以及将唯一性结果应用于全球规则性的应用。我们以关于抛物线产生的雅各布方程的注释来结束。 ARXIV版本包含对ANU开放研究存储库版本的次要更新，可从列表的DOI获得。
This is a PhD thesis about generated Jacobian equations; our purpose is twofold. First, we provide an introduction to these equations, whilst, at the same time, collating some results scattered throughout the literature. The other goal is to present the author's own results on these equations. These results all concern solutions of generated Jacobian equations, usually paired with the second boundary value problem. We prove strict convexity and $C^1$ differentiability results under optimal hypothesis in two dimensions, and the same results in higher dimensions with some additional hypothesis. We also consider uniqueness results for the second boundary value problem, and the application of the uniqueness results to global regularity. We conclude with notes on the parabolic generated Jacobian equation. The arXiv version contains minor updates to the ANU open research repository version which is available from the listed DOI.
