Geodesic
Boundary value problems for nonregular systems of differential equations on a half-plane in the class of generalized functions and functions of polynomial growth
Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 181-208 
N. E. Tovmasyan. Boundary value problems for nonregular systems of differential equations on a half-plane in the class of generalized functions and functions of polynomial growth. Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 181-208. http://geodesic.mathdoc.fr/item/SM_1988_59_1_a9/
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     title = {Boundary value problems for nonregular systems of differential equations on a~half-plane in the class of generalized functions and functions of polynomial growth},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem and the general boundary value problem are considered for nonregular systems of differential equations with constant coefficients in a half-plane. Necessary and sufficient conditions on the initial data are obtained to ensure the solvability of the Cauchy problem in the classes mentioned in the title. In the study of the general boundary value problem it is assumed that the Lopatinskii conditions hold everywhere except at a finite number of points. It is proved that in the class of functions of arbitrary polynomial growth the inhomogeneous problem is always solvable, while the homogeneous problem has a finite number of linearly independent solutions. A formula for the index is obtained. Additional conditions on the solutions are indicated, ensuring the unique solvability of the problems. At the end of the paper, the results are illustrated by the example of elliptic second order equations. Bibliography: 13 titles.

[1] Shilov G. E., Matematicheskii analiz. Vtoroi spets. kurs, Nauka, M., 1965, s. 1–327

[2] Dikopolov G. V., Shilov G. E., “O korrektnykh kraevykh zadachakh dlya uravnenii v chastnykh proizvodnykh v poluprostranstve”, Izv. AN SSSR. Ser. matem., 24:3 (1960), 369–380 | MR | Zbl

[3] Dikopolov G. V., “O kraevykh zadachakh dlya differentsialnykh uravnenii s postoyannymi koeffitsientami v poluprostranstve”, Matem. sb., 59(101) (1962), 215–228 | MR | Zbl

[4] Pavlov A. L., “Ob obschikh kraevykh zadachakh dlya differentsialnykh uravnenii s postoyannymi koeffitsientami v poluprostranstve”, Matem. sb., 103(145) (1977), 367–391 | MR | Zbl

[5] Kosheleva T. M., “Zadacha Koshi dlya obyknovennykh differentsialnykh uravnenii v klasse obobschennykh funktsii”, DAN ArmSSR, XXII:4 (1981), 238–243 | MR

[6] Tovmasyan N. E., “Zadacha Koshi dlya differentsialnykh uravnenii s postoyannymi koeffitsientami v poluprostranstve v klasse obobschennykh funktsii”, Differents. uravneniya, 18:1 (1982), 132–138 | MR | Zbl

[7] Kosheleva T. M., “Nekotorye granichnye zadachi dlya obyknovennykh differentsialnykh uravnenii $m$-go poryadka v klasse obobschennykh funktsii”, Izv. VUZov. Ser. matem., 27:3 (1983), 51–58 | MR

[8] Tovmasyan N. E., “Obschaya granichnaya zadacha dlya sistemy differentsialnykh uravnenii v poluploskosti s narusheniem usloviya Ya. B. Lopatinskogo”, Differents. uravneniya, 20:1 (1984), 132–141 | MR | Zbl

[9] Kosheleva T. M., “Obschie kraevye zadachi dlya sistemy obyknovennykh differentsialnykh uravnenii pervogo poryadka v klasse obobschennykh funktsii”, Differents. uravneniya, 20:10 (1984), 1079–1715 | MR

[10] Tovmasyan N. E., “Obschie kraevye zadachi dlya sistem differentsialnykh uravnenii s chastnymi proizvodnymi v poluprostranstve v klasse obobschennykh funktsii”, Differents. uravneniya, 20:12 (1984), 2138–2147 | MR | Zbl

[11] Daletskii Yu. L., Krein M. G., Ustoichivost resheniya differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970, 1–534 | MR

[12] Van der Warden B., Einfuhrung in die Algebraische Geometrie, Berlin, 1939

[13] Tovmasyan N. E., “Zadacha Koshi dlya neregulyarnykh differentsialnykh uravnenii s postoyannymi koeffitsientami na poluosi v klasse obobschennykh funktsii”, Seminar instituta prikladnoi matematiki im. I. N. Vekua (dokl. 17), 1983, 25–35 | MR | Zbl