On the existence and the regularity of solutions of linear pseudodifferential equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 28 (1973) no. 6 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

@article{RM_1973_28_6_a7,
     author = {L. V. H\"ormander},
     title = {On the existence and the regularity of solutions of linear pseudodifferential equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     year = {1973},
     volume = {28},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/RM_1973_28_6_a7/}
}
TY  - JOUR
AU  - L. V. Hörmander
TI  - On the existence and the regularity of solutions of linear pseudodifferential equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1973
VL  - 28
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/RM_1973_28_6_a7/
LA  - ru
ID  - RM_1973_28_6_a7
ER  - 
%0 Journal Article
%A L. V. Hörmander
%T On the existence and the regularity of solutions of linear pseudodifferential equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1973
%V 28
%N 6
%U http://geodesic.mathdoc.fr/item/RM_1973_28_6_a7/
%G ru
%F RM_1973_28_6_a7
L. V. Hörmander. On the existence and the regularity of solutions of linear pseudodifferential equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 28 (1973) no. 6. http://geodesic.mathdoc.fr/item/RM_1973_28_6_a7/

[1] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I; II”, Comm. Pure Appl. Math., 12 (1959), 623–727 ; 17 (1964), 35–94 ; S. Agmon, A. Duglis, L. Nirenberg, Otsenki reshenii ellipticheskikh uravnenii vblizi granitsy, IL, M., 1962 | DOI | MR | Zbl | DOI | MR

[2] K. G. Anderson, “Propagation of analyticity of solutions of partial differential equations with constant coefficients”, Ark. för Matematik, 8 (1970), 277–302 | DOI | MR

[3] J. M. Bony, “Une extension du theoreme de Holmgren sur l'unicite du probleme de Cauchy”, C. R. Acad. Sci. (Paris), 268 (1969), 1103–1106 | MR | Zbl

[4] R. L. Borelli, A priori estimates for a class of second order non-regular boundary problems, Thesis, U. of California, Berkeley, 1963

[5] C. Caratheodory, Variationsrechnung und partielle Differentialgleichungen Erster Ordnung, Teubner, Berlin, 1935

[6] J. J. Duistermaat, L. Hörmander, “Fourier integral operators, II”, Acta Math., 128:3–4 (1972), 183–269 | DOI | MR | Zbl

[7] Yu. V. Egorov, “O kanonicheskikh preobrazovaniyakh psevdodifferentsialnykh operatorov”, UMN, 24:5 (1969), 235–236 | MR | Zbl

[8] Yu. V. Egorov, “O subellipticheskikh psevdodifferentsialnykh operatorakh”, DAN, 188:1 (1969), 20–22 | MR | Zbl

[9] Yu. V. Egorov, “Nevyrozhdennye subellipticheskie psevdodifferentsialnye operatory”, Matem. sb., 82 (124):3 (1970), 323–342 | MR | Zbl

[10] Yu. V. Egorov, “Neobkhodimye usloviya lokalnoi razreshimosti psevdodifferentsialnykh operatorov”, Trudy MMO, 24, 1970

[11] Yu. V. Egorov, “O dostatochnykh usloviyakh lokalnoi razreshimosti psevdodifferentsialnykh uravnenii”, UMN, 25:3 (1970), 269–270 | MR | Zbl

[12] Yu. V. Egorov, “About local solvability of pseudodifferential equations”, Actes Congres Intern. Math., Nice, 1970

[13] Yu. V. Egorov, “O razreshimosti differentsialnykh uravnenii s prostymi kharakteristikami”, UMN, 26:2 (1971), 183–198 | MR | Zbl

[14] Yu. V. Egorov, O lokalnykh svoistvakh psevdodifferentsialnykh operatorov glavnogo tipa, Dissertatsiya, MGU, Moskva, 1970

[15] Yu. V. Egorov, V. A. Kondratev, “O zadache s kosoi proizvodnoi”, Matem. sb., 78 (120):1 (1969), 148–176 | MR | Zbl

