A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton-Jacobi equations. (English) Zbl 1269.49044
Loreti, Paola (ed.) et al., Hamilton-Jacobi equations: approximations, numerical analysis and applications. Based on the lectures of the CIME summer school, Cetraro, Italy, August 29–September 3, 2011. Berlin: Springer; Florence: Fondazione CIME (ISBN 978-3-642-36432-7/pbk; 978-3-642-36433-4/ebook). Lecture Notes in Mathematics 2074. CIME Foundation Subseries, 111-249 (2013).
Summary: We present an introduction to the theory of viscosity solutions of first-order partial differential equations and a review on the optimal control/dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, with the Neumann boundary condition. This article also includes some of the basics of mathematical analysis related to the optimal control/dynamical approach for easy accessibility to the topics.
For the entire collection see [Zbl 1266.65002].
For the entire collection see [Zbl 1266.65002].
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
Keywords:
first-order Hamilton-Jacobi equations; viscosity solutions; optimal control; dynamical approach; large time behavior: Neumann boundary conditionReferences:
| [1] | L. Ambrosio, P. Tilli, Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and Its Applications, vol. 25 (Oxford University Press, Oxford, 2004), viii + 133 pp · Zbl 1080.28001 |
| [2] | M. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems & Control: Foundations & Applications (Birkhäuser Boston, Inc., Boston, 1997), xviii + 570 pp · Zbl 0890.49011 |
| [3] | Barles, G.; Souganidis, PE, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31, 4, 925-939 (2000) · Zbl 0960.70015 · doi:10.1137/S0036141099350869 |
| [4] | G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, ed. by P. Loreti, N.A. Tchou. Lecture Notes in Mathematics (Springer, Berlin/Heidelberg, 2013) · Zbl 1269.49043 |
| [5] | Barles, G., Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit, Nonlinear Anal., 20, 9, 1123-1134 (1993) · Zbl 0816.35081 · doi:10.1016/0362-546X(93)90098-D |
| [6] | G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin), vol. 17 (Springer, Paris, 1994), x + 194 pp · Zbl 0819.35002 |
| [7] | G. Barles, H. Ishii, H. Mitake, A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations. Bull. Math. Sci. (to appear) · Zbl 1284.35065 |
| [8] | Barles, G.; Ishii, H.; Mitake, H., On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions, Arch. Ration. Mech. Anal., 204, 2, 515-558 (2012) · Zbl 1282.70037 · doi:10.1007/s00205-011-0484-1 |
| [9] | Barles, G.; Mitake, H., A pde approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 37, 1, 136-168 (2012) · Zbl 1243.35021 · doi:10.1080/03605302.2011.553645 |
| [10] | Barles, G.; Roquejoffre, J.-M., Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 31, 7-9, 1209-1225 (2006) · Zbl 1107.35019 · doi:10.1080/03605300500361461 |
| [11] | Barron, EN; Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Commun. Partial Differ. Equ., 15, 12, 1713-1742 (1990) · Zbl 0732.35014 · doi:10.1080/03605309908820745 |
| [12] | Bernard, P., Existence of C 1, 1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup. (4), 40, 3, 445-452 (2007) · Zbl 1133.35027 |
| [13] | Bernard, P.; Roquejoffre, J.-M., Convergence to time-periodic solutions in time-periodic Hamilton-Jacobi equations on the circle, Commun. Partial Differ. Equ., 29, 3-4, 457-469 (2004) · Zbl 1072.35055 |
| [14] | G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems. An Introduction. Oxford Lecture Series in Mathematics and Its Applications, vol. 15 (The Clarendon Press/Oxford University Press, New York, 1998), viii + 262 pp · Zbl 0915.49001 |
