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L ∞-Extremal Mappings in AMLE and Teichmüller Theory

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Fully Nonlinear PDEs in Real and Complex Geometry and Optics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2087))

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Abstract

These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L ∞ norm of an energy density rather than the more familiar L p norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.

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Notes

  1. 1.

    Implicit summation on repeated indices is used throughout the paper.

  2. 2.

    Recall that every Riemann surface is orientable and any conformal atlas yields a triangulation.

  3. 3.

    To do this however one has to assume existence of two derivatives for the solution.

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Acknowledgements

We wish to thank C. Gutierrez and E. Lanconelli for the scientific organization of the C. I. M.E. course and for inviting the author to present these lectures. We are also grateful to P. Zecca and to all the staff of C. I.M.E. for their logistic support and hospitality. The author is partially supported by the US National Science Foundation through grants DMS-1101478 and DMS-0800522

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  1. Department of Mathematical Sciences, Stratton Hall, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA, 01609, USA

    Luca Capogna

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Capogna, L. (2014). L ∞-Extremal Mappings in AMLE and Teichmüller Theory. In: Fully Nonlinear PDEs in Real and Complex Geometry and Optics. Lecture Notes in Mathematics(), vol 2087. Springer, Cham. https://doi.org/10.1007/978-3-319-00942-1_1

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