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.

- Sadakazu Aizawa,
*A semigroup treatment of the Hamilton-Jacobi equation in one space variable*, Hiroshima Math. J.**3**(1973), 367–386. MR**346300** - 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** - 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** - 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**, DOI https://doi.org/10.1016/0022-0396%2872%2990056-3 - Stanley H. Benton Jr.,
*Global variational solutions of Hamilton-Jacobi boundary value problems*, J. Differential Equations**13**(1973), 468–480. MR**402253**, DOI https://doi.org/10.1016/0022-0396%2873%2990005-3 - 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**
B. C. Burch, - 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**, DOI https://doi.org/10.1016/0022-0396%2877%2990137-1 - Julian D. Cole,
*On a quasi-linear parabolic equation occurring in aerodynamics*, Quart. Appl. Math.**9**(1951), 225–236. MR**42889**, DOI https://doi.org/10.1090/S0033-569X-1951-42889-X - 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** - 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**, DOI https://doi.org/10.2307/2373376 - 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. 2, 1–35. MR**199542** - 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** - 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** - 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**, DOI https://doi.org/10.1016/0022-0396%2872%2990022-8 - 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**, DOI https://doi.org/10.1090/S0002-9947-1974-0347383-8 - 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**, DOI https://doi.org/10.1016/0022-247X%2874%2990293-5
E. E. Feltus, - Wendell H. Fleming,
*The Cauchy problem for a nonlinear first order partial differential equation*, J. Differential Equations**5**(1969), 515–530. MR**235269**, DOI https://doi.org/10.1016/0022-0396%2869%2990091-6 - 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*, Dover Publications, Inc., New York, 1959. Six volumes bound as three. MR**0123757**
E. Hopf, - Eberhard Hopf,
*Generalized solutions of non-linear equations of first order*, J. Math. Mech.**14**(1965), 951–973. MR**0182790** - 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** - Peter D. Lax,
*Nonlinear hyperbolic equations*, Comm. Pure Appl. Math.**6**(1953), 231–258. MR**56176**, DOI https://doi.org/10.1002/cpa.3160060204 - 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** - Peter D. Lax,
*Weak solutions of nonlinear hyperbolic equations and their numerical computation*, Comm. Pure Appl. Math.**7**(1954), 159–193. MR**66040**, DOI https://doi.org/10.1002/cpa.3160070112 - P. D. Lax,
*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, DOI https://doi.org/10.1002/cpa.3160100406 - O. A. Oleĭnik,
*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737**, DOI https://doi.org/10.1090/trans2/026/05

*A semigroup approach to the Hamilton-Jacobi equation*, Tulane Univ. dissertation, New Orleans, 1975.

*Mixed problems for the Hamilton-Jacobi equation*, Tulane Univ. dissertation, New Orleans, 1975.

*The partial differential equation u*+

*uu*=

*µu*, Comm. Pure Appl. Math. 3 (1950), 201-230.

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