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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Derivation of the Aronsson equation for $ C^1$ Hamiltonians


Authors: Michael G. Crandall, Changyou Wang and Yifeng Yu
Journal: Trans. Amer. Math. Soc. 361 (2009), 103-124
MSC (2000): Primary 35J70, 49K20
Published electronically: August 12, 2008
MathSciNet review: 2439400
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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:

$\displaystyle H_{p}(Du,u,x)\cdot (H (Du,u,x))_x=0$   in$\displaystyle \,\, 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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04651-5
PII: S 0002-9947(08)04651-5
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
Article copyright: © Copyright 2008 American Mathematical Society