On the dirichlet problem for an improperly elliptic equation

Burskii, V. P.; Кириченко, Е. В.

doi:10.1007/s11253-011-0497-9

Cited by 3 publications

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“…The paper continues the investigation of boundary-value problems for improperly elliptic equations performed in [6][7][8], where the solvability of the Dirichlet , Neumann, and oblique-derivative problems was established for the analyzed equation.…”

Section: 95mentioning

confidence: 89%

“…Moreover, since the function  is expressed via the known β and the already determined quantity , in view of the imbeddings H m ⇢ (@K) ⇢ H m (@K) and H m+1 (@K) ⇢ H m (@K), we conclude that  2 H m (@K). Thus, if we solve system (7) obtained as a result of the substitution of decompositions of the unknown functions γ and  in the integral equality (6) and estimate the solution of system (10), then we conclude that γ and  belong to the same Sobolev space. Furthermore, by Theorem 1, we conclude that the original problem (2) …”

Section: Sincementioning

confidence: 89%

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On the Third Boundary-Value Problem for an Improperly Elliptic Equation in a Disk

Burskii

Лесіна

2014

Ukr Math J

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517.95We 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.The boundary-value problems for improperly elliptic equations in bounded domains are studied in the paper of one of the authors [4] in which a criterion for the Fredholm property of a general differential boundary-value problem for a scalar linear improperly elliptic equation of any order in a bounded domain with smooth boundary is obtained. The application of this criterion to the Dirichlet and Neumann problems reveals their non-Fredholm property.In the present paper, we study a second-order improperly elliptic equation in a model domain (disk) and establish the solvability of the third boundary-value problem in the ordinary Sobolev scale of spaces for the case where the right-hand side of the boundary condition belongs to a certain class of analytic functions. The paper continues the investigation of boundary-value problems for improperly elliptic equations performed in [6][7][8], where the solvability of the Dirichlet , Neumann, and oblique-derivative problems was established for the analyzed equation.In the cited papers, the solvability of the boundary-value problems is proved and, in addition, depending on the properties of the number ' 0 = ' 1 − ' 2 , which is called the angle between the characteristics of Eq. (2), the following three cases are analyzed:(i) the angle ' 0 is real and ⇡ -rational, i.e., ' 0 /⇡ 2 Q;(ii) the angle ' 0 is real and ⇡ -irrational;(iii) the angle ' 0 is not real.The first case corresponds to the violation of uniqueness of the solution to the Dirichlet problem in the case of countably many linearly independent solutions of the corresponding homogeneous problem. In the second and third cases, for the solvability in the ordinary Sobolev scale of spaces, it is necessary to introduce the spaces of analytic right-hand sides. Moreover, unlike case (iii), in case (ii), the number-theoretic properties of the number ' 0 affect the properties of the analyzed problems. In studying the third boundary-value problem, we restrict ourselves to the case of nonreal angle ' 0 .Note that results of investigations in this direction can be found in the works by Bitsadze and his colleagues [2], Tovmasyan [11,12], and Babayan (see, e.g., [1]).

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Section: 95mentioning

confidence: 89%

Section: Sincementioning

confidence: 89%