Skip to main content
SpringerLink
Log in

On some classes of inverse problems for parabolic and elliptic equations

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

We study solvability of inverse problems of finding the right-hand side together with a solution itself for vector-valued parabolic and elliptic equations. The usual boundary conditions are supplemented with the overdetermination conditions that are the values of a solution on some system of surfaces.

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. Agarwal R.P., O’Regan D., Shakhmurov V.B.: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl. 388, 970–983 (2008)

    Article  MathSciNet  Google Scholar 

  2. Agranovich M.S., Vishik M.I.: Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surv. 19, 53–157 (1964)

    Article  MATH  Google Scholar 

  3. Alekseev G.V.: Coefficient inverse extremum problems for stationary heat and mass transfer equations. Comp. Math. and Math. Phys. 47, 1007–1028 (2007)

    Article  Google Scholar 

  4. Alekseev G.V.: Solvability of control problems for stationary equations of magnetohydrodynamics of a viscous fluid. Sib. Math. J. 45, 197–213 (2004)

    Article  Google Scholar 

  5. Alekseev G.V., Kalinina E.A.: Identification of the lower coefficient for the stationary convection-diffusion-reaction equation. (Russian). Sib. Zh. Ind. Mat. 10, 3–16 (2007)

    MathSciNet  Google Scholar 

  6. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math 133, Teubner, Stuttgart. pp.9–126 (1993)

  7. Amann H.: Compact embeddings of vector-valued Sobolev and Besov spaces. Glasnik matematicki. 35(55), 161–177 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Amann, H.: Linear and quasilinear parabolic problems. 1, Birkhauser, Basel (1995)

  9. Belov, Ya.Ya.: Inverse problems for parabolic equations. VSP, Utrecht (2002)

  10. Babeshko O.M., Evdokimova O.V., Evdokimov S.M.: On taking into account the types of sources and settling zones of pollutants. Dokl. Math. 61, 283–285 (2000)

    Google Scholar 

  11. Capatina A., Stavre R.: A control problem in biconvective flow. J. Math. Kyoto Univ. 37, 585–595 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Denk R., Hieber M., Prüss J.: Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 93–224 (2007)

    Article  Google Scholar 

  13. Denk, R., Hieber, M., Prüss, J. R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type. Memoirs of the AMS. 166, no. 788 (2003)

    Google Scholar 

  14. Denk R., Krainer T.: R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators. Manuscripta Math. 124, 319–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Denk R., Saal J., Seiler J.: Inhomogeneous Symbols, the Newton Polygon, and Maximal Lp-Regularity. Russian J. of Math. Phys. 15, 171–191 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Denk R., Dore G., Hieber M., Prüss J., Venni A.: New thoughts on old results of R.T. Seeley. Math. Ann. 328, 545–583 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Efremenkova O.B.: On solvability of a parabolic inverse problem for determining an absorption coefficient of a special form. (Russian). Matem. Zametki YaGU. 13, 72–79 (2006)

    Google Scholar 

  18. Gol’dman N.L.: Properties of solutions of parabolic equations with unknown right-hand side and adjoint problems. Dokl. Math. 77, 350–355 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grisvard P.: Equations differentielles abstraites. Ann. Scient. Ec. Norm. Super. 4e-series. 2, 311–395 (1969)

    MathSciNet  MATH  Google Scholar 

  20. Grisvard P.: Equations operationnelles dans les de Banach et preblemes aux limites dans des ouverts cylindriques. Ann. Scuola Norm. Super. Pisa. 21, 308–347 (1967)

    MathSciNet  Google Scholar 

  21. Grisvard P.: Commutativit.e de deux foncteurs d’interpolation at applications. J. Math. Pures Appl. 45, 143–290 (1966)

    MathSciNet  MATH  Google Scholar 

  22. Hytonen T.P.: Anisotropic Fourier multipliers and singular integrals for vector-valued functions. Ann. Mat. Pura Appl. 186, 455–468 (2007)

    Article  MathSciNet  Google Scholar 

  23. Isakov, V.: Inverse Problems for Partial Differential Equations. Appl. Math. Sci. 127, Springer (2006)

  24. Ivanchov, M.: Inverse problems for equation of parabolic type. Math. Studies. Monograph Series. 10. WNTL Publishers, Lviv (2003)

  25. Kalinina E.A.: The numerical study of restoring the source density of the two-dimensional nonstationary convection-diffusion equation. (Russian). Dal’nevost. Matem. Zh. 5, 89–99 (2004)

    Google Scholar 

  26. Kalton N., Kunstmann P., Weis L.: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators. Math. Ann. 336, 747–801 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kriksin, Yu.A., Plyushchev, S.N., Samarskaya, E.A., Tishkin, V.F.: The inverse problem of source reconstruction for convection-diffusion equation. (Russian. English summary) Mat. Model. 7, no.11, 95–108 (1995)

    Google Scholar 

  28. Kozhanov, A.I.: Composite type equations and inverse problems. VSP, Utrecht (1999)

  29. Kunstman P.C., Weis L.: Maximal L p -regularity for Parabolic Equations, Fourier Multiplier Theorems and H ∞-functional Calculus. Lect. Notes in Math. 1855, 65–311 (2004)

    Google Scholar 

  30. Prilepko A.I., Orlovsky D.G., Vasin I.A.: Methods for solving inverse problems in Mathematical Physics. Marcel Dekker, Inc, New-York (1999)

    Google Scholar 

  31. Pyatkov S.G., Tsybikov B.N.: On evolutionary inverse problems for parabolic equations. Dokl. Math. 77, 111–113 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Triebel, H.: Interpolation Theory. Function Spaces. Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)

  33. Triebel, H. Theory of function spaces. Monographs in Mathematics. 100. Birkhauser Verlag, Basel (2006)

  34. Schmeiser H.-J., Sickel W.: Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser Verlag, Basel/Switzerland. 64, 463–472 (2005)

    Article  Google Scholar 

  35. Sobolevskii P.E.: Fractional powers of coercively positive sums of operators. Soviet. Math. Dokl. 16, 1638–1641 (1975)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ugra State University (Hanty-Mansiisk), Sobolev Institute of Mathematics (Novosibirsk), Chekhov st. 16, 628012, Hanty-Mansiisk, Russia

    S. G. Pyatkov

  2. Ugra State University (Hanty-Mansiisk), Chekhov st. 16, 628012, Hanty-Mansiisk, Russia

    B. N. Tsybikov

Authors
  1. S. G. Pyatkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. N. Tsybikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. G. Pyatkov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pyatkov, S.G., Tsybikov, B.N. On some classes of inverse problems for parabolic and elliptic equations. J. Evol. Equ. 11, 155–186 (2011). https://doi.org/10.1007/s00028-010-0087-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-010-0087-6

Mathematics Subject Classification (2000)

Keywords

Search

Navigation