Geodesic
An expression for the solution of a differential equation in terms of iterates of differential operators
Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 411-424 
A. V. Babin. An expression for the solution of a differential equation in terms of iterates of differential operators. Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 411-424. http://geodesic.mathdoc.fr/item/SM_1978_34_4_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain theorems on the expression of $A^{-1}$ in terms of iterates of the operator $A$, which is the reproducing operator of a $1$-parameter group of linear transformations of a Banach space, and whose spectrum does not surround $0$. These results are applied to first order differential equations with analytic coefficients and right-hand sides (symmetric first order systems on a compact manifold without boundary), and to second order elliptic equations (equations with a real principal part on a manifold without boundary, selfadjoint equations degenerate on the boundary of the domain, and the Dirichlet problem for a selfadjoint equation in a domain with an analytic boundary). We obtain formulas expressing the value of the solution at a point in terms of the derivatives of the coefficients and the right-hand side at this point. Bibliography: 9 titles.

[1] M. S. Agranovich, “Granichnye zadachi dlya sistem psevdodifferentsialnykh operatorov $1$-go poryadka”, Uspekhi matem. nauk, XXIV:1(145) (1963), 61–125

[2] A. V. Babin, “Formula, vyrazhayuschaya reshenie differentsialnogo uravneniya s analiticheskimi koeffitsientami na mnogoobrazii bez kraya cherez dannye zadachi”, Matem. sb., 101(143) (1976), 610–638 | MR | Zbl

[3] M. I. Vishik, “Kraevye zadachi dlya ellipticheskikh uravnenii, vyrozhdayuschikhsya na granitse oblasti”, Matem. sb., 35(77) (1954), 513–568 | Zbl

[4] N. Danford, Dzh. T. Shvarts, Lineinye operatory, t. I (Obschaya teoriya), IL, Moskva, 1962

[5] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, izd-vo “Nauka”, Moskva, 1967 | MR

[6] S. G. Mikhlin, “Vyrozhdayuschiesya ellipticheskie uravneniya”, Vestnik LGU, 1954, no. 8, 19–48

[7] R. T. Sili, “Stepeni ellipticheskogo operatora”, Matematika, 12:1 (1968), 96–112

[8] G. I. Eskin, “Zadacha Koshi dlya giperbolicheskikh uravnenii v svertkakh”, Matem. sb., 74(116) (1967), 262–297 | MR

[9] A. V. Babin, “Vyrazhenie $A^{-1}$ cherez iteratsii neogranichennogo samosopryazhennogo operatora $A$ na analiticheskikh vektorakh”, Funkts. analiz, 11:4 (1977), 3–5 | MR | Zbl