Skip to main content
SpringerLink
Log in

The Cauchy Problem for Second-Order Elliptic Systems on the Plane

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Regular solutions to second-order elliptic systems on the plane are representable in terms of A-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bitsadze A. V., Certain Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  2. Soldatov A. P., One-Dimensional Singular Operators and Boundary Value Problems of the Theory of Functions [in Russian], Vysshaya Shkola, Moscow (1991).

    Google Scholar 

  3. Arbuzov E. V. and Bukhgeim A. L., “Carleman's formulas for A-analytic functions in a half-plane,” J. Inverse Ill-Posed Probl., 5, No. 6, 491–505 (1997).

    Google Scholar 

  4. Carleman T., Les Fonctions Quasianalytiques, Gauthier-Villars, Paris (1926).

    Google Scholar 

  5. Goluzin G. M. and Krylov V. I., “A generalized Carleman formula and its application to analytic continuation of functions,” Mat. Sb., 40, No. 2, 144–149 (1933).

    Google Scholar 

  6. Lavrent 'ev M. M., On Some Ill-Posed Problems of Mathematical Physics [in Russian], Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk (1962).

    Google Scholar 

  7. Aizenberg L. A., Carleman Formulas in Complex Analysis [in Russian], Nauka, Novosibirsk (1990).

    Google Scholar 

  8. Yarmukhamedov Sh. Ya., “On continuation of a solution of the Helmholtz equation,” Dokl. Ross. Akad. Nauk, 357, No. 3, 320–323 (1997).

    Google Scholar 

  9. Bukhgeim A. L., “Inversion formulas in inverse problems,” A supplement to the monograph: M. M. Lavrentiev and L. Ya. Saveliev, Linear Operators and Ill-Posed Problems, Consultants Bureau and Nauka Publishers, New York; London; Moscow (1995).

    Google Scholar 

  10. Niezov I. È., “The Cauchy problem for a system of elasticity on the plane,” Uzbek. Mat. Zh., No. 1, 27–34 (1996).

  11. Rachele L. Z., “Boundary determination for an inverse problem in elastodynamics,” Comm. Partial Differential Equations, 25, No. 11–12, 1951–1996 (2000).

    Google Scholar 

  12. Zhura N. A., “Boundary-value problems of elliptic systems in domains with piecewise-smooth boundaries,” Differential Equations, 25, No. 5, 595–601 (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk

    E. V. Arbuzov

Authors
  1. E. V. Arbuzov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arbuzov, E.V. The Cauchy Problem for Second-Order Elliptic Systems on the Plane. Siberian Mathematical Journal 44, 1–16 (2003). https://doi.org/10.1023/A:1022034001292

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022034001292

Search

Navigation