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

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