| [15] | L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995), vi + 104 pp · Zbl 0834.35002 |
| [16] | Camilli, F.; Ley, O.; Loreti, P.; Nguyen, V. D., Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations, Nonlinear Differ. Equ. Appl. (NoDEA), 19, 6, 719-749 (2012) · Zbl 1254.49016 |
| [17] | Capuzzo-Dolcetta, I.; Lions, P.-L., Hamilton-Jacobi equations with state constraints, Trans. Am. Math. Soc., 318, 2, 643-683 (1990) · Zbl 0702.49019 · doi:10.1090/S0002-9947-1990-0951880-0 |
| [18] | Crandall, MG; Evans, LC; Lions, P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 282, 2, 487-502 (1984) · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X |
| [19] | Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. (N.S.), 27, 1, 1-67 (1992) · Zbl 0755.35015 |
| [20] | Crandall, MG; Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277, 1, 1-42 (1983) · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8 |
| [21] | Davini, A.; Siconolfi, A., A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38, 2, 478-502 (2006) · Zbl 1109.49034 · doi:10.1137/050621955 |
| [22] | W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation. Commun. Pure Appl. Math. 52(7), 811-828 (1999) · Zbl 0916.35099 |
| [23] | Evans, LC, On solving certain nonlinear partial differential equations by accretive operator methods, Isr. J. Math., 36, 3-4, 225-247 (1980) · Zbl 0454.35038 · doi:10.1007/BF02762047 |
| [24] | Evans, LC, A survey of partial differential equations methods in weak KAM theory, Commun. Pure Appl. Math., 57, 4, 445-480 (2004) · Zbl 1040.37046 · doi:10.1002/cpa.20009 |
| [25] | Fathi, A., Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324, 9, 1043-1046 (1997) · Zbl 0885.58022 · doi:10.1016/S0764-4442(97)87883-4 |
| [26] | Fathi, A., Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327, 3, 267-270 (1998) · Zbl 1052.37514 · doi:10.1016/S0764-4442(98)80144-4 |
| [27] | A. Fathi, Weak KAM theorem in Lagrangian dynamics. Cambridge University Press (to appear) · Zbl 0885.58022 |
| [28] | Fathi, A.; Siconolfi, A., Existence of C^1 critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155, 2, 363-388 (2004) · Zbl 1061.58008 · doi:10.1007/s00222-003-0323-6 |
| [29] | W.H. Fleming, H. Mete Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edn. Stochastic Modelling and Applied Probability, vol. 25 (Springer, New York, 2006), xviii + 429 pp · Zbl 1105.60005 |
| [30] | Fujita, Y.; Ishii, H.; Loreti, P., Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space, Indiana Univ. Math. J., 55, 5, 1671-1700 (2006) · Zbl 1112.35041 · doi:10.1512/iumj.2006.55.2813 |
| [31] | Y. Giga, Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99 (Birkhäuser, Basel, 2006), xii + 264 pp · Zbl 1096.53039 |
| [32] | Y. Giga, Q. Liu, H. Mitake, Singular Neumann problems and large-time behavior of solutions of non-coercive Hamiton-Jacobi equations. Tran. Amer. Math. Soc. (to appear) · Zbl 1286.35074 |
| [33] | Giga, Y.; Liu, Q.; Mitake, H., Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians, J. Differ. Equ., 252, 2, 1263-1282 (2012) · Zbl 1230.35038 · doi:10.1016/j.jde.2011.10.010 |
| [34] | Ichihara, N.; Ishii, H., Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians, Commun. Partial Differ. Equ., 33, 4-6, 784-807 (2008) · Zbl 1145.35034 · doi:10.1080/03605300701257427 |
| [35] | Ichihara, N.; Ishii, H., The large-time behavior of solutions of Hamilton-Jacobi equations on the real line, Methods Appl. Anal., 15, 2, 223-242 (2008) · Zbl 1170.35339 |
| [36] | Ichihara, N.; Ishii, H., Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194, 2, 383-419 (2009) · Zbl 1243.70017 · doi:10.1007/s00205-008-0170-0 |