[16] G. I. Eskin, “Vyrozhdayuschiesya ellipticheskie psevdodifferentsialnye uravneniya glavnogo tipa”, Matem. sb., 82 (124) (1970), 585–628 | MR

[17] G. I. Eskin, “Psevdodifferentsialnye operatory s vyrozhdeniem pervogo poryadka po prostranstvennym peremennym”, Trudy MMO, 25, 1971, 83–118 | MR | Zbl

[18] H. Flaschka, G. Strang, “The correctness of the Cauchy problem”, Advances in Math., 6:3 (1971) ; Matematika, 17:2 (1973), 74–97 | DOI | MR | Zbl | Zbl

[19] A. M. Gabrielov, “Ob odnoi teoreme Khërmandera”, Funkts. analiz, 4:2 (1970), 18–22 | MR | Zbl

[20] V. V. Grushin, “Rasprostranenie gladkosti reshenii differentsialnykh uravnenii glavnogo tipa”, DAN, 148:6 (1963), 1241–1244 | Zbl

[21] L. Hörmander, Linear parfial differential operators, Grundl. d. Math. Wiss., 116, Springer-Verlag, 1963 ; L. Khërmander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, «Mir», M., 1965 | Zbl | MR

[22] L. Hörmander, “Pseudo-differential operators”, Comm. Pure Appl. Math., 18 (1965), 501–517 ; Psevdodifferentsialnye operatory, «Mir», M., 1967 | DOI | MR | Zbl | MR

[23] L. Hörmander, “Pseudo-differential operators and non-elliptic boundary problems”, Ann. Math., 83 (1955), 129–209 ; Psevdodifferentsialnye operatory, «Mir», M., 1967 | DOI | MR | MR

[24] L. Hörmander, “Pseudo-differential operators and hypoelliptic equations”, Amer. Math. Soc. Symp. on Singular Integral Operators, 1966, 138–183 ; Psevdodifferentsialnye operatory, «Mir», M., 1967 | MR | MR

[25] L. Hörmander, “Hypoelliptic second order differential equations”, Acta Math., 119 (1967), 147–171 ; Matematika, 12:2 (1968), 88–109 | DOI | MR | Zbl | MR

[26] L. Hörmander, “The spectral function of an elliptic operator”, Acta Math., 121 (1968), 193–218 ; Matematika, 13:6 (1969), 114–137 | DOI | MR | Zbl

[27] L. Hörmander, “On the singularities of solutions of partial differential equations”, Comm. Pure Appl. Math., 23 (1970), 329–358 ; Matematika, 16:6 (1972), 33–59 | DOI | MR | Zbl

[28] L. Hörmander, “On the index of pseudo-differential operators. Elliptische Differentialgleichunge, II”, Koll. Aug., Heft 8 (1969, Berlin), Schriftenreihe der Inst. für Math. Deutsch. Akad. d. Wiss zu Berlin Reihe A; Matematika, 14:4 (1970), 78–97 | Zbl

[29] L. Hörmander, “Fourier integral operators”, Acta Math., 127 (1971), 79–183 ; Matematika, 16:1 (1972), 17–61 ; 16:2 (1972), 67–137 | DOI | MR | Zbl | Zbl

[30] L. Hörmander, “The calculus of Fourier integral operators”, Prospects in Mathematics, Princeton University Press, 1972 | MR

[31] L. Hörmander, “Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients”, Comm. Pure Appl. Math., 24 (1971) ; Matematika, 17 (1973) | MR | Zbl

[32] L. Hörmander, “A remark on Holmgren's uniqueness theorem”, J. Diff. Geom., 5 (1971)

[33] L. Hörmander, “Linear differential operators”, Actes Congres Intern. Math., Nice, 1970, 1 | MR

