A Pogorelov-type counterexample to the discreteness and openness of gradient mappings

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a domain, and suppose that $u\in W_{loc}^{2,n}(\Omega )$ satisfies the following ineqiality \[ \det D^2u\geq δ>0\qquad\text{a.e. in }Ω. \] A question of Guerra--Tione \cite[Question 5.5]{GuerraTione} asks whether the gradient mapping $Du$ must be open and discrete. In this paper, we give an explicit Pogorelov-type construction showing that the answer is negative in every dimension $n\ge 4$: there exists $u\in W_{loc}^{2,n}(\Omega )$ satisfying the above lower bound for the Hessian determinant, with $D^{2}u>0$ a.e., such that $Du$ collapses an entire line segment to a single point and hence is not discrete. We also show that the same construction has a logarithmic divergence when $n=3$ and therefore does not directly settle the three-dimensional case.