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

Abstract: Let $Ω\subset\mathbb{R}<sup>{n}$ be a domain, and suppose that $u\in W<sup>{2,n}_{loc}(Ω)$ 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 $n\geq4$: there exists $u\in W<sup>{2,n}_{loc}(Ω)$ satisfying the above lower bound for the Hessian determinant, with $D<sup>2u&gt;0$ 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.