Abstract
We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.
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Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.
Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Izdat. Moskov. Gos. Univ., 1999.
Lifanov, I.K., Metod singulyarnykh integral’nykh uravnenii i chislennyi eksperiment, Moscow: Yanus, 1995. Engl. transl.: Singular integral equations and discrete vortices, VSP, The Nethederlands, 1996.
Vainikko, G.M., Lifanov, I.K., and Poltavskii, L.N., Chislennye metody v gipersingulyarnykh integral’nykh uravneniyakh i ikh prilozheniya (Numerical Methods in Hypersingular Integral Equations and Their Applications), Moscow, 2001.
Gutnikov, V.A., Kiryakin, V.Yu., Lifanov, I.K., et al., Numerical Solution to a Two-Dimensional Hypersingular Integral Equation and Sound Propagation in Urban Areas, Comput. Math. Math. Phys., 2007, vol. 47, no. 12, pp. 2002–2013.
Zakharov, E.V., Ryzhakov, G.V., and Setukha, A.V., Numerical Solution of 3D Problems of Electromagnetic Wave Diffraction on a System of Ideally Conducting Surfaces by the Method of Hypersingular Integral Equations, Differ. Equ., 2014, vol. 50, no. 9, pp. 1240–1251.
Pisarev, I.V. and Setukha, A.V., Transferring the Boundary Conditions to the Middle Surface for the Numerical Solution of a Boundary Value Problem in the Linear Wing Theory, Vychisl. Metody Programmir.: Novye Vychisl. Tekhnol., 2014, vol. 15, pp. 109–120.
Colton, D. and Kress, R., Integral Methods in Scattering Theory, New York: John Willey & Sons, 1983. Translated under the title Metody integral’nykh uravnenii v teorii rasseyaniya, Moscow: Mir, 1987.
Setukha, A.V., Construction of Fundamental Solutions of the Neumann Boundary Value Problem in a Domain outside an Open Plane Surface, Differ. Equ., 2002, vol. 38, no. 4, pp. 528–540.
Setukha, A.V., The Three-Dimensional Neumann Problem with Generalized Boundary Conditions and the Prandtl Equation, Differ. Equ., 2003, vol. 39, no. 9, pp. 1249–1262.
Setukha, A.V., The Neumann Problem with Boundary Condition on an Open Plane Surface, Differ. Equ., 2001, vol. 37, no. 10, pp. 1376–1398.
Mikhlin, S.G., Mnogomernye singulyarnye integraly i integral’nye uravneniya (Higher-Dimensional Singular Integrals and Integral Equations), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1962.
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Original Russian Text © A.V. Setukha, D.A. Yukhman, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 9, pp. 1231–1241.
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Setukha, A.V., Yukhman, D.A. On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative. Diff Equat 52, 1188–1198 (2016). https://doi.org/10.1134/S001226611609010X
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DOI: https://doi.org/10.1134/S001226611609010X