Geodesic
Boundary values of solutions of some classes of differential equations
Sbornik. Mathematics, Tome 31 (1977) no. 1, pp. 109-133 Cet article a éte moissonné depuis la source Math-Net.Ru

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Differential equations of the following forms are considered: $$ y'+Ay=0\quad\text{and}\quad-y''+A^2y=0, $$ where $A$ is a positive selfadjoint operator in a Hilbert space $H$. The question of whether the solutions of such equations have boundary values at the end points of the interval $(a,b)$ on which they are considered is investigated, as well as the problem of recovering a solution from its boundary values. A characterization of the boundary values is given in terms of the behavior of the solution near the end points $a$ and $b$. A number of examples are cited in which $A$ is realized as a differential operator in various function spaces. When applied to these concrete situations, the abstract theorems yield the existence and characteristics of the boundary values for certain classes of elliptic and parabolic equations; in particular, the well-known results of F. Riesz, Köthe and Komatsu are obtained and sharpened in this way. The approach is based on the spectral theory of selfadjoint operators. Bibliography: 18 titles. 
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V. I. Gorbachuk; M. L. Gorbachuk. Boundary values of solutions of some classes of differential equations. Sbornik. Mathematics, Tome 31 (1977) no. 1, pp. 109-133. http://geodesic.mathdoc.fr/item/SM_1977_31_1_a6/

[1] I. I. Privalov, Granichnye svoistva odnoznachnykh analiticheskikh funktsii, Gostekhizdat, Moskva–Leningrad, 1950

[2] F. Riesz, “Über die Randwerte einer analytischen Funktion”, Math. Z., 18 (1922), 87–95 | DOI | MR

[3] G. Köthe, “Die Randverteilungen analytischer Funktionen”, Math. Z., 57 (1952), 13–33 | DOI | MR | Zbl

[4] H. G. Tillman, “Distributionen als Randverteilungen analytischer Funktionen”, Math. Z., 76 (1964), 5–21 | DOI | MR

[5] H. Komatsu, “An introduction to the theory of hyperfunctions”, Lecture Notes in Mathematics, 287 (1973), 3–41 | DOI | MR

[6] H. Komatsu, “Ultradistributions and hyperfunctions”, Lecture Notes in Math., 287 (1973), 180–192 | DOI | MR

[7] Yu. M. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, izd-vo «Naukova dumka», Kiev, 1965 | MR

[8] A. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, Moskva, 1959

[9] E. Nelson, “Analiticheskie vektory”, Matematika, 6:3 (1962), 89–133

[10] M. L. Gorbachuk, “Samosopryazhennye granichnye zadachi dlya differentsialnogo uravneniya vtorogo poryadka s neogranichennym operatornym koeffitsientom”, Funkts. analiz, 5:1 (1971), 10–21 | MR | Zbl

[11] Zh.-L. Lions, E. Madzhenes, Neodnorodnye granichnye zadachi i ikh prilozheniya, izd-vo «Mir», Moskva, 1971

[12] V. I. Gorbachuk, “O granichnykh znacheniyakh obobschennykh reshenii odnorodnogo uravneniya Shturma-Liuvillya v prostranstve vektor-funktsii”, Matem. zametki, 18:2 (1975), 243–252 | MR | Zbl

[13] L. Schwartz, Theorie des distributions, I, II, Paris, 1957 | MR

[14] J. L. Lions, E. Magenes, Problèmes aux Limites non Homogènes et Applications, v. 3, Dunod, Paris, 1970 | MR | Zbl

[15] I. M. Gelfand, G. E. Shilov, Prostranstva osnovnykh i obobschennykh funktsii, Fizmatgiz, Moskva, 1958

[16] N. Viner, R. Peli, Preobrazovanie Fure v kompleksnoi oblasti, izd-vo «Nauka», Moskva, 1964 | MR

[17] Ya. A. Roitberg, “O znacheniyakh na granitse oblasti obobschennykh reshenii ellipticheskikh uravnenii”, Matem. sbornik, 86 (128) (1971), 248–267 | MR | Zbl

[18] V. P. Mikhailov, “O granichnykh svoistvakh reshenii ellipticheskikh uravnenii”, DAN SSSR, 226:6 (1976), 1264–1267 | MR