Neumann boundary condition for a nonlocal biharmonic equation

B. Kh. Turmetov   ; V. V. Karachik

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 2, pp. 51-58

Résumé

The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained. First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.

Détail

Keywords: nonlocal operator, the Neumann boundary condition, biharmonic equation, solvability conditions, Green's function.

Bibliographie

[1] A.M. Nakhushev, Uravneniya matematicheskoi biologii, Vyssh. shk., M., 1995, 301 pp.

[2] A.A. Andreev, “Ob analogakh klassicheskikh kraevykh zadach dlya odnogo differentsialnogo uravneniya vtorogo poryadka s otklonyayuschimsya argumentom”, Differents. uravneniya, 40:8 (2004), 1126–1128 | MR | Zbl

[3] A. Ashyralyev, A.M. Sarsenbi, “Well-posedness of a parabolic equation with involution”, Numerical Functional Analysis and Optimization, 38:10 (2017), 1295–1304 | DOI | MR | Zbl

[4] A. Ashyralyev, A.M. Sarsenbi, “Well-posedness of an elliptic equation with involution”, Electronic Journal of Differential Equations, 2015, no. 284, 1–8 | MR

[5] V.V. Karachik, A.M. Sarsenbi, B.Kh. Turmetov, “On the solvability of the main boundary value problems for a nonlocal Poisson equation”, Turkish Journal of Mathematics, 43:3 (2019), 1604–1625 | DOI | MR | Zbl

[6] M. Kirane, N. Al-Salti, “Inverse problems for a nonlocal wave equation with an involution perturbation”, Journal of Nonlinear Sciences and Applications, 9:3 (2016), 1243–1251 | DOI | MR | Zbl

[7] A.L. Skubachevskii, “Nonclassical boundary value problems. I”, Journal of Mathematical Sciences, 155:2 (2008), 199–334 | DOI | MR | Zbl

[8] A.L. Skubachevskii, “Nonclassical boundary-value problems. II”, Journal of Mathematical Sciences, 166:4 (2010), 377–561 | DOI | MR | Zbl

[9] D. Przeworska-Rolewicz, “Some boundary value problems with transformed argument”, Commentationes Mathematicae, 17:2 (1974), 451–457 | MR | Zbl

[10] V.V. Karachik, “Ob odnoi zadache tipa Neimana dlya bigarmonicheskogo uravneniya”, Matematicheskie trudy, 19:2 (2016), 86–108 | Zbl

[11] M.A. Sadybekov, A.A. Dukenbayeva, “On boundary value problems of the Samarskii-Ionkin type for the Laplace operator in a ball”, Complex Variables and Elliptic Equations, 2020, 1–15 | MR

[12] V.V. Karachik, B.Kh. Turmetov, “On solvability of some nonlocal boundary value problems for biharmonic equation”, Mathematica Slovaca, 70:2 (2020), 329–342 | DOI | MR | Zbl

[13] V.V. Karachik, B.Kh. Turmetov, “Solvability of one nonlocal Dirichlet problem for the Poisson equation”, Novi Sad Journal of Mathematics, 50:1 (2020), 67–88 | Zbl

[14] B.Kh. Turmetov, V.V. Karachik, “O zadache Dirikhle dlya nelokalnogo poligarmonicheskogo uravneniya”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Seriya: Matematika. Mekhanika. Fizika, 13:2 (2021), 37–45 | Zbl

[15] V.V. Karachik, “Greens function of Dirichlet problem for biharmonic equation in the ball”, Complex Variables and Elliptic Equations, 64:9 (2019), 1500–1521 | DOI | MR | Zbl

[16] V.V. Karachik, “Funktsii Grina zadach Nave i Rike-Neimana dlya bigarmonicheskogo uravneniya v share”, Differentsialnye uravneniya, 57:5 (2021), 673–686 | Zbl

[17] Karachik V., Green's functions of some boundary value problems for the biharmonic equation. Complex Variables and Elliptic Equations, 2021 (Online) | MR

[18] V. Karachik, “Dirichlet and Neumann boundary value problems for the polyharmonic equation in the unit ball”, Mathematics, 9:16 (2021), 1907 | DOI | MR

[19] V.V. Karachik, B.K. Turmetov, “On the Green's function for the third boundary value problem”, Siberian Advances in Mathematics, 29:1 (2019), 32–43 | DOI | MR

[20] M.A. Sadybekov, B.T. Torebek, B.K. Turmetov, “Representation of Green's function of the Neumann problem for a multi-dimensional ball”, Complex Variables and Elliptic Equations, 61:1 (2016), 104–123 | DOI | MR | Zbl