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Solvability of the Neumann problem in a disk for fourth order properly elliptic equations. (Ukrainian, English) Zbl 1349.35096

Mat. Metody Fiz.-Mekh. Polya 57, No. 1, 7-17 (2014); translation in J. Math. Sci., New York 212, No. 1, 1-15 (2016).
In the paper the authors establish and study sufficient conditions of solvability of the Neumann problem for fourth order properly elliptic equations of general form in a unit disk \(K\) in the space \(C^4(K)\cap C^{3,\alpha}(\overline K)\).

MSC:

35J30 Higher-order elliptic equations
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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[1] A. O. Babayan, “On the Dirichlet problem for a fourth-order improperly elliptic equation,” in: Nonclassical Equations of Mathematical Physics [in Russian], Institute of Mathematics, Siberian Branch, Russ. Acad. Sci., Novosibirsk (2007), pp. 56-68. · Zbl 1286.34038
[2] A. V. Bitsadze, “On the uniqueness of solution to the Dirichlet problem for partial elliptic equations,” Usp. Mat. Nauk,3, No. 6(28), 211-212 (1948). · Zbl 0041.21703
[3] V. P. Burskii, Methods for the Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).
[4] V. P. Burskii, “Breakdown of uniqueness of solutions of the Dirichlet problem for elliptic systems in a disc,” Mat. Zametki,48, No. 3, 32-36 (1990); English translation:Math. Notes,48, No. 3, 894-897 (1990). · Zbl 0732.35012
[5] V. P. Burskii and K. A. Buryachenko, “On the breakdown of the uniqueness of a solution of the Dirichlet problem for typeless differential equations of arbitrary even order in a disk,” Ukr. Mat. Visnyk,9, No. 4, 477-514 (2012); English translation:J. Math. Sci.,190, No. 4, 539-566 (2013). · Zbl 1293.35374
[6] V. P. Burskii and E. V. Lesina, “Neumann problem for second-order improperly elliptic equations,” in: Trans. Inst. Appl. Math. Mech. [in Russian], Vol. 23 (2011), pp. 10-16. · Zbl 1324.35012
[7] E. A. Buryachenko, “Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles,” Ukr. Mat. Zh.,62, No. 5, 591-603 (2010); English translation:Ukr. Math. J.,62, No. 5, 676-690 (2010). · Zbl 1224.35068
[8] K. O. Buryachenko, “Solvability of inhomogeneous boundary-value problems for fourth-order differential equations,” Ukr. Mat. Zh.,63, No. 8, 1011-1020 (2011); English translation:Ukr. Math. J.,63, No. 8, 1165-1175 (2012). · Zbl 1257.35073
[9] A. I. Markushevich, Recurrent Sequences [in Russian], Gostekhteorizdat, Moscow (1950).
[10] B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
[11] A. P. Soldatov, “On the first and second boundary-value problems for elliptic systems on a plane,” Differents. Uravn.,39, No. 5, 674-686 (2003). · Zbl 1123.35319
[12] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin (1988). · Zbl 0679.46057 · doi:10.1007/978-3-642-88201-2
[13] A. R. Aliev and A. L. Elbably, “Well-posedness of a boundary value problem for a class of third-order operator-differential equations,” Bound. Value Probl.,2013, 140 (2013). · Zbl 1297.34071 · doi:10.1186/1687-2770-2013-140
[14] S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Springer, New York (2001). · Zbl 0959.31001 · doi:10.1007/978-1-4757-8137-3
[15] A. O. Babayan, “Dirichlet problem for properly elliptic equation in unit disk,” J. Contemp. Math. Anal. (=Izv. NAN Armen., Matematika),38, No. 6, 39-48 (2003). · Zbl 1160.35390
[16] A. O. Babayan, “On unique solvability of Dirichlet problem for fourth order properly elliptic equation,” Izv. NAN Armen., Matematika,34, No. 5, 3-18 (1999). · Zbl 1075.35516
[17] H. Begehr and A. Kumar, “Boundary-value problems for the inhomogeneous polyanalytic equation. I,” Analysis: Int. Math. J. Anal. & Its Appl.,25, No. 1, 55-71 (2005). · Zbl 1077.30034
[18] A. L. Beklaryan, “On the existence of solutions of the first boundary-value problem for elliptic equations on unbounded domains,” Int. J. Pure Appl. Math.,88, No. 4, 499-522 (2013). · Zbl 1288.35222 · doi:10.12732/ijpam.v88i4.5
[19] G. Bonanno and P. F. Pizzimenti, “Neumann boundary-value problems with not coercive potential,” Mediter. J. Math.,9, No. 4, 601-609 (2012). · Zbl 1260.34041 · doi:10.1007/s00009-011-0136-6
[20] B. Brown, G. Grubb, and I. G. Wood, “M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,” Math. Nachr.,282, No. 3, 314-347 (2009). · Zbl 1167.47057 · doi:10.1002/mana.200810740
[21] B. Brown, M. Marletta, S. Naboko, and I. G. Wood, “Boundary triplets and M-functions for nonself-adjoint operators, with applications to elliptic PDEs and block operator matrices,” J. London Math. Soc.,77, No. 3, 700-718 (2008). · Zbl 1148.35053 · doi:10.1112/jlms/jdn006
[22] G. Grubb, Distributions and Operators (Graduate Texts in Mathematics), Vol. 252, Springer, New York (2009). · Zbl 1171.47001
[23] L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Springer, Berlin (1983). · Zbl 0521.35001
[24] N. B. Kerimov and U. Kaya, “Spectral properties of some regular boundary-value problems for fourth-order differential operators,” Cent. Eur. J. Math.,11, No. 1, 94-111 (2013). · Zbl 1272.34025 · doi:10.2478/s11533-012-0059-x
[25] M. Mokhtarzadeh, M. Pournaki, and A. Razani, “An existence-uniqueness theorem for a class of boundary-value problems,” Fixed Point Theory,13, No. 2, 583-591 (2012). · Zbl 1286.34038
[26] A. Posilicano, “Self-adjoint extensions of restrictions,” Operat. Matrices,2, No. 4, 483-506 (2008). · Zbl 1175.47025 · doi:10.7153/oam-02-30
[27] N. E. Tovmasyan, Non-Regular Differential Equations and Calculations of Electromagnetic Fields, World Scientific, Singapore (1998). · Zbl 0954.35003 · doi:10.1142/9789812816627
[28] N. E. Tovmasian and V. S. Zakarian, “Dirichlet problem for properly elliptic equations in multiply connected domains,” J. Contemp. Math. Anal. (=Izv. NAN Armen., Matematika),37, No. 6, 2-34 (2002). · Zbl 1086.35503
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