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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 5, Pages 1070–1100 (Mi izv981)  

This article is cited in 13 scientific papers (total in 13 papers)

A function theory method in boundary value problems in the plane. I. The smooth case

A. P. Soldatov

Vladimir State Pedagogical University

Abstract: A general (not necessarily local) boundary value problem is considered for an elliptic $(l\times l)$ system on the plane of $n$th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic $(s\times s)$ systems of first order
$$ \frac{\partial\Phi}{\partial y}-J\frac{\partial\Phi}{\partial x}=0 $$
with a constant triangular matrix $J=(J_{ij})_1^s$, $\operatorname{Im}J_{ij}>0$; this problem is reduced to an equivalent system of integrofunctional equations on the boundary. In particular, a criterion that the problem be Noetherian and a formula for its index are obtained in this way. All considerations are carried out in the smooth case when the boundary of the domain has no corner points, while the boundary operators act in spaces of continuous functions.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 39:2, 1033–1061

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J55; Secondary 35M99
Received: 29.05.1990

Citation: A. P. Soldatov, “A function theory method in boundary value problems in the plane. I. The smooth case”, Izv. Akad. Nauk SSSR Ser. Mat., 55:5 (1991), 1070–1100; Math. USSR-Izv., 39:2 (1992), 1033–1061

Citation in format AMSBIB
\by A.~P.~Soldatov
\paper A function theory method in boundary value problems in the plane. I.~The~smooth case
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1991
\vol 55
\issue 5
\pages 1070--1100
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 2
\pages 1033--1061

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    This publication is cited in the following articles:
    1. A. P. Soldatov, “A function theorety method in elliptic problems in the plane. II. The piecewise smooth case”, Russian Acad. Sci. Izv. Math., 40:3 (1993), 529–563  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Mitin S. Soldatov A., “Solvability of the Generalized Mixed Problem”, Differ. Equ., 32:3 (1996), 388–392  mathnet  mathscinet  zmath  isi
    3. M. M. Sirazhudinov, “Boundary-value problems for general elliptic systems in the plane”, Izv. Math., 61:5 (1997), 1031–1068  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Soldatov A.P., “The algebra of singular operators with terminal symbol on a piecewise smooth curve: II. Basic constructions”, Differential Equations, 37:6 (2001), 866–879  mathnet  crossref  mathscinet  zmath  isi  elib
    5. V. A. Oganyan, “Zadacha Dirikhle dlya slabo svyazannykh ellipticheskikh differentsialnykh uravnenii vtorogo poryadka s razryvnymi granichnymi usloviyami”, Uch. zapiski EGU, ser. Fizika i Matematika, 2003, no. 3, 16–24  mathnet
    6. Soldatov A.P., “The Bitsadze-Samarskii problem for Douglis analytic functions”, Differential Equations, 41:3 (2005), 416–428  mathnet  crossref  mathscinet  zmath  isi  elib
    7. A. P. Soldatov, “Second-order elliptic systems in the half-plane”, Izv. Math., 70:6 (2006), 1233–1264  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Vashchenko O.V. Soldatov A.P., “The Hardy Space of Solutions of the Generalized Beltrami System”, Differ. Equ., 43:4 (2007), 503–506  crossref  mathscinet  zmath  isi
    9. Radzhabov N.R., Rasulov A.B., “O korrektnoi postanovke zadach dlya sistemy bitsadze so sverkhsingulyarnoi tochkoi i okruzhnostyu”, Nauchnye vedomosti belgorodskogo gosudarstvennogo universiteta, 25:23 (2011), 96–101  elib
    10. A. Soldatov, “The Neumann problem for elliptic systems on a plane”, Journal of Mathematical Sciences, 202:6 (2014), 897–910  mathnet  crossref
    11. A. P. Soldatov, “Generalized potentials of double layer in plane theory of elasticity”, Eurasian Math. J., 5:2 (2014), 78–125  mathnet
    12. A. P. Soldatov, “K teorii anizotropnoi ploskoi uprugosti”, Trudy Sedmoi Mezhdunarodnoi konferentsii po differentsialnym i funktsionalno-differentsialnym uravneniyam (Moskva, 22–29 avgusta, 2014). Chast 3, SMFN, 60, RUDN, M., 2016, 114–163  mathnet
    13. Yu. A. Bogan, “The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane”, Math. Notes, 104:5 (2018), 636–641  mathnet  crossref  crossref  isi  elib
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