[34] F. John, “Continuous dependence on data for solutions of partial differential equations with a prescribed bound”, Comm. Pure Appl. Math., 13 (1960), 551–585 | DOI | MR | Zbl

[35] T. Kawai, “Construction of elementary solutions for $\mathrm{I}$-hyperbolic operators and solutions with swall singularities”, Proc. Jap. Acad., 46 (1970), 912–915 | DOI | MR

[36] T. Kawai, “Construction of a local elementary solution for linear partial differential operators. I; II”, Proc. Jap. Acad., 47 (1971), 19–23 | DOI | MR | Zbl

[37] M. Kline, I. W. Kay, Electromagnetic theory and geometrical optics, Pure and Applied Mathematics, XII, Interscience Publ., New York, 1965 | MR | Zbl

[38] J. J. Kohn, L. Nirenberg, “On the algebra of pseudo-differential operators”, Comm. Pure Appl. Math., 18 (1965), 269–305 ; Psevdodifferentsialnye operatory, «Mir», M., 1967 | DOI | MR | Zbl | MR

[39] H. Kumano-go, “Algebras of pseudo-differential operators”, J. Fac. Sci. Univ. Tokyo, 17 (1970), 31–50 | MR | Zbl

[40] P. D. Lax, “Asymptotic solutions of oscillatory initial value problems”, Duke Math. J., 24 (1957), 627–646 | DOI | MR | Zbl

[41] P. D. Lax, L. Nirenberg, “On stability for difference schemes; a sharp form of Garding's inequality”, Comm. Pure Appl. Math., 19 (1966), 473–492 ; Matematika, 11:6 (1967), 3–20 | DOI | MR | Zbl | MR

[42] H. Lewy, “An example of a smooth linear partial differential equation without solutions”, Ann. Math., 66 (1957), 155–158 | DOI | MR

[43] B. Malgrange, “Existence et approximation des solutions des équations aux derivées partielles et des equations de convolution”, Ann. Inst. Fourier Grenoble, 6 (1955–1956), 271–355 | MR

[44] B. Malgrange, “Sur les ouverts convexes par rapport a un opérateur différentiel”, C.R. Acad. Sci. (Paris), 254 (1962), 614–615 | MR | Zbl

[45] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo MGU, M., 1965

[46] A. Melin, “Lower bounds for pseudo-differential operators”, Ark. for Matematik, 9 (1971) | MR | Zbl

[47] S. Mizohata, “Solutions nulles et solutions non analytiques”, J. Math. Kyoto Univ., 1 (1962), 271–302 | MR | Zbl

[48] S. Mizohata, Y. Ohya, “Sur la condition de E. E. Levi concernant des équations hyperboliques”, Publ. Res. Inst. Math. Sci. Kyoto Univ. A, 4 (1968), 511–526 | DOI | MR | Zbl

[49] L. Nirenberg, F. Treves, “Solvability of a first order linear partial differential equation”, Comm. Pure Appl. Math., 14 (1963), 331–351 | DOI | MR

[50] L. Nirenberg, F. Treves, “On local solvability of linear partial differential equations. I: Necessary conditions”, Comm. Pure Appl. Math., 23 (1970), 1–38 ; “II: Sufficient conditions”, 459–510 ; Matematika, 15:3 (1971), 142–172 ; 15:4 (1971), 68–110 | DOI | MR | Zbl | MR | Zbl | Zbl

[51] L. Nirenberg, F. Treves, “A correction to: On local solvability of linear partial differential equations. II: Sufficient conditions”, Comm. Pure Appl. Math., 23 (1971), 279–288 ; Matematika, 16:4 (1972), 149–152 | MR | Zbl

[52] E. V. Radkevich, “Ob odnoi teoreme Khërmandera”, UMN, 24:2 (1969), 233–234 | MR

[53] E. V. Radkevich, “Apriornye otsenki i gipoellipticheskie operatory s kratnymi kharakteristikami”, DAN, 18:2 (1969), 271–274

