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 $H\in C^1.$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