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Triangulation of diffeomorphisms

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Abstract

Is it possible to approximate a diffeomorphism of Euclidean domains with piecewise affine homeomorphisms, locally uniformly up to the first order derivatives? The answer is yes. However, any effort to provide a rigorous and clear proof reveals the complexity of this question, especially in higher dimensions. It is the objective of the present paper to formulate this question in its greatest generality, as well as to provide all details for the affirmative answer, Theorem 1.1. A novelty, which has broader applications, is the construction of selfsimilar isotropic triangulation of the Euclidean domains, Theorem 1.2.

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Acknowledgments

We thank the reviewer \( \sharp 1 \) for his/her thorough review and highly appreciate the comments, which contributed to improving the quality of the paper.

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Correspondence to Tadeusz Iwaniec.

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Communicated by Y. Giga.

T. Iwaniec was supported by the United States NSF Grant DMS-1301558. J. Onninen was supported by the United States NSF Grant DMS-1301570.

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Iwaniec, T., Onninen, J. Triangulation of diffeomorphisms. Math. Ann. 368, 1133–1169 (2017). https://doi.org/10.1007/s00208-016-1426-x

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  • DOI: https://doi.org/10.1007/s00208-016-1426-x

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