Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings
Year: 2026
Type: preprint
Abstract: Let $Ω\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(Ω)$ satisfy \[ \det D^2u\geqδ>0\qquad\text{a.e. in }Ω. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k
Source: arXiv (Cornell University)
Author: Zhong, Deguang
Institution: Shenzhen Polytechnic University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4343562384, G553955554
OpenAlex ID: W7203710542