[54] O. A. Oleinik, E. V. Radkevich, “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Matematicheskii analiz 1969, Itogi nauki, VINITI, M., 1971

[55] M. Sato, “Hyperfunctions and partial differential equations”, Proc. Int. Gonf. on Funct. Anal., Tokyo, 1969, 91–94

[56] M. Sato, “Regularity of hyperfunction solutions of partial differential equations”, Actes Congres Intern. Math., Nice, 1970 | MR

[57] M. Sato, T. Kawai, “Structure of hyperfunctions” (Katata, 1969), Symp. Alg. Geom. and Hyperfunction Theory (Japanese)

[58] M. Sato, M. Kashiwara, “Structure of hyperfunctions”, Sugaku no Ayumi, 15 (1970), 9–72 (Japanese)

[59] L. Schwartz, “Theorie des distributions a valeurs vectorielles, I”, Ann. Inst. Fourier Grenoble, 7 (1957), 1–141 | MR | Zbl

[60] J. Sjostrand, “Sur une classe d'operateurs pseudo-differentiels de type principal”, C.R. Acad. Sci. (Paris), 271 (1970), 781–783 | MR

[61] J. Sjostrand, “Operators of principal type with interior boundary conditions”, Acta Math., 130:1–2 (1973), 1–51 | DOI | MR

[62] F. Treves, Linear partial differential equations with constant coefficients, Gordon and Breach, New York, 1966 | MR | Zbl

[63] F. Treves, “On the local solvability of linear partial differential equations in two independent variables”, Amer. J. Math., 92 (1970), 174–204 | DOI | MR | Zbl

[64] F. Treves, “Local solvability in $L^2$ of first order linear PDE's”, Amer. J. Math., 92 (1970), 369–380 | DOI | MR | Zbl

[65] F. Treves, “Hypoelliptic partial differential equations of principal type with analytic coefficients”, Comm. Pure Appl. Math., 23 (1970), 637–651 | DOI | MR | Zbl

[66] F. Treves, “Analytic hypoelliptic partial differential equations of principal type”, Comm. Pure Appl. Math., 24 (1971), 537–570 | DOI | MR | Zbl

[67] F. Treves, “A new method of proof of the subelliptic estimates”, Comm. Pure Appl. Math., 24 (1971), 71–115 | DOI | MR | Zbl

[68] F. Treves, “Hypoelliptic partial differential equations of principal types. Sufficient conditions and necessary conditions”, Comm. Pure Appl. Math., 24 (1971), 631–670 | DOI | MR | Zbl

[69] F. Treves, “Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters”, Amer. J. Math., 84 (1962), 561–577 | DOI | MR | Zbl

[70] F. Treves, “Equations aux derivees partielles inhomogenes a coefficients constants dependant de parametres”, Ann. Inst. Fourier Grenoble, 13 (1963), 123–138 | MR | Zbl

[71] R. Vallancourt, “A simple proof of Lax-Nirenberg theorems”, Comm. Pure Appl. Math., 23 (1970), 151–163 | DOI | MR

[72] E. C. Zachmanoglou, “Uniqueness of the Gauchy problem when the initial surface contains characteristic points”, Arch. Rat. Mach. Anal., 23 (1966), 317–326 | DOI | MR | Zbl

[73] E. C. Zachmanoglou, “Non-uniqueness of the Gauchy problem for linear partial differential equations with variable coefficients”, Arch. Rat. Mech. Anal., 27 (1968), 373–384 | DOI | MR

[74] E. C. Zachmanoglou, “Propagation of zeros and uniqueness in the Cauchy problem for first order partial differential equations”, Arch. Rat. Mech. Anal., 38 (1970), 178–188 | DOI | MR | Zbl

[75] M. Zerner, “Solutions singulieres d'equations aux derivees partielles”, Bull. Soc. Math. France, 91 (1963), 203–226 | MR | Zbl