On splitting up singularities of fundamental solutions to elliptic equations in ℂ2

T. Savina

Open Mathematics (2007)

  • Volume: 5, Issue: 4, page 733-740
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

T. Savina. "On splitting up singularities of fundamental solutions to elliptic equations in ℂ2." Open Mathematics 5.4 (2007): 733-740. <http://eudml.org/doc/269735>.

@article{T2007,
abstract = {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.},
author = {T. Savina},
journal = {Open Mathematics},
keywords = {Complex Variables; Elliptic Equations; Fundamental Solution; complex variables; elliptic equations; fundamental solution; analytic coefficients; singularities on characteristics},
language = {eng},
number = {4},
pages = {733-740},
title = {On splitting up singularities of fundamental solutions to elliptic equations in ℂ2},
url = {http://eudml.org/doc/269735},
volume = {5},
year = {2007},
}

TY - JOUR
AU - T. Savina
TI - On splitting up singularities of fundamental solutions to elliptic equations in ℂ2
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 733
EP - 740
AB - 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.
LA - eng
KW - Complex Variables; Elliptic Equations; Fundamental Solution; complex variables; elliptic equations; fundamental solution; analytic coefficients; singularities on characteristics
UR - http://eudml.org/doc/269735
ER -

References

top
  1. [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 Zbl0165.44402
  2. [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 Zbl0041.06203
  3. [3] F. John: Plane waves and spherical means applied to partial differential equations, Springer-Verlag, New York-Berlin, 1981. Zbl0464.35001
  4. [4] D. Khavinson: Holomorphic partial differential equations and classical potential theory, Universidad de La Laguna, 1996. 
  5. [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 Zbl0098.29601
  6. [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. [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. Zbl0965.35028
  8. [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. [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. Zbl0818.35003
  10. [10] I.N. Vekua: New methods for solving elliptic equations, North Holland, 1967. 
  11. [11] I.N. Vekua: Generalized analytic functions, Second edition, Nauka, Moscow, 1988. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.