Book Review
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MathSciNet review:
1567628
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Book Information:
Author:
Sigeru Mizohata
Title:
On the Cauchy problem
Additional book information:
Science Press, Beijing, and Academic Press, Orlando, 1985, 177 pp., $36.00 (paper). ISBN 0-12-501660-3.
A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16–36. MR 104925, DOI 10.2307/2372819
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
A.-P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901–921. MR 100768, DOI 10.2307/2372441
4. A. Cauchy, Mémoire sur un théorème fondamental dans le calcul intégral, C. R. Acad. Sci. Paris 14 (1842), 1020.
5. A. Cauchy, Mémoire sur l'emploi du calcul des limites dans l'integration des équations aux dérivées partielles, C. R. Acad. Sci. Paris 15 (1842), 44.
6. A. Cauchy, Mémoire sur l'application du calcul des limites à rintegration d'un système d 'equations aux dérivées partielles, C. R. Acad. Sci. Paris 15 (1842), 85.
7. A. Cauchy, Mémoire sur les systèmes d'equations aux dérivées partielles d'ordres quelconques, et sur leur réduction à des systèmes d'equations linéaires du premier ordre, C. .R. Acad. Sci. Paris 15 (1842), 131.
Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
G. B. Folland, Lectures on partial differential equations, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 70, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1983. MR 696577, DOI 10.1007/978-3-662-00729-7
10. M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, J. Math. Pures Appl. 9(6) (1913), 305-475.
11. M. Gevrey, Sur les fonctions indéfinement dérivables de classe donnée et leur rôle dans la théorie des équations partielles, C. R. Acad. Sci. Paris 157 (1913), 1121-1124.
12. J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, New Haven, CT, 1923.
13. E. Holmgren, Über Systeme von linearen partiellen Differentialgleichungen, Öfversigt af Kongl. Vetenskap-Akad. Förh. 58 (1901), 91-103.
Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183. MR 0383152
Lars Hörmander, Linear differential operators, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 121–133. MR 0513000
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. MR 176362, DOI 10.1002/cpa.3160180121
18. S. Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. Reine Angew. Math. 80 (1875), 1-32.
Peter D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627–646. MR 97628
S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (Russian). MR 0080799
Sigeru Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ. 1 (1961/62), 109–127. MR 170112, DOI 10.1215/kjm/1250525109
22. I. G. Petrowsky, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Mat. Sb. 2(44) (1937), 815-870.
23. I. G. Petrowsky, Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen im Gebiete der nichtanalytischen Funktionen, Bull. Univ. Moscow Ser. Int. 1(7) (1938), 1-74.
Mikio Sato, Regularity of hyperfunctions solutions of partial differential equations, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 785–794. MR 0650826
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, University Series in Mathematics, Plenum Press, New York-London, 1980. Fourier integral operators. MR 597145
- 1.
- A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 0104925
- 2.
- A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 52553
- 3.
- A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901-921. MR 100768
- 4.
- A. Cauchy, Mémoire sur un théorème fondamental dans le calcul intégral, C. R. Acad. Sci. Paris 14 (1842), 1020.
- 5.
- A. Cauchy, Mémoire sur l'emploi du calcul des limites dans l'integration des équations aux dérivées partielles, C. R. Acad. Sci. Paris 15 (1842), 44.
- 6.
- A. Cauchy, Mémoire sur l'application du calcul des limites à rintegration d'un système d 'equations aux dérivées partielles, C. R. Acad. Sci. Paris 15 (1842), 85.
- 7.
- A. Cauchy, Mémoire sur les systèmes d'equations aux dérivées partielles d'ordres quelconques, et sur leur réduction à des systèmes d'equations linéaires du premier ordre, C. .R. Acad. Sci. Paris 15 (1842), 131.
- 8.
- J. Chazarain and A. Piriou, Introduction to the theory of linear partial differential equations, North-Holland, Amsterdam, 1982. MR 678605
- 9.
- G. Folland, Lectures on partial differential equations, Tata Institute of Fundamental Research, Springer-Verlag, Berlin and New York, 1983. MR 696577
- 10.
- M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, J. Math. Pures Appl. 9(6) (1913), 305-475.
- 11.
- M. Gevrey, Sur les fonctions indéfinement dérivables de classe donnée et leur rôle dans la théorie des équations partielles, C. R. Acad. Sci. Paris 157 (1913), 1121-1124.
- 12.
- J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, New Haven, CT, 1923.
- 13.
- E. Holmgren, Über Systeme von linearen partiellen Differentialgleichungen, Öfversigt af Kongl. Vetenskap-Akad. Förh. 58 (1901), 91-103.
- 14.
- L. Hörmander, Pseudo-differential operators and hypoelliptic equations, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 138-183. MR 383152
- 15.
- L. Hörmander, Linear differential operators, Actes Congr. Int. Math. Nice 1 (1970), 121-133. MR 513000
- 16.
- L. Hörmander, The analysis of linear partial differential operators, vols. I-IV, Springer-Verlag, Berlin and New York, 1983-1985. MR 717035
- 17.
- J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269-305. MR 176362
- 18.
- S. Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. Reine Angew. Math. 80 (1875), 1-32.
- 19.
- P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627-646. MR 97628
- 20.
- S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956), 701-703. (Russian) MR 80799
- 21.
- S. Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ. 1 (1961), 109-127. MR 170112
- 22.
- I. G. Petrowsky, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Mat. Sb. 2(44) (1937), 815-870.
- 23.
- I. G. Petrowsky, Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen im Gebiete der nichtanalytischen Funktionen, Bull. Univ. Moscow Ser. Int. 1(7) (1938), 1-74.
- 24.
- M. Sato, Regularity of hyperfunction solutions of partial differential equations, Actes Congr. Int. Math. Nice 2 (1970), 785-794. MR 650826
- 25.
- M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, N. J., 1981. MR 618463
- 26.
- F. Treves, Introduction to pseudodifferential and Fourier integral operators. Vol. I: Pseudodifferential operators; Vol. 2: Fourier integral operators, Plenum Press, New York, 1980. MR 597145
Review Information:
Reviewer:
Michael Beals
Journal:
Bull. Amer. Math. Soc.
17 (1987), 169-176
DOI:
https://doi.org/10.1090/S0273-0979-1987-15552-2