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

HTML articles powered by AMS MathViewer

- 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 $H\in C^1.$## References

- Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\,F(x,\,f(x),\,f^{\prime } (x))$*, Ark. Mat.**6**(1965), 33β53 (1965). MR**196551**, DOI 10.1007/BF02591326 - Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\, F(x, f(x),f^\prime (x))$. II*, Ark. Mat.**6**(1966), 409β431 (1966). MR**203541**, DOI 10.1007/BF02590964 - Gunnar Aronsson,
*Extension of functions satisfying Lipschitz conditions*, Ark. Mat.**6**(1967), 551β561 (1967). MR**217665**, DOI 10.1007/BF02591928 - Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\,F(x,\,f(x),\,f^{\prime } \,(x))$. III*, Ark. Mat.**7**(1969), 509β512. MR**240690**, DOI 10.1007/BF02590888 - Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen,
*A tour of the theory of absolutely minimizing functions*, Bull. Amer. Math. Soc. (N.S.)**41**(2004), no.Β 4, 439β505. MR**2083637**, DOI 10.1090/S0273-0979-04-01035-3 - E. N. Barron and R. Jensen,
*Minimizing the $L^\infty$ norm of the gradient with an energy constraint*, Comm. Partial Differential Equations**30**(2005), no.Β 10-12, 1741β1772. MR**2182310**, DOI 10.1080/03605300500299976 - E. N. Barron, R. R. Jensen, and C. Y. Wang,
*The Euler equation and absolute minimizers of $L^\infty$ functionals*, Arch. Ration. Mech. Anal.**157**(2001), no.Β 4, 255β283. MR**1831173**, DOI 10.1007/PL00004239 - Thierry Champion and Luigi De Pascale,
*Principles of comparison with distance functions for absolute minimizers*, J. Convex Anal.**14**(2007), no.Β 3, 515β541. MR**2341302** - Michael G. Crandall,
*An efficient derivation of the Aronsson equation*, Arch. Ration. Mech. Anal.**167**(2003), no.Β 4, 271β279. MR**1981858**, DOI 10.1007/s00205-002-0236-3 - 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. - Michael G. Crandall and L. C. Evans,
*A remark on infinity harmonic functions*, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (ViΓ±a del Mar-Valparaiso, 2000) Electron. J. Differ. Equ. Conf., vol. 6, Southwest Texas State Univ., San Marcos, TX, 2001, pp.Β 123β129. MR**1804769** - M. G. Crandall, L. C. Evans, and R. F. Gariepy,
*Optimal Lipschitz extensions and the infinity Laplacian*, Calc. Var. Partial Differential Equations**13**(2001), no.Β 2, 123β139. MR**1861094**, DOI 10.1007/s005260000065 - Michael G. Crandall, Gunnar Gunnarsson, and Peiyong Wang,
*Uniqueness of $\infty$-harmonic functions and the eikonal equation*, Comm. Partial Differential Equations**32**(2007), no.Β 10-12, 1587β1615. MR**2372480**, DOI 10.1080/03605300601088807 - Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions,
*Userβs guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc. (N.S.)**27**(1992), no.Β 1, 1β67. MR**1118699**, DOI 10.1090/S0273-0979-1992-00266-5 - Albert Fathi and Antonio Siconolfi,
*PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians*, Calc. Var. Partial Differential Equations**22**(2005), no.Β 2, 185β228. MR**2106767**, DOI 10.1007/s00526-004-0271-z - Ronald Gariepy, Changyou Wang, and Yifeng Yu,
*Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers*, Comm. Partial Differential Equations**31**(2006), no.Β 7-9, 1027β1046. MR**2254602**, DOI 10.1080/03605300600636788 - Robert Jensen,
*Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient*, Arch. Rational Mech. Anal.**123**(1993), no.Β 1, 51β74. MR**1218686**, DOI 10.1007/BF00386368 - R. Jensen, C. Wang, Y. Yu,
*Uniqueness for Viscosity Solutions of Aronsson Equations,*Arch. Ration. Mech. Anal., to appear. - Petri Juutinen,
*Minimization problems for Lipschitz functions via viscosity solutions*, Ann. Acad. Sci. Fenn. Math. Diss.**115**(1998), 53. Dissertation, University of JyvΓ€skulΓ€, JyvΓ€skulΓ€, 1998. MR**1632063** - Pierre-Louis Lions,
*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669** - Yifeng Yu,
*$L^\infty$ variational problems and Aronsson equations*, Arch. Ration. Mech. Anal.**182**(2006), no.Β 1, 153β180. MR**2247955**, DOI 10.1007/s00205-006-0424-7 - Y. Yu,
*$L^{\infty }$ variational problems, Aronsson equations and weak KAM theory*, Ph.D. dissertation, U.C. Berkeley, 2005.

## 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