Derivation of the Aronsson equation for Hamiltonians

Authors:
Michael G. Crandall, Changyou Wang and Yifeng Yu

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

Published electronically:
August 12, 2008

MathSciNet review:
2439400

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved herein that any absolute minimizer for a suitable Hamiltonian is a viscosity solution of the Aronsson equation:

**1.**G. ARONSSON,*Minimization problem for the functional ,*Ark. Mat. 6 (1965), 33-53. MR**0196551 (33:4738)****2.**G. ARONSSON,*Minimization problem for the functional . II,*Ark. Mat. 6 (1966), 409-431. MR**0203541 (34:3391)****3.**G. ARONSSON,*Extension of functions satisfying Lipschitz conditions,*Ark. Mat. 6 (1967), 551-561. MR**0217665 (36:754)****4.**G. ARONSSON,*Minimization problem for the functional . III,*Ark. Mat. 7 (1969), 509-512. MR**0240690 (39:2035)****5.**G. ARONSSON, M. G. CRANDALL, P. JUUTINEN,*A Tour of the Theory of Absolutely Minimizing Functions,*Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439-505 (electronic). MR**2083637 (2005k:35159)****6.**BARRON, E. N., JENSEN, R.,*Minimizing the norm of the gradient with an energy constraint*, Comm. Partial Differential Equations 30 (2005), no. 12, 1741-1772. MR**2182310 (2006h:35047)****7.**E. N. BARRON, R. R. JENSEN, C. Y. WANG,*The Euler equation and absolute minimizers of functionals,*Arch. Ration. Mech. Anal. 157 (2001), no. 4, 255-283. MR**1831173 (2002m:49006)****8.**CHAMPION, T., DE PASCALE, L.,*Principles of comparison with distance functions for absolute minimizers*, J. Convex Anal. 14 (2007), no 3, 515-541. MR**2341302****9.**M. G. CRANDALL,*An efficient derivation of the Aronsson equation,*Arch. Ration. Mech. Anal. 167 (2003), no. 4, 271-279. MR**1981858 (2004b:35053)****10.**M. G. CRANDALL,*A Visit with the -Laplace Equation*, in Calculus of Variations and Non-Linear Partial Differential Equations, (C.I.M.E. Summer School, Cetraro, 2005), Lecture Notes in Math., vol. 1927, Springer, Berlin, 2008.**11.**M. G. CRANDALL, L. C. EVANS,*A remark on infinity harmonic functions,*123-129 (electronic), Electron. J. Differ. Equ. Conf., 6. MR**1804769 (2001j:35076)****12.**M. G. CRANDALL, L. C. EVANS, R. GARIEPY,*Optimal Lipschitz Extensions and the Infinity Laplacian,*Cal. Var. Partial Differential Equations 13 (2001), no.2, 123-139. MR**1861094 (2002h:49048)****13.**M. G. CRANDALL, G. GUNNARSSON, P. Y. WANG,*Uniqueness of -harmonic functions and the eikonal equation,*Comm. Partial Differential Equations 32 (2007), 1587-1615. MR**2372480****14.**M. G. CRANDALL, H. ISHII, P. L. LIONS,*User's guide to viscosity solutions of second order partial differential equations,*Bull. AMS 27 (1992), 1-67. MR**1118699 (92j:35050)****15.**A. FATHI, A. SICONOLFI,*PDE aspects of Aubry-Mather theory for continuous Hamiltonians*, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185-228.. MR**2106767 (2006f:35023)****16.**R. GARIEPY, C. WANG, Y. YU,*Generalized Cone Comparison, Aronsson equation, and Absolute Minimizers,*Comm. Partial Differential Equations, 36 (2006), no. 7-9, 1027-1046. MR**2254602 (2007h:35089)****17.**R. JENSEN,*Uniqueness of Lipschitz extensions minimizing the sup-norm of the gradient*, Archive for Rational Mechanics and Analysis 123 (1993), 51-74. MR**1218686 (94g:35063)****18.**R. JENSEN, C. WANG, Y. YU,*Uniqueness for Viscosity Solutions of Aronsson Equations,*Arch. Ration. Mech. Anal., to appear.**19.**P. JUUTINEN,*Minimization problems for Lipschitz functions via viscosity solutions,*Academiae Scientiarum Fennicae, Mathematica Dissertationes 115, 1998. MR**1632063 (2000a:49055)****20.**P. L. LIONS,*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics 69, Pitman, Boston, MA, 1982. MR**667669 (84a:49038)****21.**Y. YU,*variational problems and the Aronsson equations*, Archive for Rational Mechanics and Analysis (2006), 153-180. MR**2247955 (2007k:49066)****22.**Y. YU,*variational problems, Aronsson equations and weak KAM theory*, Ph.D. dissertation, U.C. Berkeley, 2005.

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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:
https://doi.org/10.1090/S0002-9947-08-04651-5

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