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

TL;DR

This paper demonstrates that, in a generic sense, most positive continuous functions cannot be represented as the Jacobian of a bi-Lipschitz homeomorphism in the plane, highlighting limitations in the realizability of certain densities.

Contribution

It establishes that, generically, positive continuous functions are not Jacobians of bi-Lipschitz plane homeomorphisms, revealing a fundamental restriction in geometric analysis.

Findings

Positive continuous functions are not generally Jacobians of bi-Lipschitz maps.

The non-realizability result extends to bounded functions.

Most densities cannot be realized as Jacobians of bi-Lipschitz maps.

Abstract

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.