Abstract
In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.
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REFERENCES
B. V. Shabat, ‘‘Examples of solving the Dirichlet problem for a mixed type equation,’’ Dokl. Akad. Nauk SSSR 112, 386–389 (1957).
A. V. Bitsadze, ‘‘Incorrectness of the Dirichlet problem for equations of mixed type,’’ Dokl. Akad. Nauk SSSR 122, 167–170 (1958).
J. R. Cannon, ‘‘Dirichlet problem for an equation of mixed type with a discontinious coefficient,’’ Ann. Math. Pure Appl. 62, 371–377 (1963).
A. M. Nakhushev, ‘‘The uniqueness criterion for the Dirichlet problem for a mixed type equation in a cylindrical region,’’ Differ. Uravn. 6, 190–191 (1970).
M. M. Khachev, ‘‘The Dirichlet problem for the generalized Lavrentiev-Bitsadze equation in a rectangular domain,’’ Differ. Uravn. 14, 136–139 (1978).
A. P. Soldatov, ‘‘Dirichlet type problems for the Lavrentiev-Bitsadze equation. I, II’’ Dokl. Akad. Nauk 332, 696–698 (1993); Dokl. Akad. Nauk 333, 16–18 (1993).
K. B. Sabitov, To the Theory of Mixed-Type Equations (Fizmalit, Moscow, 2014) [in Russian].
K. B. Sabitov, ‘‘Dirichlet problem for mixed-type equation in a rectangular domain,’’ Dokl. Math. 75, 193–196 (2007).
K. B. Sabitov and A. Kh. Suleymanova, ‘‘The Dirichlet problem for a mixed-type equation of the second kind in a rectangular region,’’ Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 45–53 (2007).
K. B. Sabitov and O. G. Sidorenko, ‘‘Problem with periodicity conditions for a degenerating equation of mixed type,’’ Differ. Equat. 46, 108–116 (2010).
K. B. Sabitov and E. V. Vagapova, ‘‘Dirichlet problem for an equation of mixed type with two degeneration lines in a rectangular domain,’’ Differ. Equat. 49, 68–78 (2013).
R. S. Khairullin, ‘‘Solvability of the Dirichlet problem for a mixed-type equation of the second kind,’’ Differ. Equat. 53, 677–685 (2017).
A. A. Gimaltdinova, ‘‘The Dirichlet problem for the Lavrent’ev–Bitsadze with two type-change lines in a rectangular domain,’’ Dokl. Math. 460 (3), 41–46 (2015).
A. A. Gimaltdinova, ‘‘The Dirichlet problem for a mixed type equation with two transition lines in a rectangular region,’’ Vestn. Samar. Tekh. Univ., Ser.: Fiz.-Mat. Nauki 19, 634–649 (2015).
A. A. Gimaltdinova, ‘‘Neumann problem for the Lavrent’ev–Bitsadze with two type-change lines in a rectangular domain,’’ Dokl. Math. 466, 1–6 (2016).
F. Tricomi, ‘‘Sulle equazioni lineari alle derivate parziali del secondo ordine di tipo misto,’’ Atti R. Accad. Naz. Lincei, Mem. Cl. sci. fis. mat. nat. 14, 133–247 (1923).
M. M. Smirnov, Equations of Mixed Type (Vyssh. Shkola, Moscow, 1985; Am. Math. Soc., 1978).
K. B. Sabitov, ‘‘On the Tricomi problem for the Lavrentiev–Bitsadze equation with a spectral parameter,’’ Differ. Uravn 22, 1977–1984 (1986).
R. Beals, ‘‘Indefinite Sturm–Liouville problems and half-range completness,’’ J. Differ. Equat. 56, 391–407 (1985).
I. E. Egorov, S. G. Pyatkov, and S. V. Popov, Nonclassical Differential Operator Equations (Nauka, Novosibirsk, 2000) [in Russian].
I. M. Karabash and L. S. Kostenko. ‘‘Similarity of \({\textrm{sgn}}x(-\frac{d^{2}}{dx^{2}}+c\delta)\) type operators to normal and self-adjoint operators,’’ Math. Notes 74, 134–139 (2003).
J. Behrndt, Q. Katatbeh, and C. Trunk, ‘‘Non-real Eigenvalues of singular indefinite Sturm–Liouville operators,’’ Proc. Am. Math. Soc. 137, 3797–3806 (2009).
M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
N. G. de Bruijn, Asymptotic Methods in Analysis (Dover, New York, 2010).
A. A. Gimaltdinova and K. V. Kurman, ‘‘On the completeness of one pair of biorthogonally conjugate systems of functions,’’ Vestn. Samar. Tekh. Univ., Ser.: Fiz.-Mat. Nauki 19 (1), 7–18 (2015).
I. S. Lomov, ‘‘Nonsmooth eigenfunctions in problems of mathematical phisics,’’ Differ. Equat. 47, 355–362 (2011).
E. I. Moiseev, ‘‘The solution of a nonlocal boundary value problem by the spectral method,’’ Differ. Equat. 35, 1105–1112 (1999).
V. I. Arnold, ‘‘Small denominators. I. About circle mappings on yourself,’’ Izv. Akad. Nauk USSR 25, 21–86 (1961).
A. Ya. Khinchin, Continued Fractions (Nauka, Moscow, 1978; Dover, New York, 1997).
A. A. Bukhshtab, Number Theory (Moscow, 1966) [in Russian].
K. B. Sabitov and E. M. Safin, ‘‘The inverse problem for an equation of mixed parabolic-hyperbolic type,’’ Math. Notes 87, 880–889 (2010).
K. B. Sabitov, ‘‘The Dirichlet problem for higher-order partial differential equations,’’ Math. Notes 97, 255–267 (2015).
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Gimaltdinova, A. The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change. Lobachevskii J Math 41, 2155–2167 (2020). https://doi.org/10.1134/S1995080220110098
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DOI: https://doi.org/10.1134/S1995080220110098