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

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 Omega subset of R-2 be an open set containing 0 with polygonal boundary. Let h : (Omega) over bar -> R-2 be a Lipschitz homeomorphism such that h h(-1) is also Lipschitz, h is of class C-2 in Omega \ {0} and vertical bar vertical bar D(2)h(x)vertical bar vertical bar = O(vertical bar x vertical bar(-1)) as x -> 0.Then, for all 1 <= p < 2, the function h can be approximated in the norm of the intersection space L-infinity boolean AND W-1,W-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 Omega that coincides with h at the vertices of the triangulation is not always a homeomorphism.