Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Michael G. Crandall; Changyou Wang; Yifeng Yu
Journal: Trans. Amer. Math. Soc. 361 (2009), 103-124.
MSC (2000): Primary 35J70, 49K20
Posted: August 12, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.$

References:

1.
G. ARONSSON, Minimization problem for the functional $ sup_xF(x,f(x), f'(x))$, Ark. Mat. 6 (1965), 33-53. MR 0196551 (33:4738)

2.
G. ARONSSON, Minimization problem for the functional $ sup_xF(x,f(x), f'(x))$. 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 $ sup_xF(x,f(x), f'(x))$. 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 $ L^\infty$ 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 $ L^{\infty}$ 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 $ \infty$-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 $ \infty$-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, $ L^{\infty}$ variational problems and the Aronsson equations, Archive for Rational Mechanics and Analysis (2006), 153-180. MR 2247955 (2007k:49066)

22.
Y. YU, $ L^{\infty}$ variational problems, Aronsson equations and weak KAM theory, Ph.D. dissertation, U.C. Berkeley, 2005.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J70, 49K20

Retrieve articles in all Journals with MSC (2000): 35J70, 49K20

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: 10.1090/S0002-9947-08-04651-5
PII: S 0002-9947(08)04651-5
Received by editor(s): October 20, 2006
Posted: 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 of article: Copyright 2008, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google