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Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

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Abstract

We show that an absolutely minimizing function with respect to a convex Hamiltonian \({H : \mathbb{R}^{n} \rightarrow \mathbb{R}}\) is uniquely determined by its boundary values under minimal assumptions on H. Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in L ∞.

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Acknowledgements

The third author was partially supported by the Academy of Finland, project #129784

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, USA

    Scott N. Armstrong

  2. Department of Mathematics, University of California, Santa Barbara, USA

    Michael G. Crandall

  3. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Vesa Julin

  4. Department of Mathematics, University of California, Berkeley, USA

    Charles K. Smart

Authors
  1. Scott N. Armstrong
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  2. Michael G. Crandall
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  3. Vesa Julin
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  4. Charles K. Smart
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Corresponding author

Correspondence to Scott N. Armstrong.

Additional information

Communicated by P. Rabinowitz

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Armstrong, S.N., Crandall, M.G., Julin, V. et al. Convexity Criteria and Uniqueness of Absolutely Minimizing Functions. Arch Rational Mech Anal 200, 405–443 (2011). https://doi.org/10.1007/s00205-010-0348-0

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  • DOI: https://doi.org/10.1007/s00205-010-0348-0

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