Approximation by piecewise affine homeomorphisms of Sobolev homeomorphisms that are smooth outside a point

• Autores: Carlos Mora Corral
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 35, Nº 2, 2009, págs. 515-539
• Idioma: inglés
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• Resumen

  • 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 O be a subset of R² be an open set containing 0 with polygonal boundary. Let h: O ? R² be a Lipschitz homeomorphism such that h-1 is also Lipschitz, h is of class C2 in O \ {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 L8 n W1,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 O that coincides with h at the vertices of the triangulation is not always a homeomorphism.