×

zbMATH — the first resource for mathematics

On splitting up singularities of fundamental solutions to elliptic equations in \(\mathbb C^{2}\). (English) Zbl 1145.35308
Summary: 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 \(\mathbb C^{2}\). In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.

MSC:
35A20 Analyticity in context of PDEs
35A08 Fundamental solutions to PDEs
32S70 Other operations on complex singularities
35J15 Second-order elliptic equations
32W50 Other partial differential equations of complex analysis in several variables
35C15 Integral representations of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[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.
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.