$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in $W^{2,1}$ together with quantitative preservation of the Jacobian. The resulting maps are $C^{1}$ on the whole domain and smooth inside each cell of the partition; in particular they are $C^{2}$ away from the interfaces. These local constructions are combined into a global smoothing theorem: any piecewise quadratic $C^{1}$-compatible planar homeomorphism satisfying a quantitative bi-Lipschitz condition can be approximated in $W^{2,1}$ by maps that are $C^{1}$, injective, and have positive Jacobian. As a consequence, we show that the general $W^{2,1}$ approximation problem reduces to a purely geometric question: the construction of piecewise quadratic approximations with quantitative injectivity and nondegeneracy.