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STATEMENT OF THE PROBLEMLet D ⊆ R 2 be a domain bounded by a piecewise smooth contour Γ without return points. We choose a finite subset F ⊆ Γ that contains all corner points of the contour and consider a continuously differentiable mapping α : Γ\F → D. Suppose that the function α and its derivative α (with respect to the arc length) are piecewise continuous on Γ, i.e., have one-sided limits at points τ ∈ F ; moreover, α(τ ± 0) ∈ D ∪ F and α (τ ± 0) = 0 for τ ∈ F . In addition, the curve α(Γ\F ) is not tangent to Γ. The Bitsadze-Samarskii problem. Find an analytic functionwhere the coefficients a and a 0 are piecewise continuous on Γ. Furthermore, in the case of an unbounded domain D, we assume that the function φ is bounded at infinity. One can also consider the case in which the coefficient a 0 and the shift α are defined on some part Γ of the curve Γ and the boundary condition splits accordingly asBy continuing α to the entire curve Γ\F and by completing the definition of a 0 by zero, one can always reduce this problem to the form (1). If a 0 ≡ 0, then we obtain the classical Riemann-Hilbert problem.Obviously, the Bitsadze-Samarskii problem [1] for the Laplace equation [1] can be reduced to problem (1). The Bitsadze-Samarskii problem was comprehensively studied [2-15] for general elliptic equations. In the present paper, we consider problem (1) for Douglis analytic functions, that is, solutions φ = (φ 1 , . . . , φ l ) of the first-order canonical systemwith matrix J ∈ C l×l whose eigenvalues ν ∈ σ(J) lie in the upper half-plane Im ν > 0. Therefore, the coefficients a and a 0 in (1) are piecewise continuous l × l matrix functions. If J = i, then system (2) becomes the Cauchy-Riemann system and problem (1), (2) corresponds to the problem for analytic vector functions. The interest in the statement of problem (1), (2) is due to the fact that the Bitsadze-Samarskii problem for elliptic equations and systems with constant (and only leading) coefficients [16,17] can be reduced to it.We note the special case of problem (1 ), (2) in which Γ is a smooth arc and the image α (Γ ) splits D into two subdomains. As indicated in [18], in this case, the problem can be reduced to the generalized Riemann-Hilbert problem, to which general results in [17] can be applied. In this 0012-2661/05/4103-0416 c 2005 Pleiades Publishing, Inc.
STATEMENT OF THE PROBLEMLet D ⊆ R 2 be a domain bounded by a piecewise smooth contour Γ without return points. We choose a finite subset F ⊆ Γ that contains all corner points of the contour and consider a continuously differentiable mapping α : Γ\F → D. Suppose that the function α and its derivative α (with respect to the arc length) are piecewise continuous on Γ, i.e., have one-sided limits at points τ ∈ F ; moreover, α(τ ± 0) ∈ D ∪ F and α (τ ± 0) = 0 for τ ∈ F . In addition, the curve α(Γ\F ) is not tangent to Γ. The Bitsadze-Samarskii problem. Find an analytic functionwhere the coefficients a and a 0 are piecewise continuous on Γ. Furthermore, in the case of an unbounded domain D, we assume that the function φ is bounded at infinity. One can also consider the case in which the coefficient a 0 and the shift α are defined on some part Γ of the curve Γ and the boundary condition splits accordingly asBy continuing α to the entire curve Γ\F and by completing the definition of a 0 by zero, one can always reduce this problem to the form (1). If a 0 ≡ 0, then we obtain the classical Riemann-Hilbert problem.Obviously, the Bitsadze-Samarskii problem [1] for the Laplace equation [1] can be reduced to problem (1). The Bitsadze-Samarskii problem was comprehensively studied [2-15] for general elliptic equations. In the present paper, we consider problem (1) for Douglis analytic functions, that is, solutions φ = (φ 1 , . . . , φ l ) of the first-order canonical systemwith matrix J ∈ C l×l whose eigenvalues ν ∈ σ(J) lie in the upper half-plane Im ν > 0. Therefore, the coefficients a and a 0 in (1) are piecewise continuous l × l matrix functions. If J = i, then system (2) becomes the Cauchy-Riemann system and problem (1), (2) corresponds to the problem for analytic vector functions. The interest in the statement of problem (1), (2) is due to the fact that the Bitsadze-Samarskii problem for elliptic equations and systems with constant (and only leading) coefficients [16,17] can be reduced to it.We note the special case of problem (1 ), (2) in which Γ is a smooth arc and the image α (Γ ) splits D into two subdomains. As indicated in [18], in this case, the problem can be reduced to the generalized Riemann-Hilbert problem, to which general results in [17] can be applied. In this 0012-2661/05/4103-0416 c 2005 Pleiades Publishing, Inc.
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