Distinct solutions to generated Jacobian equations cannot intersect

Densities non-realizable as the Jacobian of a 2-dimensional bi-Lipschitz map are generic

It is shown that, generically, a positive continuous functions cannot be written as the derivative of a bi-Lipschitz plane homeomorphism. The same is proved for generic bounded functions.

Strict $g$-convexity for generated Jacobian equations with applications to global regularity

This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the same problem under stronger hypothesis. These are related because the strict $g$-convexity is essential for the proof of the global regularity. The assumptions for the strict $g$-convexity are the natural extension of those used by Chen and Wang in the optimal transport case. They improve the existing domain conditions though at the expense of requiring a $…

曲面下扰动中的节点域：上半连续性、Courant尖锐性与边界相交

Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections

研究在闭曲面上Schrödinger算子的特征函数节点域数量在光滑扰动下的变化，证明节点域计数的上半连续性，并构造可任意有限水平的Courant尖锐度度量及边界相交控制。