Skip to main content
SpringerLink
Log in

Completeness of the Asymmetric Products of Solutions of a Second-Order Elliptic Equation and the Uniqueness of the Solution of an Inverse Problem for the Wave Equation

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We establish the property of completeness in the space \(L_2(D) \), where \(D \) is a domain in \(\mathbb {R}^n \), \(n\geq 3\), of the products of all possible regular solutions of the equation \(\Delta u-\kappa ^2 u=0 \), \(\kappa \in \mathbb {R}_{+}\cup i\mathbb {R}_{-} \), by the fundamental solutions of this equation with singularities on a straight line that does not meet \(\overline {D}\). This result is used when establishing the uniqueness of the solution of a coefficient inverse problem of wave tomography in a nonoverdetermined statement.

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.

Similar content being viewed by others

REFERENCES

  1. Calderón, A.-P., On an inverse boundary value problem, Seminar on Numerical Analysis and Its Applications to Continuum Physics, (Soc. Bras. Mat., Rio de Janeiro, 1980), pp. 65–73.

  2. Isakov, V., Inverse Problems for Partial Differential Equations, New York: Springer, 2006.

    MATH  Google Scholar 

  3. Kenig, C. and Salo, M., Recent progress in the Calderon problem with partial data, in Contemp. Math. Vol. 615. Inverse Problems and Applications, Providence, RI: Am. Math. Soc., 2014, pp. 193–222.

  4. Lavrent’ev, M.M., On one inverse problem for the wave equation, Dokl. Akad. Nauk SSSR, 1964, vol. 157, no. 3, pp. 520–521.

    Google Scholar 

  5. Lavrent’ev, M.M., On one class of inverse problems for differential equations, Dokl. Akad. Nauk SSSR, 1965, vol. 160, no. 1, pp. 32–35.

    MATH  Google Scholar 

  6. Bakushinskii, A.B, Kozlov, A.I., and Kokurin, M.Yu., On one inverse problem for three-dimensional wave equation, Comput. Math. Math. Phys., 2003, vol. 43, no. 8, pp. 1149–1158.

    MathSciNet  MATH  Google Scholar 

  7. Vainberg, M.M., Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki (Asymptotic Methods in Equations of Mathematical Physics), Moscow: Izd. Mosk. Gos. Univ., 1982.

    Google Scholar 

  8. Romanov, V.G., On smoothness of a fundamental solution to a second order hyperbolic equation, Sib. Math. J., 2009, vol. 50, no. 4, pp. 700–705.

    Article  MathSciNet  Google Scholar 

  9. Maz’ya, V. and Schmidt, G., Approximate Approximations, Providence, RI: Am. Math. Soc., 2007.

    Book  Google Scholar 

  10. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Natl. Bur. Stand., 1964. Translated under the title: Spravochnik po spetsial’nym funktsiyam, Moscow: Nauka, 1979.

  11. Lavrent’ev, M.M., Romanov, V.G., and Shishatskii, S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-Posed Problems of Mathematical Physics and Analysis), Moscow: Nauka, 1980.

    Google Scholar 

  12. Ramm, A.G., Multidimensional Inverse Scattering Problems, New York: John Wiley & Sons, 1992. Translated under the title: Mnogomernye obratnye zadachi rasseyaniya, Moscow: Mir, 1994.

    MATH  Google Scholar 

  13. Kokurin, M.Yu. and Paimerov, S.K., Inverse coefficient problem for a wave equation in a bounded domain, Comput. Math. Math. Phys., 2008, vol. 48, no. 1, pp. 109–120.

    Article  MathSciNet  Google Scholar 

  14. Kokurin, M.Yu., On the completeness of products of harmonic functions and the uniqueness of the solution of the inverse acoustic sounding problem, Math. Notes, 2018, vol. 104, no. 5, pp. 689–695.

    Article  MathSciNet  Google Scholar 

  15. Stein, I.M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton: Princeton Univ. Press, 1971. Translated under the title: Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Moscow: Mir, 1974.

    MATH  Google Scholar 

  16. Bateman, H. and Erdélyi, A., Tables of Integral Transforms. Vol. 1, New York–Toronto–London: McGraw–Hill Book Co., 1954. Translated under the title: Tablitsy integral’nykh preobrazovanii. T. I. Preobrazovaniya Fur’e, Laplasa, Mellina, Moscow: Nauka, 1969.

    Google Scholar 

  17. Kipriyanov, I.A., Singulyarnye ellipticheskie kraevye zadachi (Singular Elliptic Boundary Value Problems), Moscow: Nauka, 1997.

    MATH  Google Scholar 

  18. Schwartz, L., Analyse mathématique, Paris: Hermann, 1967. Translated under the title: Analiz. T. I , Moscow: Mir, 1972.

    MATH  Google Scholar 

  19. Akhiezer, N.I., Lektsii po teorii approksimatsii (Lectures on Approximation Theory), Moscow: Nauka, 1965.

    Google Scholar 

  20. Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities, New York: Springer, 1995.

    Book  Google Scholar 

  21. Ditkin, V.A. and Prudnikov, A.P., Integral’nye preobrazovaniya i operatsionnoe ischislenie (Integral Transformations and Operator Calculus), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1961.

    Google Scholar 

  22. Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer, 1983. Translated under the title: Ellipticheskie differentsial’nye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Moscow: Nauka, 1989.

    MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation, project no. 20-11-20085.

Author information

Authors and Affiliations

  1. Mari State University, Yoshkar-Ola, Mari El, 424001, Russia

    M. Yu. Kokurin

Authors
  1. M. Yu. Kokurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Yu. Kokurin.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kokurin, M.Y. Completeness of the Asymmetric Products of Solutions of a Second-Order Elliptic Equation and the Uniqueness of the Solution of an Inverse Problem for the Wave Equation. Diff Equat 57, 241–250 (2021). https://doi.org/10.1134/S0012266121020129

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266121020129

Search

Navigation