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Variational Methods for Discrete Geometric Functionals

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

While consistent discrete notions of curvatures and differential operators have been widely studied, the question of whether the resulting minimizers converge to their smooth counterparts still remains open for various geometric functionals. Building on tools from variational analysis, and in particular using the notion of Kuratowski convergence, we offer a general framework for treating convergence of minimizers of (discrete) geometric functionals. We show how to apply the resulting machinery to minimal surfaces and Euler elasticae.

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Acknowledgements

The authors gratefully acknowledge support from DFG-project 282535003: Geometric curvature functionals: energy landscape and discrete methods.

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Authors and Affiliations

  1. RWTH Aachen University, Institute for Mathematics, Aachen, Germany

    Henrik Schumacher

  2. Institute of Numerical and Applied Mathematics, University of Göttingen, Göttingen, Germany

    Max Wardetzky

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  1. Henrik Schumacher
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Correspondence to Max Wardetzky .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Schumacher, H., Wardetzky, M. (2020). Variational Methods for Discrete Geometric Functionals. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_5

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