Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians
HTML articles powered by AMS MathViewer
- by Atsushi Nakayasu PDF
- Proc. Amer. Math. Soc. 147 (2019), 701-710 Request permission
Abstract:
Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton–Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen’s inequality. The second proof is completely different with elementary calculations. It is based on the convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.References
- Scott N. Armstrong and Panagiotis E. Souganidis, Stochastic homogenization of level-set convex Hamilton-Jacobi equations, Int. Math. Res. Not. IMRN 15 (2013), 3420–3449. MR 3089731, DOI 10.1093/imrn/rns155
- Scott N. Armstrong, Hung V. Tran, and Yifeng Yu, Stochastic homogenization of a nonconvex Hamilton-Jacobi equation, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1507–1524. MR 3396421, DOI 10.1007/s00526-015-0833-2
- Scott N. Armstrong, Hung V. Tran, and Yifeng Yu, Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, J. Differential Equations 261 (2016), no. 5, 2702–2737. MR 3507985, DOI 10.1016/j.jde.2016.05.010
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- Martino Bardi and Italo Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. MR 1484411, DOI 10.1007/978-0-8176-4755-1
- E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 15 (1990), no. 12, 1713–1742. MR 1080619, DOI 10.1080/03605309908820745
- E. N. Barron, R. Jensen, and W. Liu, Hopf-Lax-type formula for $u_t+H(u,Du)=0$, J. Differential Equations 126 (1996), no. 1, 48–61. MR 1382056, DOI 10.1006/jdeq.1996.0043
- E. N. Barron, R. R. Jensen, and C. Y. Wang, Lower semicontinuity of $L^\infty$ functionals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 4, 495–517 (English, with English and French summaries). MR 1841130, DOI 10.1016/S0294-1449(01)00070-1
- F. Cagnetti, D. Gomes, and H. V. Tran, Aubry-Mather measures in the nonconvex setting, SIAM J. Math. Anal. 43 (2011), no. 6, 2601–2629. MR 2873233, DOI 10.1137/100817656
- Fabio Camilli, Italo Capuzzo Dolcetta, and Diogo A. Gomes, Error estimates for the approximation of the effective Hamiltonian, Appl. Math. Optim. 57 (2008), no. 1, 30–57. MR 2373005, DOI 10.1007/s00245-007-9006-9
- Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI 10.1090/S0002-9947-1975-0367131-6
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
- G. Contreras, R. Iturriaga, G. P. Paternain, and M. Paternain, Lagrangian graphs, minimizing measures and Mañé’s critical values, Geom. Funct. Anal. 8 (1998), no. 5, 788–809. MR 1650090, DOI 10.1007/s000390050074
- S. S. Dragomir and C. E. M. Pearce, Jensen’s inequality for quasiconvex functions, Numer. Algebra Control Optim. 2 (2012), no. 2, 279–291. MR 2929452, DOI 10.3934/naco.2012.2.279
- Lawrence C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159–177. MR 1986317, DOI 10.1007/s00526-002-0164-y
- L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal. 157 (2001), no. 1, 1–33. MR 1822413, DOI 10.1007/PL00004236
- L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. II, Arch. Ration. Mech. Anal. 161 (2002), no. 4, 271–305. MR 1891169, DOI 10.1007/s002050100181
- L. C. Evans and D. Gomes, Linear programming interpretations of Mather’s variational principle, ESAIM Control Optim. Calc. Var. 8 (2002), 693–702. A tribute to J. L. Lions. MR 1932968, DOI 10.1051/cocv:2002030
- Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185–228. MR 2106767, DOI 10.1007/s00526-004-0271-z
- Luke Finlay, Vladimir Gaitsgory, and Ivan Lebedev, Duality in linear programming problems related to deterministic long run average problems of optimal control, SIAM J. Control Optim. 47 (2008), no. 4, 1667–1700. MR 2421325, DOI 10.1137/060676398
- Yoshikazu Giga, Qing Liu, and Hiroyoshi Mitake, Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1905–1941. MR 3152717, DOI 10.1090/S0002-9947-2013-05905-3
- Diogo Aguiar Gomes, A stochastic analogue of Aubry-Mather theory, Nonlinearity 15 (2002), no. 3, 581–603. MR 1901094, DOI 10.1088/0951-7715/15/3/304
- Diogo A. Gomes and Adam M. Oberman, Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim. 43 (2004), no. 3, 792–812. MR 2114376, DOI 10.1137/S0363012902417620
- Luca Granieri, A finite dimensional linear programming approximation of Mather’s variational problem, ESAIM Control Optim. Calc. Var. 16 (2010), no. 4, 1094–1109. MR 2744164, DOI 10.1051/cocv/2009039
- Nao Hamamuki, Atsushi Nakayasu, and Tokinaga Namba, On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and their application to homogenization problems, J. Differential Equations 259 (2015), no. 11, 6672–6693. MR 3397334, DOI 10.1016/j.jde.2015.08.003
- P.-L. Lions, G. C. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton–Jacobi equations, unpublished.
- Songting Luo, Yifeng Yu, and Hongkai Zhao, A new approximation for effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations, Multiscale Model. Simul. 9 (2011), no. 2, 711–734. MR 2818417, DOI 10.1137/100799885
- Hiroyoshi Mitake and Hung V. Tran, Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates, Arch. Ration. Mech. Anal. 211 (2014), no. 3, 733–769. MR 3158806, DOI 10.1007/s00205-013-0685-x
- Gawtum Namah and Jean-Michel Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations 24 (1999), no. 5-6, 883–893. MR 1680905, DOI 10.1080/03605309908821451
- Adam M. Oberman, Ryo Takei, and Alexander Vladimirsky, Homogenization of metric Hamilton-Jacobi equations, Multiscale Model. Simul. 8 (2009), no. 1, 269–295. MR 2575055, DOI 10.1137/080743019
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- M. Rorro, An approximation scheme for the effective Hamiltonian and applications, Appl. Numer. Math. 56 (2006), no. 9, 1238–1254. MR 2244974, DOI 10.1016/j.apnum.2006.03.006
- Etsuro Yokoyama, Yoshikazu Giga, and Piotr Rybka, A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation, Phys. D 237 (2008), no. 22, 2845–2855. MR 2514066, DOI 10.1016/j.physd.2008.05.009
Additional Information
- Atsushi Nakayasu
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan
- MR Author ID: 1057585
- Email: ankys@ms.u-tokyo.ac.jp
- Received by editor(s): December 21, 2014
- Received by editor(s) in revised form: May 8, 2018
- Published electronically: October 12, 2018
- Additional Notes: The work of the author was supported by a Grant-in-Aid for JSPS Fellows No. 25-7077 and the Program for Leading Graduate Schools, MEXT, Japan.
- Communicated by: Yingfei Yi
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 701-710
- MSC (2010): Primary 35F21; Secondary 49L25, 26B25, 26B05
- DOI: https://doi.org/10.1090/proc/14280
- MathSciNet review: 3894909