# Abstract

In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

(Communicated by Giuseppe Di Fazio )

### References

[1] Agmon, S.—Douglis, A.—Nirenberg, L.: *Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions*, Comm. Pure Appl. Math. **12** (1959), 623–727.Search in Google Scholar

[2] Andersson, L−*E*.—Elfving, T.—Golub, G. H.: *Solution of biharmonic equations with application to radar imaging*, J. Comput. Appl. Math. **94** (1998), 153–180.Search in Google Scholar

[3] Begerh, H.—Vu, T. N. H.—Zhang, Z. X.: *Polyharmonic Dirichlet problems*, Proc. Steklov Inst. Math. **255** (2006), 13–34.Search in Google Scholar

[4] Bitsadze, A. V.—Samarskii, A. A.: *Some elementary generalizations of linear elliptic boundary value problems*, Dokl. Akad. Nauk SSSR **185** (1969), 739–740, (in Russian).Search in Google Scholar

[5] Bitsadze A. V.: *On a class of conditionally solvable nonlocal boundary-value problems for harmonic functions*, Soviet Physics Doklady **280** (1985), 521–524.Search in Google Scholar

[6] Bitsadze, A. V.: *Some properties of polyharmonic functions*, Differ. Equ. **24** (1988), 825–831.Search in Google Scholar

[7] Boggio, T.: *Sulle funzioni di Green ďordine m*, Rend. Circ. Mat. Palermo **20** (1905), 97–135.Search in Google Scholar

[8] Criado, F.—Criado, F. J.—Odishelidze, N.: *On the solution of some non-local problems*, Czechoslovak Math. J. **54** (2004), 487–498.Search in Google Scholar

[9] Ehrlich, L. N.—Gupta, M. M.: *Some difference schemes for the biharmonic equation*, SIAM J. Numer. Anal. **12** (1975), 773–790.Search in Google Scholar

[10] Kadirkulov, B. J.—Kirane, M.: *On solvability of a boundary value problem for the Poisson equation with a nonlocal boundary operator*, Acta Math. Sci. **35** (2015), 970–980.Search in Google Scholar

[11] Karachik, V. V.: *Normalized system of functions with respect to the Laplace operator and its applications*, J. Math. Anal. Appl. **287** (2003), 577–592.Search in Google Scholar

[12] Karachik, V. V.: *Solvability conditions for the Neumann problem for the homogeneous polyharmonic equation*, Differ. Equ. **50** (2014), 1449–1456.Search in Google Scholar

[13] Karachik V. V.—Torebek, B. T.: *On the Dirichlet-Riquier problem for biharmonic equations*, Math. Notes **102** (2017), 31–42.Search in Google Scholar

[14] Karachik, V. V.—Turmetov, B. K.: *On solvability of some Neumann-type boundary value problems for biharmonic equation*, Electron. J. Differential Equations **2017** (2017), 218.Search in Google Scholar

[15] Karachik, V. V.—Antropova, N. A.: *Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball*, Differ. Equ. **49** (2013), 251–256.Search in Google Scholar

[16] Karachik, V. V.—Turmetov, B. K.—Bekaeva, A. E.: *Solvability conditions of the biharmonic equation in the unit ball*, Int. J. Pure Appl. Math. **81** (2012), 487–495.Search in Google Scholar

[17] Karachik, V. V.: *On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball*, J. Appl. Ind. Math. **8** (2014), 63–75.Search in Google Scholar

[18] Karachik, V. V.: *Construction of polynomial solutions to some boundary value problems for Poisson’s equation*, Comput. Math. Math. Phys. **51** (2011), 1567–1587.Search in Google Scholar

[19] Karachik, V. V.: *A Neumann-type problem for the biharmonic equation*, Siberian Adv. Math. **27** (2017), 103–118.Search in Google Scholar

[20] Karachik, V. V.: *Green’s function of Dirichlet problem for biharmonic equation in the ball*, Complex Var. Elliptic Equ. **64** (2019), 1500–1521.Search in Google Scholar

[21] Kirane, M.—Torebek, B. T.: *On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions*, Math. Methods Appl. Sci. **39** (2016), 1121–1128.Search in Google Scholar

[22] Kishkis, K. Y.: *On some nonlocal problem for harmonic functions in multiply connected domain*, Differ. Equ. **23** (1987), 174–177.Search in Google Scholar

[23] Koshanova, M. D.—Turmetov, B. K.—Usmanov, K. I.: *About solvability of some boundary value problems for Poisson equation with Hadamard type boundary operator*, Electron. J. Differential Equations **2016** (2016), 1–12.Search in Google Scholar

[24] Lai, M-C.—Liu, H-C.: *Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows*, Appl. Math. Comput. **164** (2005), 679–695.Search in Google Scholar

[25] Love, A. E. H.: *Biharmonic analysis, especially in a rectangle, and its application to the theory of elasticity*, J. Lond. Math. Soc. **3** (1928), 144–156.Search in Google Scholar

[26] Muratbekova, M. A.—Shinaliyev, K. M.—Turmetov, B. K.: *On solvability of a nonlocal problem for the Laplace equation with the fractional-order boundary operator*, Bound. Value Probl. **2014** (2014), 1–13.Search in Google Scholar

[27] Sadybekov, M. A.—Turmetov, B. K.: *On analogues of periodic boundary value problems for the Laplace operator in ball*, Eurasian Math. J. **3** (2012), 143–146.Search in Google Scholar

[28] Sadybekov, M. A.—Turmetov, B. K.: *On an analog of periodic boundary value problems for the Poisson equation in the disk*, Differ. equ. **50** (2014), 268–273.Search in Google Scholar

[29] Skubachevskii, A. L.: *Nonclassical boundary value problems I*, J. Math. Sci. **155** (2008), 199–334.Search in Google Scholar

[30] Skubachevskii, A. L. *Nonclassical boundary value problems II*, J. Math. Sci. **166** (2010), 377–561.Search in Google Scholar

[31] Turmetov, B. K.—Karachik, V. V.: *On solvability of some boundary value problems for a biharmonic equation with periodic conditions*, Filomat **32** (2018), 947–953.Search in Google Scholar

[32] Turmetov, B. K.—Ashurov, R. R.: *On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation*, British J. Math. Comput. Sci. **4** (2014), 557–571.Search in Google Scholar

[33] Turmetov, B. K.—Ashurov, R. R.: *On solvability of the Neumann boundary value problem for a non-homogeneous polyharmonic equation in a ball*, Bound. Value Probl. **2013** (2013), 1–15.Search in Google Scholar

[34] Zaremba, S.: *Sur ľintegration de ľequation biharmonique*, Bulletin international de ľAcademie des sciences de Cracovie (1908), 1–29.Search in Google Scholar

**Received:**2018-09-23

**Accepted:**2019-09-24

**Published Online:**2020-03-10

**Published in Print:**2020-04-28

© 2020 Mathematical Institute Slovak Academy of Sciences