Viscosity solutions of Hamilton-Jacobi equations

Authors:
Michael G. Crandall and Pierre-Louis Lions

Journal:
Trans. Amer. Math. Soc. **277** (1983), 1-42

MSC:
Primary 35F20

DOI:
https://doi.org/10.1090/S0002-9947-1983-0690039-8

MathSciNet review:
690039

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Problems involving Hamilton-Jacobi equations--which we take to be either of the stationary form or of the evolution form , where is the spatial gradient of --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.

**[1]**Sadakazu Aizawa,*A semigroup treatment of the Hamilton-Jacobi equation in several space variables*, Hiroshima Math. J.**6**(1976), no. 1, 15–30. MR**0393779****[2]**Viorel Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843****[3]**Anatole Beck,*Uniqueness of flow solutions of differential equations*, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer, Berlin, 1973, pp. 30–50. Lecture Notes in Math., Vol. 318. MR**0409997****[4]**Stanley H. Benton Jr.,*The Hamilton-Jacobi equation*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. A global approach; Mathematics in Science and Engineering, Vol. 131. MR**0442431****[5]**Jean-Michel Bony,*Principe du maximum dans les espaces de Sobolev*, C. R. Acad. Sci. Paris Sér. A-B**265**(1967), A333–A336 (French). MR**0223711****[6]**E. D. Conway and E. Hopf,*Hamilton’s theory and generalized solutions of the Hamilton-Jacobi equation*, J. Math. Mech.**13**(1964), 939–986. MR**0182761****[7]**Michael G. Crandall,*An introduction to evolution governed by accretive operators*, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 131–165. MR**0636953****[8]**Michael G. Crandall and Pierre-Louis Lions,*Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 3, 183–186 (French, with English summary). MR**610314****[9]**-,*Two approximations of solutions of Hamilton-Jacobi equations*(to appear).**[10]**Avron Douglis,*The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data*, Comm. Pure Appl. Math.**14**(1961), 267–284. MR**0139848**, https://doi.org/10.1002/cpa.3160140307**[11]**Lawrence C. Evans,*On solving certain nonlinear partial differential equations by accretive operator methods*, Israel J. Math.**36**(1980), no. 3-4, 225–247. MR**597451**, https://doi.org/10.1007/BF02762047**[12]**Lawrence C. Evans,*Application of nonlinear semigroup theory to certain partial differential equations*, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 163–188. MR**513818****[13]**Wendell H. Fleming,*The Cauchy problem for a nonlinear first order partial differential equation*, J. Differential Equations**5**(1969), 515–530. MR**0235269**, https://doi.org/10.1016/0022-0396(69)90091-6**[14]**-,*Nonlinear partial differential equations--probabilistic and game theoretic methods*, Problems in Nonlinear Analysis, CIME, Ed. Cremonese, Roma, 1971.**[15]**Wendell H. Fleming,*The Cauchy problem for degenerate parabolic equations*, J. Math. Mech.**13**(1964), 987–1008. MR**0179473****[16]**Avner Friedman,*The Cauchy problem for first order partial differential equations*, Indiana Univ. Math. J.**23**(1974), 27–40. MR**0326136**, https://doi.org/10.1512/iumj.1973.23.23004**[17]**Eberhard Hopf,*On the right weak solution of the Cauchy problem for a quasilinear equation of first order*, J. Math. Mech.**19**(1969/1970), 483–487. MR**0251357****[18]**S. N. Kružkov,*Generalized solution of the Hamilton-Jacobi equations of Eikonal type*. I, Math. USSR-Sb.**27**(1975), 406-446.**[19]**-,*Generalized solutions of nonlinear first order equations and certain quasilinear parabolic equations*, Vestnik Moscov. Univ. Ser. I Mat. Meh.**6**(1964), 67-74. (Russian)**[20]**S. N. Kružkov,*Generalized solutions of nonlinear equations of the first order with several variables. I*, Mat. Sb. (N.S.)**70 (112)**(1966), 394–415 (Russian). MR**0199543****[21]**-,*First order quasilinear equations with several space variables*, Math. USSR-Sb.**10**(1970), 217-243.**[22]**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****[23]**P.-L. Lions,*Control of diffusion processes in 𝑅^{𝑁}*, Comm. Pure Appl. Math.**34**(1981), no. 1, 121–147. MR**600574**, https://doi.org/10.1002/cpa.3160340106**[24]**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737****[25]**M. B. Tamburro,*The evolution operator approach to the Hamilton-Jacobi equations*, Israel J. Math.**[26]**A. I. Vol′pert,*Spaces 𝐵𝑉 and quasilinear equations*, Mat. Sb. (N.S.)**73 (115)**(1967), 255–302 (Russian). MR**0216338****[27]**M. G. Crandall, L. C. Evans, and P.-L. Lions,*Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 487–502. MR**732102**, https://doi.org/10.1090/S0002-9947-1984-0732102-X

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35F20

Retrieve articles in all journals with MSC: 35F20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0690039-8

Keywords:
Hamilton-Jacobi equations,
uniqueness criteria

Article copyright:
© Copyright 1983
American Mathematical Society