Abstract:
In a finite domain D of the complex plane bounded by a smooth contour Γ, we consider the Riemann–Hilbert boundary-value problem
ReCU+=f
for the first-order elliptic system
∂U∂y−A∂U∂x+a(z)U(z)+b(z)¯U(z)=F(z)
with constant leading coefficients. Here + denotes the boundary value of the function U on Γ, the constant matrices A1,A2∈Cl×l and (l×l)-matrix coefficients a and b belong to the Hölder class Cμ, 0<μ<1, and (l×l)-matrix function C belongs to the class Cμ(Γ). We prove that in the class U∈Cμ(¯D)∩C1(D), this problem is a Fredholm problem and its index is given by the formula
ϰ=−m∑j=11π[argdetG]Γj+(2−m)l.
Keywords:
elliptic systems, Riemann–Hilbert problem, index formula, Fredholm operator.
Citation:
A. P. Soldatov, O. V. Chernova, “Riemann–Hilbert Problem for First-Order Elliptic Systems with Constant Leading Coefficients on the Plane”, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 149, VINITI, Moscow, 2018, 95–102; J. Math. Sci. (N. Y.), 250:5 (2020), 811–818
\Bibitem{SolChe18}
\by A.~P.~Soldatov, O.~V.~Chernova
\paper Riemann--Hilbert Problem for First-Order Elliptic Systems with Constant Leading Coefficients on the Plane
\inbook Proceedings of the International Conference ``Actual Problems of Applied Mathematics and Physics,'' Kabardino-Balkaria, Nalchik, May 17--21, 2017
\serial Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz.
\yr 2018
\vol 149
\pages 95--102
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into322}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3847728}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2020
\vol 250
\issue 5
\pages 811--818
\crossref{https://doi.org/10.1007/s10958-020-05046-y}
Linking options:
https://www.mathnet.ru/eng/into322
https://www.mathnet.ru/eng/into/v149/p95
This publication is cited in the following 5 articles:
Ali Darya, Nasir Taghizadeh, “SCHWARZ AND DIRICHLET PROBLEMS FOR COMPLEX PARTIAL DIFFERENTIAL EQUATIONS IN THE PARTIAL ECLIPSE DOMAIN”, J Math Sci, 2024
A. P. Soldatov, “On a boundary problem for a fourth-order elliptic equation on a plane”, Comput. Math. Math. Phys., 62:4 (2022), 599–607
A. P. Soldatov, O. V. Chernova, “Zadacha lineinogo sopryazheniya dlya ellipticheskikh sistem na ploskosti”, Materialy mezhdunarodnoi konferentsii po matematicheskomu modelirovaniyu v prikladnykh naukakh “International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19”. Belgorod, 20–24 avgusta 2019 g., Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 195, VINITI RAN, M., 2021, 108–117
B. D. Koshanov, A. P. Soldatov, “O razreshimosti obobschennoi zadachi Neimana dlya ellipticheskogo uravneniya vysokogo poryadka v beskonechnoi oblasti”, Posvyaschaetsya 70-letiyu prezidenta RUDN V.M. Filippova, SMFN, 67, no. 3, Rossiiskii universitet druzhby narodov, M., 2021, 564–575
S. P. Mitin, A. P. Soldatov, “Solution of the Dirichlet Problem for the Inhomogeneous Lamé System with Lower Order Coefficients”, J Math Sci, 255:6 (2021), 732