Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
Dear authors!
Submission of all materials is carried out only electronically through Online Submission System in
personal account.
DOI: 10.23671/VNC.2017.3.7130
A Boundary Value Problem for Higher Order Elliptic Equations in Many Connected Domain on the Plane
Soldatov A. P.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 3.
Abstract: For the elliptic equation of \(2l\)th order with constant (and leading) coefficients boundary value a problem with normal derivatives of the \((k_j-1)-\)order, \(j=1,\ldots,l\) considered. Here \(1\le k_1 <\ldots< k_l\le 2l\). When \(k_j=j\) it moves to the Dirichlet problem, and when \)k_j = j + 1\) it corresponds to the Neumann problem. The sufficient condition of the Fredholm problem and index formula are given.
Keywords: elliptic equation, boundary value problem, normal derivatives, many connected domain, smooth contour, Fredholm property, index formula
For citation: Soldatov A. P. A boundary value problem for higher order elliptic equations
in many connected domain on the plane Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 51-58.
DOI 10.23671/VNC.2017.3.7130
1. Bitsadze A.V. To Neumann problem for harmonic functions. Dokl. AN
SSSR, 1990, vol. 311, no. 1.
2. Malahova N.A., Soldatov A.P. On boundary value problem for a
elliptic equation of high order. Differentsial’nye Uravneniya [Differential Equations], 2008,
vol. 44, no. 8, pp. 1077-1083 (in Russian).
3. Koshanov B. D., Soldatov A. P. Boundary Value Problem with
Normal Derivatives for a Higher-Order Elliptic Equation on the
Plane. Differentsial’nye Uravneniya [Differential Equations], 2016,
vol. 52, no. 12, pp. 1666–1681 (in Russian).
4. Vaschenko O. V., Soldatov A. P. Integral representation of
solutions of the generalized Beltramy system, Nauchnie vedomosti
BelGU, 2006, vol. 6, no. 1(21), pp. 3-6 (in Russian).
5. Soldatov A. P. Singular integral operators and elliptic boundary
value problems. Sovremen. matematika. Fundament. napravleniya
[Contemporary Mathematics. Fundamental Directions], 2016, vol. 63,
pp. 1-179 (in Russian).
6. Soldatov A. P. Hyperanalytic functions and their applications, J.
Math. Sciences, 2004, vol. 17, P. 1-111.
7. Abapolova E. A., Soldatov A. P. To the theory of singular
integral equations on a smooth contour, Nauchnie vedomosti BelGU
[Belgorod State University Scientific Bulletin], 2010, no. 5(76),
issue 18, pp. 6-20.
8. Mushelishvili N. I. Singular Integral Equations. Noordhoff,
Groningen, 1953.
9. Soldatov A. P. A function theory method in boundary value
problems in the plane. I. A smooth case. Izvestiya AN USSR. Ser.
Matem., 1992, vol. 39, no. 2, pp. 1070-1100 (in Russian).
10. Soldatov A.P. The Schvarts problem for Douglis analytic
functions, Sovremennaya Matematika i ee Prilozheniya, 2010, vol. 67,
pp. 99-102 (in Russian).
11. Palais P. Seminar on the Atiya-Singer Theorem. Princeton,
Univers. Press Princeton, 1965.
