It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ2. In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.
@article{bwmeta1.element.doi-10_2478_s11533-007-0027-z,
author = {T. Savina},
title = {On splitting up singularities of fundamental solutions to elliptic equations in $\mathbb{C}$2},
journal = {Open Mathematics},
volume = {5},
year = {2007},
pages = {733-740},
zbl = {1145.35308},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0027-z}
}
T. Savina. On splitting up singularities of fundamental solutions to elliptic equations in ℂ2. Open Mathematics, Tome 5 (2007) pp. 733-740. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0027-z/
[1] D. Colton and R.P. Gilbert: “Singularities of solutions to elliptic partial differential equations with analytic coefficients”, Quart. J. Math. Oxford Ser. 2, Vol. 19, (1968), pp. 391–396. http://dx.doi.org/10.1093/qmath/19.1.391 | Zbl 0165.44402
[2] F. John: “The fundamental solution of linear elliptic differential equations with analytic Coefficients”, Comm. Pure Appl. Math., Vol. 3, (1950), pp. 273–304. http://dx.doi.org/10.1002/cpa.3160030305 | Zbl 0041.06203
[3] F. John: Plane waves and spherical means applied to partial differential equations, Springer-Verlag, New York-Berlin, 1981. | Zbl 0464.35001
[4] D. Khavinson: Holomorphic partial differential equations and classical potential theory, Universidad de La Laguna, 1996.
[5] D. Ludwig: “Exact and Asymptotic solutions of the Cauchy problem/rd, Comm. Pure Appl. Math., Vol. 13, (1960), pp. 473–508. http://dx.doi.org/10.1002/cpa.3160130310 | Zbl 0098.29601
[6] T.V. Savina: “On a reflection formula for higher-order elliptic equations/rd, Math. Notes, Vol. 57, no. 5–6, (1995), pp. 511–521. http://dx.doi.org/10.1007/BF02304421
[7] T.V. Savina: “A reflection formula for the Helmholtz equation with the Neumann Condition/rd, Comput. Math. Math. Phys., Vol. 39, no. 4, (1999), pp. 652–660. | Zbl 0965.35028
[8] T.V. Savina, B.Yu. Sternin and V.E. Shatalov: “On a reflection formula for the Helmholtz equation”, Radiotechnika i Electronica, (1993), pp. 229–240.
[9] B.Yu. Sternin and V.E. Shatalov: Differential equations on complex manifolds, Mathematics and its Applications, Vol. 276, Kluwer Academic Publishers Group, Dordrecht, 1994. | Zbl 0818.35003
[10] I.N. Vekua: New methods for solving elliptic equations, North Holland, 1967.
[11] I.N. Vekua: Generalized analytic functions, Second edition, Nauka, Moscow, 1988.