Authors

Michael G. Crandall, Pierre-Louis Lions

Abstract

Problems involving Hamilton-Jacobi equations—which we take to be either of the stationary form H ( x , u , D u ) = 0 H(x,u,Du) = 0 or of the evolution form u t + H ( x , t , u , D u ) = 0 {u_{t}[[:space:]]} + H(x,t,u,Du) = 0 , where D u Du is the spatial gradient of u u —arise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.

Citation

  • Journal: Transactions of the American Mathematical Society
  • Year: 2010
  • Volume: 277
  • Issue: 1
  • Pages: 1–42
  • Publisher: American Mathematical Society (AMS)
  • DOI: 10.1090/s0002-9947-1983-0690039-8

BibTeX

@article{Crandall_1983,
  title={{Viscosity solutions of Hamilton-Jacobi equations}},
  volume={277},
  ISSN={1088-6850},
  DOI={10.1090/s0002-9947-1983-0690039-8},
  number={1},
  journal={Transactions of the American Mathematical Society},
  publisher={American Mathematical Society (AMS)},
  author={Crandall, Michael G. and Lions, Pierre-Louis},
  year={1983},
  pages={1--42}
}

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References

  • Aizawa, Sadakazu, A semigroup treatment of the Hamilton-Jacobi equation in several space variables. Hiroshima Math. J. (1976)
  • Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces. (Springer Netherlands, 1976). doi:10.1007/978-94-010-1537-0 – 10.1007/978-94-010-1537-0
  • Beck, Anatole, Uniqueness of flow solutions of differential equations. (1973)
  • Benton, Stanley H., Jr., The Hamilton-Jacobi equation (1977)
  • Bony, Jean-Michel, Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris S'{e}r. A-B (1967)
  • Conway, E. D., Hamilton’s theory and generalized solutions of the Hamilton-Jacobi equation. J. Math. Mech. (1964)
  • Crandall, Michael G., An introduction to evolution governed by accretive operators. (1976)
  • Crandall, Michael G., Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. C. R. Acad. Sci. Paris S'{e}r. I Math. (1981)
  • Douglis, A. The continuous dependence of generalized solutions of non‐linear partial differential equations upon initial data. Communications on Pure and Applied Mathematics vol. 14 267–284 (1961) – 10.1002/cpa.3160140307
  • Evans, L. C. On solving certain nonlinear partial differential equations by accretive operator methods. Israel Journal of Mathematics vol. 36 225–247 (1980) – 10.1007/bf02762047
  • Evans, Lawrence C., Application of nonlinear semigroup theory to certain partial differential equations. (1978)
  • Fleming, W. H. The Cauchy problem for a nonlinear first order partial differential equation. Journal of Differential Equations vol. 5 515–530 (1969) – 10.1016/0022-0396(69)90091-6
  • Fleming, Wendell H., The Cauchy problem for degenerate parabolic equations. J. Math. Mech. (1964)
  • Friedman, A. The Cauchy Problem for First Order Partial Differential Equations. Indiana University Mathematics Journal vol. 23 27–40 (1973) – 10.1512/iumj.1973.23.23004
  • Hopf, E. On the Right Weak Solution of the Cauchy Problem for a Quasilinear Equation of First Order. Indiana University Mathematics Journal vol. 19 483–487 (1969) – 10.1512/iumj.1970.19.19045
  • Kružkov, S. N. GENERALIZED SOLUTIONS OF THE HAMILTON-JACOBI EQUATIONS OF EIKONAL TYPE. I. FORMULATION OF THE PROBLEMS; EXISTENCE, UNIQUENESS AND STABILITY THEOREMS; SOME PROPERTIES OF THE SOLUTIONS. Mathematics of the USSR-Sbornik vol. 27 406–446 (1975) – 10.1070/sm1975v027n03abeh002522
  • Kružkov, S. N., Generalized solutions of nonlinear equations of the first order with several variables. I. Mat. Sb. (N.S.) (1966)
  • Kružkov, S. N. FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES. Mathematics of the USSR-Sbornik vol. 10 217–243 (1970) – 10.1070/sm1970v010n02abeh002156
  • Lions, Pierre-Louis, Generalized solutions of Hamilton-Jacobi equations (1982)
  • Lions, P. L. Control of diffusion processes in RN. Communications on Pure and Applied Mathematics vol. 34 121–147 (1981) – 10.1002/cpa.3160340106
  • Oleĭnik, O. A. Discontinuous solutions of non-linear differential equations. American Mathematical Society Translations: Series 2 95–172 (1963) doi:10.1090/trans2/026/05 – 10.1090/trans2/026/05
  • Vol′pert, A. I., Spaces 𝐵𝑉 and quasilinear equations. Mat. Sb. (N.S.) (1967)
  • Crandall, M. G., Evans, L. C. & Lions, P.-L. Some properties of viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society vol. 282 487–502 (1984) – 10.2307/1999247