Exact Solution to the Dirichlet Problem for Degenerating on the Boundary Elliptic Equation of Tricomi - Keldysh Type in the Half-Space

In the paper we solve the Dirichlet problem for a multidimensional equation by means of Fourier transform method and similarity method. The problem is a generalization of the Tricomi, Gellerstedt and Keldysh equations in the half-space, the equation is of an elliptic type with the boundary condition on the boundary hyperplane where equation degenerates. We present the solution in the form of an integral with a simple kernel. It is an approximation to the identity and self-similar solution of Tricomi type equation. In particular, this formula contains a Poisson’s formula, which gives the solution of the Dirichlet problem for the Laplace equation for the half-space. If the given boundary value is a generalized function of slow growth, the solution of the Dirichlet problem can be presented as a convolution of this function with the kernel (if a convolution exists).

References

[1] Keldysh M.V. On some cases of degenerate elliptic equations on the boundary of a domain. Dokl. Akad. nauk USSR [Proc. Acad. Sci. USSR], 1951, vol. 77, no. 2, pp. 181-183 (in Russ.).

[2] Bers L. Mathematical aspects of subsonic and transonic gas dynamics. N.Y., Wiley, 1958.

[3] Otway T.H. Dirichlet problem for elliptic-hyperbolic equations of Keldych type. Berlin, Heidelberg, Springer-Verlag, 2012. 214 p.

[4] Bitsadze A.V. Equations of mathematical physics. Moscow, Mir Publ., 1980. 318 p.

[5] Stein E. Singular integrals and differentiability properties of functions. N.J., Princeton University Press, 1970.

[6] Parasyuk L.S. Boundary problems for elliptic differential equations that degenerate on the boundary of domain. Ukrainska akademia drukarstva. Naukovi zapysky, 1961, no. 13, pp. 6575 (in Ukrainian).

[7] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator. Duke Math. J. 1999, vol. 98(3), pp. 465-483.

[8] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, II. Duke Math. J., 2002, vol. 111(3), pp. 561-584.

[9] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, III. Duke Math. J., 2005, vol. 128(1), pp. 119-140.

[10] Barros-Neto J., Cardoso F. Bessel integrals and fundamental solutions for a generalized Tricomi operator. Journal of Functional Analysis, 2001, vol. 183, pp. 472-497. DOI: 10.1006/jfan.2001.3749

[11] Algazin O.D., Kopaev A.V. Solution to the mixed boundary-value problem for Laplace equation in multidimentional 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

[12] Algazin O.D., Kopaev A.V. Dirichlet problem solution for Poisson equation in a multidimensional infinite layer. Mat. i mat. model. MGTU im. N.E. Baumana [Mathematics and Mathematical Modelling of the Bauman MSTU. Electron. Journ.], 2015, no. 4. DOI: 10.7463/mathm.0415.0812943 Available at: http://mathmjournal.ru/en/doc/812943.html

[13] Vladimirov V.S. Obobshchennye funktsii v matematicheskoy fizike [Generalized functions in mathematical physics]. Moscow, Nauka Publ., 1979. 320 p.

[14] Ryshik I.M., Gradstein I.S. Tables of integrals, series, and products. N.Y., Academic Press, 2007.

[15] Barenblatt G.I. Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, 2002.

[16] Polyanin A.D., Zaytsev V.F., Zhurov A.I. Metody resheniya nelineynykh uravneniy matematicheskoy fiziki i mekhaniki [Methods for solving nonlinear equations of mathematical physics and mechanics]. Moscow, Fizmatlit Publ., 2005. 256 p.