Book Review

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MathSciNet review: 1567454

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Book Information:

Author: P. L. Lions

Title: Generalized solutions of Hamilton-Jacobi equations

Additional book information: Research Notes in Mathematics, Vol. 69, Pitman Advanced Publishing Program, Boston, 1982, 317 pp., $24.95. ISBN 0-2730-8556-5.

**1.**Sadakazu Aizawa,*A semigroup treatment of the Hamilton-Jacobi equation in one space variable*, Hiroshima Math. J.**3**(1973), 367–386. MR**346300****2.**Sadakazu Aizawa,*A semigroup treatment of the Hamilton-Jacobi equation in several space variables*, Hiroshima Math. J.**6**(1976), no. 1, 15–30. MR**393779****3.**Sadakazu Aizawa and Norio Kikuchi,*A mixed initial and boundary-value problem for the Hamilton-Jacobi equation in several space variables*, Funkcial. Ekvac.**9**(1966), 139–150. MR**211056****4.**Stanley H. Benton Jr.,*A general space-time boundary value problem for the Hamilton-Jacobi equation*, J. Differential Equations**11**(1972), 425–435. MR**298196**, https://doi.org/10.1016/0022-0396(72)90056-3**5.**Stanley H. Benton Jr.,*Global variational solutions of Hamilton-Jacobi boundary value problems*, J. Differential Equations**13**(1973), 468–480. MR**402253**, https://doi.org/10.1016/0022-0396(73)90005-3**6.**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****7.**B. C. Burch,*A semigroup approach to the Hamilton-Jacobi equation*, Tulane Univ. dissertation, New Orleans, 1975.**8.**B. C. Burch,*A semigroup treatment of the Hamilton-Jacobi equation in several space variables*, J. Differential Equations**23**(1977), no. 1, 107–124. MR**440183**, https://doi.org/10.1016/0022-0396(77)90137-1**9.**Julian D. Cole,*On a quasi-linear parabolic equation occurring in aerodynamics*, Quart. Appl. Math.**9**(1951), 225–236. MR**0042889**, https://doi.org/10.1090/S0033-569X-1951-42889-X**10.**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****11.**M. G. Crandall and T. M. Liggett,*Generation of semi-groups of nonlinear transformations on general Banach spaces*, Amer. J. Math.**93**(1971), 265–298. MR**287357**, https://doi.org/10.2307/2373376**12.**Avron Douglis,*Solutions in the large for multi-dimensional, non-linear partial differential equations of first order*, Ann. Inst. Fourier (Grenoble)**15**(1965), no. fasc., fasc. 2, 1–35. MR**199542****13.**Avron Douglis,*Layering methods for nonlinear partial differential equations of first order*, Ann. Inst. Fourier (Grenoble)**22**(1972), no. 3, 141–227 (English, with French summary). MR**358089****14.**Robert J. Elliott and Nigel J. Kalton,*The existence of value in differential games*, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 126. MR**0359845****15.**Robert J. Elliott and Nigel J. Kalton,*The existence of value in differential games of pursuit and evasion*, J. Differential Equations**12**(1972), 504–523. MR**359846**, https://doi.org/10.1016/0022-0396(72)90022-8**16.**Robert J. Elliott and Nigel J. Kalton,*Cauchy problems for certain Isaacs-Bellman equations and games of survival*, Trans. Amer. Math. Soc.**198**(1974), 45–72. MR**347383**, https://doi.org/10.1090/S0002-9947-1974-0347383-8**17.**Robert J. Elliott and Nigel J. Kalton,*Boundary value problems for nonlinear partial differential operators*, J. Math. Anal. Appl.**46**(1974), 228–241. MR**395887**, https://doi.org/10.1016/0022-247X(74)90293-5**18.**E. E. Feltus,*Mixed problems for the Hamilton-Jacobi equation*, Tulane Univ. dissertation, New Orleans, 1975.**19.**Wendell H. Fleming,*The Cauchy problem for a nonlinear first order partial differential equation*, J. Differential Equations**5**(1969), 515–530. MR**235269**, https://doi.org/10.1016/0022-0396(69)90091-6**20.**Andrew Russell Forsyth,*Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations*, Six volumes bound as three, Dover Publications, Inc., New York, 1959. MR**0123757****21.**E. Hopf,*The partial differential equation u*+*uu*=*µu*, Comm. Pure Appl. Math. 3 (1950), 201-230.**22.**Eberhard Hopf,*Generalized solutions of non-linear equations of first order*, J. Math. Mech.**14**(1965), 951–973. MR**0182790****23.**S. N. Kružkov,*Generalized solutions of nonlinear equations of the first order with several independent variables. II*, Mat. Sb. (N.S.)**72 (114)**(1967), 108–134 (Russian). MR**0204847****24.**Peter D. Lax,*Nonlinear hyperbolic equations*, Comm. Pure Appl. Math.**6**(1953), 231–258. MR**56176**, https://doi.org/10.1002/cpa.3160060204**25.**P. D. Lax,*The initial value problem for nonlinear hyperbolic equations in two independent variables*, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., 1954, pp. 211–229. MR**0068093****26.**Peter D. Lax,*Weak solutions of nonlinear hyperbolic equations and their numerical computation*, Comm. Pure Appl. Math.**7**(1954), 159–193. MR**66040**, https://doi.org/10.1002/cpa.3160070112**27.**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, https://doi.org/10.1002/cpa.3160100406**28.**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737**, https://doi.org/10.1090/trans2/026/05

Review Information:

Reviewer: Stanley H. Benton

Journal: Bull. Amer. Math. Soc.

**9**(1983), 252-256

DOI: https://doi.org/10.1090/S0273-0979-1983-15174-1