Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms

2011 - JC Bellido, C Mora Corral

Houston Journal of Mathematics 37, 449-500 (2011)

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authors IMACI

Abtract

This paper is concerned with the problem of approximating a homeomorphism by piecewise
affine homeomorphisms. The main result is as follows: every homeomorphism from a planar
domain with a polygonal boundary to R2 that is globally H¨older continuous of exponent
? ? (0, 1], and whose inverse is also globally H¨older continuous of exponent ? can be approximated
in the H¨older norm of exponent ? by piecewise affine homeomorphisms, for some
? ? (0, ?) that only depends on ?. The proof is constructive. We adapt the proof of simplicial
approximation in the supremum norm, and measure the side lengths and angles of the
triangulation over which the approximating homeomorphism is piecewise affine. The approximation
in the supremum norm, and a control on the minimum angle and on the ratio between
the maximum and minimum side lengths of the triangulation suffice to obtain approximation in
the H¨older norm