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The unique solvability of a Dirichlet problem for fourth order properly elliptic equation
Abstract
The paper studies the unique solvability of a Dirichlet problem for some class of properly elliptic fourth-order equations. Necessary and sufficient conditions as well as some sufficient conditions rendering the corresponding problem uniquely solvable are found.
... For higher-order equations and, in particular, for fourth-order equations and later for equations of arbitrary even order 2m , m ≥ 2 , the Dirichlet problem was studied by Babayan [1,15,16] and Buryachenko [5,7,8]. As for the Neumann problem, some conditions of its solvability for the second-order equations without lower terms in a disk were obtained in a recent work by Burskii and Lesina [6]. ...
... Moreover, the determinant of the form (16) has no meaning for the Laplace operator because this operator is an example of the so-called degenerate case of properly elliptic equations (whose characteristic equations may have roots ± i ). These equations, as shown in [5] and [1,15,16], require additional investigations. ...
... because we consider the case of different roots λ j , j = 1,…, 4 , of the characteristic equation for the properly elliptic equation (1). ...
We establish and study sufficient conditions of solvability of the Neumann problem for fourth-order properly elliptic equations of the general form in a unit disk K in the space (Formula presented.).
The Dirichlet problem for a class of properly elliptic sixth-order equations in the unit disk is considered. Formulas for determining the defect numbers of this problem are obtained. Linearly independent solutions of the homogeneous problem and conditions for the solvability of the inhomogeneous problem are given explicitly.
We study the problem of solvability of the inhomogeneous third boundary-value problem in a bounded domain for a scalar improperly elliptic differential equation with complex coefficients and homogeneous symbol. It is shown that this problem has a unique solution in the Sobolev space over the circle for special classes of boundary data from the spaces of functions with exponentially decreasing Fourier coefficients.
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.
The nontrivial solvability of the homogeneous Dirichlet problem in a unit disk
in positive Sobolev spaces is studied for a typeless differential equation of arbitrary even order 2m, m ≥ 2, with constant complex-valued coefficients and homogeneous symbol. The detailed proofs of the criteria of nontrivial solvability of the problem are given in various cases that form a complete picture. The example considered by A. V. Bitsadze is generalized for the equations of arbitrary even order 2m, m ≥ 2.
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.
In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order 2m with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases m=1, 2, 3 are studied separately. For the case m=2, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions.
We consider the homogeneous Dirichlet problem in the unit disk K ⊂ R
2 for a general typeless differential equation of any even order 2m, m ≥ 2, with constant complex coefficients whose characteristic equation has multiple roots ± i. For each value of multiplicity of the roots i and – i, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely
the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any
even order.
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