# The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

Article

First Online:

- 5 Downloads

## Abstract

A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Hölder boundary, the differential properties of the solution are studied.

## Keywords

ellipticity simple complex characteristics## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Ya. B. Lopatinskii, “On a method of reducing boundary problems for a system of differential equations of elliptic type to regular integral equations,” Ukrain.Mat. Zh.
**5**, 123–151 (1953).MathSciNetGoogle Scholar - 2.A. V. Bitsadze,
*Boundary-Value Problems for Elliptic Second-Order Equations*(Nauka, Moscow, 1966).Google Scholar - 3.I. N. Vekua,
*Generalized Analytic Functions*(Fizmatlit, Moscow, 1959) [in Russian].zbMATHGoogle Scholar - 4.G. Fichera, “Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anisotropic inhomogeneous elasticity,” in
*Partial Differential Equations and Continuum Mechanics*(Wisconsin Press, Madison, 1961), pp. 55–80.Google Scholar - 5.S. Agmon, “Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane,” Comm. Pure Appl.Math.
**10**, 197–239 (1957).MathSciNetCrossRefzbMATHGoogle Scholar - 6.O. I. Panich, “Equivalent regularization and solvability of normally solvable boundary value problems with a zero index for polyharmonic equations and highly elliptical second-order plane systems,” Sibirsk. Mat. Zh.
**7**, 591–619 (1966).zbMATHGoogle Scholar - 7.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), 1070–1100 (1991) [Math. USSR-Izv.**39**(2), 1033–1061 (1992)].zbMATHGoogle Scholar - 8.A. P. Soldatov, “A function theory method in elliptic problems in the plane. II. The piecewise smooth case,” Izv. Akad. Nauk SSSR Ser. Mat.
**56**(3), 566–604 (1992) [Math. USSR-Izv.**40**(3), 529–563 (1993)].Google Scholar - 9.S. G. Mikhlin
*Integral Equations and Their Applications to Problems of Mechanics, of Mathematical Physics, and Technology*(OGIZ, Moscow, 1947) [in Russian].Google Scholar - 10.D. I. Sherman, “On the solution of a plane problem of the theory of elasticity for an isotropic medium,” Prikl. Mat. Mekh.
**6**, 509–514 (1942).MathSciNetzbMATHGoogle Scholar - 11.A. I. Kostrikin,
*Introduction to Algebra*(Nauka, Moscow, 1977) [in Russian].zbMATHGoogle Scholar - 12.G. E. Shilov,
*Mathematical Analysis. Second Special Course*(Nauka, Moscow, 1965) [in Russian].Google Scholar

## Copyright information

© Pleiades Publishing, Ltd. 2018