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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Derivation of the Aronsson equation for $C^1$ Hamiltonians
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by Michael G. Crandall, Changyou Wang and Yifeng Yu PDF
Trans. Amer. Math. Soc. 361 (2009), 103-124 Request permission

Abstract:

It is proved herein that any absolute minimizer $u$ for a suitable Hamiltonian $H\in C^1(\mathbb {R}^n \times \mathbb {R}\times U)$ is a viscosity solution of the Aronsson equation: \[ H_{p}(Du,u,x)\cdot (H (Du,u,x))_x=0 \quad \text {in} U. \] The primary advance is to weaken the assumption that $H\in C^2,$ used by previous authors, to the natural condition that $H\in C^1.$
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Additional Information
  • Michael G. Crandall
  • Affiliation: Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106
  • Email: crandall@math.ucsb.edu
  • Changyou Wang
  • Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
  • Email: cywang@ms.uky.edu
  • Yifeng Yu
  • Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
  • Email: yifengyu@math.utexas.edu
  • Received by editor(s): October 20, 2006
  • Published electronically: August 12, 2008
  • Additional Notes: The first author was supported by NSF Grant DMS-0400674
    The second author was supported by NSF Grant DMS-0601162
    The third author was supported by NSF Grant DMS-0601403
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 103-124
  • MSC (2000): Primary 35J70, 49K20
  • DOI: https://doi.org/10.1090/S0002-9947-08-04651-5
  • MathSciNet review: 2439400