Any piecewise quadratic C^1-compatible planar homeomorphism with quantitative bi-Lipschitz condition can be approximated in W^{2,1} by C^1 injective maps with positive Jacobian, resolving the local analytical component of the approximation problem.
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$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms
Any piecewise quadratic C^1-compatible planar homeomorphism with quantitative bi-Lipschitz condition can be approximated in W^{2,1} by C^1 injective maps with positive Jacobian, resolving the local analytical component of the approximation problem.