Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with $L^p$ Data

TL;DR

This paper constructs bi-Sobolev planar mappings with prescribed Jacobian bounds in $L^p$, providing solutions with specific regularity and boundary preservation, including bi-Lipschitz maps for bounded data.

Contribution

It introduces a method to construct bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with $L^p$ data, extending regularity and boundary control.

Findings

Constructed bi-Sobolev solutions for $f \,\in L^p$

Developed Lipschitz stretching maps for compact sets

Provided bi-Lipschitz solutions for bounded $f$

Abstract

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$, i.e. bi-Sobolev solutions for the prescribed Jacobian inequality in the plane for right-hand sides $f\in L^p$. More precisely, for any $1<q<(p+1)/2$ we construct a solution which belongs to $W^{1,q}$ and which preserves the boundary pointwise. For bounded right-hand sides $f\in L^{\infty}$, we provide bi-Lipschitz solutions. The basic building block of our construction are Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the covering result of Alberti, Cs\"ornyei, and Preiss for measurable planar sets in the case of compact sets.