APPROXIMATION BY PIECEWISE AFFINE HOMEOMORPHISMS OF SOBOLEV HOMEOMORPHISMS THAT ARE SMOOTH OUTSIDE A POINT

Abstract. This paper is concerned with the problem of approximating in the Sobolev norm a homeomorphism by piecewise affine homeomorphisms. The homeomorphism we want to approximate is supposed to be smooth except at one point. As a corollary of our main result, we prove the following: Let Ω ⊂ R2 be an open set containing 0 with polygonal boundary. Let h: Ω̄ → R2 be a Lipschitz homeomorphism such that h−1 is also Lipschitz, h is of class C2 in Ω \ {0} and ‖D2h(x) ‖ = O(|x|−1) as x → 0. Then, for all 1 ≤ p < 2, the function h can be approximated in the norm of the intersection space L ∞ ∩W 1,p by a piecewise affine homeomorphism f. Several results in the same spirit are also proved, where we suppose that h and h−1 are smooth except at one point, and their derivatives may have one singularity. The construction of f is explicit. We also show examples of functions satisfying the assumptions of the main theorem of the paper and for which the piecewise affine function on a regular triangulation of Ω that coincides with h at the vertices of the triangulation is not always a homeomorphism. 1