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General boundary value problem for elliptic systems of second order with constant coefficients. II
... The classical method of studying boundary value problems for elliptic equations and systems on a plane is based on representing the solutions of elliptic equations in terms of analytic functions, which allows one to reduce the matter to the study of the boundary value problems of function theory. For elliptic systems with constant leading coefficients, a method was developed by A. V. Bitsadze [1] (see also [2,3]). In the Bitsadze representation of solutions to elliptic systems, along with analytical functions, its derivatives up to a certain order are also involved. ...
... Vishik [18], who called them strongly elliptic. They are determined by the positive definiteness condition for the matrix p(t) given in (2) for an arbitrary t ∈ R. Note that this condition is equivalent [19] to det(a 0 α + 2a 1 β + a 2 γ) = 0 for β 2 − αγ < 0. ...
... with the eigenvalues being coincided with the roots of the characteristic polynomial (2). It reduces to the Jordan form ...
The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method.
... The second method is based on representing solutions of elliptic equations through analytic functions, which makes it possible to reduce the task to studying boundary-value problems in terms of the theory of functions. For elliptic equations in a plane with real analytical coefficients, this method was developed by Vekua (see [22]); for elliptic systems with constant coefficients -by Bitsadze [6] (see also [18,21]). ...
In a reversible hydrodynamic approximation, a closed system of second order dynamic equation with respect to the displacement vector of an elastic porous body and pore pressure has been obtained. The Cauchy problem for the obtained system of poroelasticity equations in the stationary case is considered. The Carleman formula for the Cauchy problem under consideration has been constructed.
The Dirichlet problem for sixth order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of homogeneous problem and the number of linearly independent solvability conditions of in-homogeneous problem are obtained. Solutions of homogeneous problem and solvability conditions of in-homogeneous problem are obtained in explicit form.
The Dirichlet problem for sixth-order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of the homogeneous problem and the number of linearly independent solvability conditions of the inhomogeneous problem are determined. Solutions of the homogeneous problem and solvability conditions of the inhomogeneous problem are obtained in explicit form.
The present article is concerned with research in the last five to ten years on systems of linear partial differential equations. The total number of published works in this area, of course, is too great to cover each one in sufficient detail. While consciously refraining from undertaking such a task, I have endeavored to focus a proportionate amount of attention on each facet of the topic insofar as it embodies the characteristics which distinguish the theory of systems from the analogous theory of one equation in one unknown function. For example, considerable space is accorded the -Neumann problem, which is endowed with a specialized character and whose solution has contributed a great deal that is conceptually new to the general theory. On the other hand, the highly-developed theory of boundary-value problems is scarcely touched at all, as its methods pertain by and large to scalar theory.
We consider the Dirichlet problem in a unit disk for the main-type partial differential equations with constant complex-valued coefficients whose symbols are forms of any even order. We establish the nonuniqueness conditions for the solution in terms of the coefficients of equation for the case where the angles of inclination of the complex characteristics exist. We construct some examples of equations for which the Dirichlet problem in a disk has nontrivial solutions and is not Noetherian.
Second order elliptic systems with constant leading coefficients are considered. It is shown that the Bitsadze definition of weakly connected elliptic systems is equivalent to the known Shapiro–Lopatinskiy condition with respect to the Dirichlet problem for weakly connected elliptic systems. An analogue of potentials of double layer for these systems is introduced in the frame of functional theoretic approach. With the help of these potentials all solutions are described in the Holder and Hardy classes as well as in the class of all continuous functions.
We consider second-order elliptic systems on the plane with constant (and only leading) matrix coefficients. We show that for these systems the notion of being weakly coupled (in the sense of A.V. Bitsadze) is equivalent to the well-known complementarity condition for the Dirichlet problem. In the framework of the function theoretic approach, we introduce analogs of double-layer potentials for solutions of weakly coupled systems. By using these potentials, we obtain a complete description of solutions of weakly elliptic systems in Hölder classes as well as in the Hardy classes h
p
(D) and
.
We obtain a criterion of nontrivial solvability of the homogeneous Dirichlet problem in a unit disk K for a general equation of even order 2m, m > 2 , with constant complex coefficients and a homogeneous degenerate symbol. The dependence between the multiplicity of
roots of the characteristic equation and the existence of a nontrivial solution of the problem from the space in the case where the roots are not equal to ±
i
is established.
The Dirichlet problem for the elliptic system of second order with constant and only leading coefficients is considered in a piecewise smooth domain on the plane. The Fredholm criterion and index formula for this problem in Holder spaces are given. In particular, the examples of the elliptic systems are found such that the index of Dirichlet problem is not equal to 0.
Generalized double layer potentials are considered. Their boundary properties on a plane are studied. Conditions when such
potentials are continuously extended onto the boundary of the domain are obtained. A formula for limit values of such potentials
is also obtained.
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