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On a nonlocal generalization of the biharmonic Dirichlet problem. (English. Russian original) Zbl 1195.35133

Differ. Equ. 46, No. 3, 321-328 (2010); translation from Differ. Uravn. 46, No. 3, 318-325 (2010).
Summary: We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space \(W _{2}^{2}\). We show that the problem can be viewed as a generalization of the Dirichlet problem.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35D30 Weak solutions to PDEs
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[1] Cannon, J.R., The Solution of the Heat Equation Subject to the Specification of Energy, Quart. Appl. Math., 1963, vol. 21, no. 2, pp. 155–160. · Zbl 0173.38404 · doi:10.1090/qam/160437
[2] Karaushev, A.V., Rechnaya gidravlika (Fluvial Hidraulics), Leningrad, 1969.
[3] Alekseev, A.S. and Kirillov, Yu.P., Investigation of a Class of Partial Differential Equations with Integral Boundary Conditions, Diff. Integr. Uravn., Gorkii, 1979, no. 3, pp. 118–122.
[4] Sapagovas, M.P., A Difference Scheme for Two-Dimensional Elliptic Problems with an Integral Condition, Litovsk. Mat. Sb., 1983, vol. 23, no. 3, pp. 155–159. · Zbl 0535.65065
[5] Paneyakh, B.P., Some Nonlocal Boundary Value Problems for Linear Differential Operators, Mat. Zametki, 1984, vol. 35, no. 3, pp. 425–434. · Zbl 0553.35020
[6] Skubachevskii, A.L. and Steblov, G.M., On the Spectrum of Differential Operators with Domain That Is Not Dense in L 2(0, 1), Dokl. Akad. Nauk SSSR, 1991, vol. 321, no. 6, pp. 1158–1163.
[7] Gushchin, A.K. and Mikhailov, V.P., On the Solvability of Nonlocal Problems for a Second-Order Elliptic Equation, Mat. Sb., 1994, vol. 185, no. 1, pp. 121–160.
[8] Chipot, M. and Lovat, B., Some Remarks on Nonlocal Elliptic and Parabolic Problems, Nonlinear Anal., 1997, vol. 30, no. 7, pp. 4619–4627. · Zbl 0894.35119 · doi:10.1016/S0362-546X(97)00169-7
[9] Davitashvili, T. and Gordeziani, D., Mathematical Model with Nonlocal Boundary Conditions for the Atmospheric Pollution, Bull. Georgian Acad. Sci., 2000, vol. 161, no. 3, pp. 435–437. · Zbl 1028.86001
[10] De Schepper, H. and Slodička, M., Recovery of the Boundary Data for a Linear Second Order Elliptic Problem with a Nonlocal Boundary Condition, ANZIAM J., 2000, vol. 42, part C, pp. C518–C535.
[11] Gordeziani, D. and Avalishvili, G., Investigation of the Nonlocal Initial Boundary Value Problems for Some Hyperbolic Equations, Hiroshima Math. J., 2001, vol. 31, no. 3, pp. 345–366. · Zbl 1008.35037
[12] Mesloub, S., Bouziani, A., and Kechkar, N., A Strong Solution of an Evolution Problem with Integral Conditions, Georgian Math. J., 2002, vol. 9, no. 1, pp. 149–159. · Zbl 1040.35037
[13] Berikelashvili, G., To a Nonlocal Generalization of the Dirichlet Problem, J. Inequal. Appl., 2006, vol. 2006, article ID 93858, pp. 1–6. · Zbl 1108.35040 · doi:10.1155/JIA/2006/93858
[14] Kufner, A. and Sändig, A.-M., Some Applications of Weighted Sobolev Spaces, Teubner-Texte Math., vol. 100, Leipzig, 1987.
[15] Nekvinda, A. and Pick, L., A Note on the Dirichlet Problem for the Elliptic Linear Operator in Sobolev Spaces with Weight d M , Comment. Math. Univ. Carolin., 1988, vol. 29, no. 1, pp. 63–71. · Zbl 0662.35033
[16] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978. Translated under the title Metod konechnykh elementov dlya ellipticheskikh zadach, Moscow: Mir, 1980. · Zbl 0383.65058
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