| [37] | Ishii, H., Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 25, 2, 231-266 (2008) · Zbl 1145.35035 · doi:10.1016/j.anihpc.2006.09.002 |
| [38] | Ishii, H., Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions, Calc. Var. Partial Differ. Equ., 42, 1-2, 189-209 (2011) · Zbl 1222.35031 · doi:10.1007/s00526-010-0385-4 |
| [39] | Ishii, H., Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl. (9), 95, 1, 99-135 (2011) · Zbl 1213.35184 |
| [40] | Ishii, H.; Mitake, H., Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians, Indiana Univ. Math. J., 56, 5, 2159-2183 (2007) · Zbl 1136.35016 · doi:10.1512/iumj.2007.56.3048 |
| [41] | S. Koike, A Beginner’s Guide to the Theory of Viscosity Solutions. MSJ Memoirs, vol. 13 (Mathematical Society of Japan, Tokyo, 2004), viii + 123 pp · Zbl 1056.49027 |
| [42] | P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69 (Pitman, Boston, 1982), iv + 317 pp · Zbl 0497.35001 |
| [43] | Lions, P.-L., Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J., 52, 4, 793-820 (1985) · Zbl 0599.35025 · doi:10.1215/S0012-7094-85-05242-1 |
| [44] | Lions, P.-L.; Sznitman, A.-S., Stochastic differential equations with reflecting boundary conditions, Commun. Pure Appl. Math., 37, 4, 511-537 (1984) · Zbl 0598.60060 · doi:10.1002/cpa.3160370408 |
| [45] | Lions, P.-L.; Trudinger, NS, Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation, Math. Z., 191, 1, 1-15 (1986) · Zbl 0593.35046 · doi:10.1007/BF01163605 |
| [46] | Mather, J. N., Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43, 5, 1349-1386 (1993) · Zbl 0803.58019 |
| [47] | Mather, JN, Total disconnectedness of the quotient Aubry set in low dimensions, Dedicated to the memory of Jürgen K. Moser. Commun. Pure Appl. Math., 56, 8, 1178-1183 (2003) · Zbl 1046.37039 · doi:10.1002/cpa.10091 |
| [48] | Mitake, H., Asymptotic solutions of Hamilton-Jacobi equations with state constraints, Appl. Math. Optim., 58, 3, 393-410 (2008) · Zbl 1178.35137 · doi:10.1007/s00245-008-9041-1 |
| [49] | H. Mitake, The large-time behavior of solutions of the Cauchy-Dirichlet problem. Nonlinear Differ. Equ. Appl. (NoDEA) 15(3), 347-362 (2008) · Zbl 1171.35332 |
| [50] | Mitake, H., Large time behavior of solutions of Hamilton-Jacobi equations with periodic boundary data, Nonlinear Anal., 71, 11, 5392-5405 (2009) · Zbl 1184.35061 · doi:10.1016/j.na.2009.04.028 |
| [51] | Mitake, H.; Tran, HV, Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations, Asymptot. Anal., 77, 1-2, 43-70 (2012) · Zbl 1241.35020 |
| [52] | H. Mitake, H.V. Tran, A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations. J. Math. Pures Appl. (to appear) · Zbl 1278.35020 |
| [53] | Namah, G.; Roquejoffre, J.-M., Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 24, 5-6, 883-893 (1999) · Zbl 0924.35028 · doi:10.1080/03605309908821451 |
| [54] | Roquejoffre, J.-M., Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80, 1, 85-104 (2001) · Zbl 0979.35033 |
| [55] | Soner, HM, Optimal control with state-space constraint, I. SIAM J. Control Optim., 24, 3, 552-561 (1986) · Zbl 0597.49023 · doi:10.1137/0324032 |
| [56] | M. Spivak, Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus (W.A. Benjamin, Inc., New York, 1965), xii + 144 pp · Zbl 0141.05403 |
| [57] | E.M. Stein, R. Shakarchi, Real Analysis. Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. III (Princeton University Press, Princeton, 2005), xx + 402 pp · Zbl 1081.28001 |
| [58] | Yokoyama, E.; Giga, Y.; Rybka, P., A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation, Phys. D, 237, 22, 2845-2855 (2008) · Zbl 1375.82118 · doi:10.1016/j.physd.2008.05.009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
