Openness and discreteness of gradient mappings in three dimensions

Determine whether every gradient mapping f = Du associated with a function u ∈ W^{2,3}_{loc}(Ω) satisfying det D^2u ≥ δ > 0 almost everywhere in a domain Ω ⊂ R^3 is necessarily open and discrete.

Background

The paper studies whether a Sobolev gradient mapping f = Du must be open and discrete when u ∈ W^{2,n}_{loc}(Ω) and its Hessian determinant satisfies the uniform lower bound det D^2u ≥ δ > 0 almost everywhere. The authors construct a Pogorelov-type counterexample showing that this assertion is false in every dimension n ≥ 4: the Hessian is positive definite almost everywhere, yet Du collapses a line segment to a point and is therefore neither discrete nor open.

For n = 3, the same Pogorelov-type construction has only logarithmic integrability at the critical exponent and does not belong to W^{2,3}_{loc}. Consequently, the construction does not resolve whether the original Guerra–Tione question has an affirmative or negative answer in three dimensions.