Abstract:
A general (not necessarily local) boundary value problem is considered for an elliptic (l×l) system on the plane of nth order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic (s×s) systems of first order
∂Φ∂y−J∂Φ∂x=0
with a constant triangular matrix J=(Jij)s1, ImJij>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.
Citation:
A. P. Soldatov, “A function theory method in boundary value problems in the plane. I. The smooth case”, Math. USSR-Izv., 39:2 (1992), 1033–1061
\Bibitem{Sol91}
\by A.~P.~Soldatov
\paper A function theory method in boundary value problems in the plane. I.~The~smooth case
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 2
\pages 1033--1061
\mathnet{http://mi.mathnet.ru/eng/im981}
\crossref{https://doi.org/10.1070/IM1992v039n02ABEH002236}
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This publication is cited in the following 20 articles:
S. Aghekyan, “On a Riemann Boundary Value Problem in the Space of p-Summable Functions with Infinite Index”, J. Contemp. Mathemat. Anal., 57:6 (2022), 323
M. Otelbaev, A. P. Soldatov, “Integral representations of vector functions based on the parametrix of first-order elliptic systems”, Comput. Math. Math. Phys., 61:6 (2021), 964–973
Alexandre Soldatov, INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2020), 2325, INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2020), 2021, 020025
A. H. Babayan, “On a Dirichlet problem for one improperly elliptic equation”, Complex Variables and Elliptic Equations, 64:5 (2019), 825
Armenak H. Babayan, Seyran H. Abelyan, Springer Proceedings in Mathematics & Statistics, 291, Modern Methods in Operator Theory and Harmonic Analysis, 2019, 317
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
A. P. Soldatov, O. V. Chernova, “Riemann–Hilbert Problem for First-Order Elliptic Systems with Constant Leading Coefficients on the Plane”, J. Math. Sci. (N. Y.), 250:5 (2020), 811–818
N. Manjavidze, “Riemann–Hilbert Problems on a Cut Plane”, J Math Sci, 235:5 (2018), 632
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
A. P. Soldatov, “Generalized potentials of double layer in plane theory of elasticity”, Eurasian Math. J., 5:2 (2014), 78–125
A. Soldatov, “The Neumann problem for elliptic systems on a plane”, Journal of Mathematical Sciences, 202:6 (2014), 897–910
Vashchenko O.V. Soldatov A.P., “The Hardy Space of Solutions of the Generalized Beltrami System”, Differ. Equ., 43:4 (2007), 503–506
A. P. Soldatov, “Second-order elliptic systems in the half-plane”, Izv. Math., 70:6 (2006), 1233–1264
Soldatov A.P., “The Bitsadze-Samarskii problem for Douglis analytic functions”, Differential Equations, 41:3 (2005), 416–428
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
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
M. M. Sirazhudinov, “Boundary-value problems for general elliptic systems in the plane”, Izv. Math., 61:5 (1997), 1031–1068
Mitin S. Soldatov A., “Solvability of the Generalized Mixed Problem”, Differ. Equ., 32:3 (1996), 388–392
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