The solution of the mixed boundary value problem of Dirichlet- Neumann for the Poisson equation in a multidimensional infinite layer

In this research by the method of Fourier transform we solve the mixed boundary value problem of Dirichlet - Neumann for the Poisson equation in the domain bounded by two parallel hyperplanes in Rⁿ. The solution is represented as a sum of integrals whose kernels are found in the final form. In particular, we constructed Green’s function of the Laplace operator for the mixed boundary value problem of Dirichlet - Neumann, by which the solution of the problem is written. If the given boundary values are tempered distributions, the solution of the mixed boundary value problem for the homogeneous equation (Laplace) is written as the convolution of the kernels with these distributions.

References

[1] Algazin O.D., Kopaev A.V. Solution to the mixed boundary-value problem for Laplace equation in multidimensional infinite layer. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2015, no. 1, pp. 3-13 (in Russ.). DOI: 10.18698/1812-3368-2015-1-3-13

[2] Komech A.I. Linear differential equations in partial derivatives with constant coefficients. Itogi Nauki Tekhn., 1988, vol. 31, pp. 127-261 (in Russ.).

[3] Polyanin A.D. Spravochnik po lineynym uravneniyam matematicheskoy fiziki [Linear Equations of Mathematical Physics. Handbook]. Moscow, Fizmatlit Publ., 2001. 576 p.

[4] Kas’yanov E.Yu., Kopaev A.V. On the solution of the Dirichlet problem for some multidimensional domains by the method of reproducing kernels. Soviet Mathematics [Izv. Vyssh. Uchebn. Zaved. Mat.], 1991, vol. 35, no. 6, pp. 16-18.

[5] Vladimirov V.S. Obobshchennye funktsii v matematicheskoy fizike [Generalized Functions in Mathematical Physics]. Moscow, Nauka Publ., 1979. 320 p.

[6] Bochner S. Vorlesungen uber Fouriersche Integrale. Leipzig, Akademische Verlagsgesellschaft m.b. H. VIII, 1932. 229 p.

[7] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow, MGU im. M.V. Lomonosova Publ., 1999. 798 p.

[8] Ditkin V.A., Prudnikov A.P. Integral’nye preobrazovaniya i operatsionnoe is-chislenie [Integral Transformations and Operational Calculus]. Moscow, Fizmatgiz Publ., 1961. 524 p.

[9] Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of Integrals, Sums, Series and Products]. Moscow, Nauka Publ., 1971. 1108 p.