Skip to main content
SpringerLink
Log in
Book cover

Analysis and Applications — ISAAC 2001 pp 199–236Cite as

Carleman Estimates for a Plate Equation on a Riemann Manifold with Energy Level Terms

  • Chapter

Part of the book series: International Society for Analysis, Applications and Computation ((ISAA,volume 10))

Abstract

We provide Carleman estimates for an Euler-Bernoulli type plate equation with energy level terms, defined on an open bounded set Ω of a finite-dimensional Riemann manifold (M, g). The energy level for this problem is H 3(Ω) × H 1(Ω). The basic assumption made is the existence of a strictly convex function on Ω. Carleman estimates are also a critical springboard from which one may derive the a-priori inequalities of continuous observability/uniform stabilization of interest in control theory of PDEs.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Eller, M.:A remark on a theorem by Hörmander. Proceedings of 3rd International ISAAC Congress, Berlin, August 2001. World Scientific, Singapore, to appear.

    Google Scholar 

  2. Eller, M., Lasiecka, I., Triggiani, R.: Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficients. Lecture Notes in Pure and Applied Mathematics 216, Marcel Dekker, New York, 2001, 109–230.

    MATH  Google Scholar 

  3. Eller, M., Lasiecka, I., Triggiani, R.: Unique continuation for over-determined Kirchhoff plate equations and related thermo-elastic systems. J. Inv. and Ill-Posed Problems 9 (2) (2001), 103–148.

    MathSciNet  MATH  Google Scholar 

  4. Gulliver, R., Littman, W.: The use of geometric tools in the boundary control of partial differential equations. Proceedings of 3rd International ISAAC Congress, Berlin, August 2001. World Scientific, Singapore, to appear.

    Google Scholar 

  5. Greene, R.-E., Wu, H.: C∞ convex functions and manifolds of positive curvature. Acta Math. 137 (1976), 209–245.

    Article  MathSciNet  Google Scholar 

  6. Hebey, E.: Sobolev spaces on Riemannian manifolds. Lecture Notes in Mathematics, Springer, Berlin, 1996.

    Google Scholar 

  7. Hörmander, L.: On the characteristic Cauchy problem. Ann. Math. (2) 88 (1968), 341–370.

    Article  MATH  Google Scholar 

  8. Hörmander, L.: On the uniqueness of the Cauchy problem, II, Math. Scand. 7 (1959), 177–190.

    MathSciNet  MATH  Google Scholar 

  9. Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin etc. 1963.

    Book  MATH  Google Scholar 

  10. Isakov, V.: On uniqueness on a lateral Cauchy problem with multiple characteristics. J. Diff. Eqns. 134 (1997), 134–147.

    Article  MathSciNet  MATH  Google Scholar 

  11. Isakov, V.: Inverse problems for partial differential equations. Springer, Berlin etc. 1998.

    MATH  Google Scholar 

  12. Lasiecka, I., Triggiani, R.: Exact controllability of the Euler-Bernoulli equation with Lz(E)-control only in the Dirichlet B.C. Atti Accad. Naz. Lincei Rend. Cl. Sc. Fis. Mat. Nat. Vol. LXXXII, No. 1 (1988), Rome.

    Google Scholar 

  13. Lasiecka, I., Triggiani, R.: Exact controllability of the Euler-Bernoulli equations with controls in the Dirichlet and Neumann B.C.: A non-conservative case. SIAM J. Control Optim. 27 (1989), 330–373.

    Article  MathSciNet  MATH  Google Scholar 

  14. Lasiecka, I., Triggiani, R.: Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment. JMAA 146 (1990), 1–33.

    MathSciNet  MATH  Google Scholar 

  15. Lasiecka, I., Triggiani, R.: Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometric conditions. Appl. Math. Optim. 25 (1992), 189–224.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lasiecka, I., Triggiani, R.: A-priori observability inequalities. Chapter 1, Lecture Notes, University of Virginia, May 1995.

    Google Scholar 

  17. Lasiecka, I., Triggiani, R.: Carleman estimates and exact boundary controllability for a system of coupled, non-conservative second-order hyperbolic equations. Lecture Notes in Pure and Applied Mathematics 188, eds. J.P. Zolesio and G. Da Prato. Marcel Dekker, New York, 1997, 125–243.

    Google Scholar 

  18. Lasiecka, I., Triggiani, R., Yao, P.F.: Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms. Nonlinear Analysis: Theory, Methods, Applications 30 (1) (1997), 111–122.

    MathSciNet  MATH  Google Scholar 

  19. Lasiecka, I., Triggiani, R., Yao, P.F.: Inverse/observability estimates for second order hyperbolic equations with variable coefficients. JMAA 235 (1999), 13–57.

    MathSciNet  MATH  Google Scholar 

  20. Lions, J.L.: Controllabilite exacte, stabilization et perturbation des systemes distribues. 1, Masson, Paris, 1988.

    Google Scholar 

  21. Nirenberg, L.: Uniqueness in Cauchy problems for differential equations with constant leading coefficients. Comm. Pure Appl. Math. 10 (1957), 89–105.

    MathSciNet  MATH  Google Scholar 

  22. Shirota, T.: A remark on the unique continuation theorem for certain fourth order elliptic equations. Proc. Japan Acad. 36 (1960), 571–573.

    Article  MathSciNet  MATH  Google Scholar 

  23. Tataru, D.: A-priori pseudoconvexity energy estimates in domains with boundary and applications to exact boundary controllability for conservative PDEs. Ph.D. Thesis, University of Virginia, May 1992.

    Google Scholar 

  24. Tataru, D.: Boundary controllability for conservative PDEs. Appl. Math. and Optimiz. 31 (1995), 257–295.

    Article  MathSciNet  MATH  Google Scholar 

  25. Tataru, D.: A-priori estimates of Carleman’s type in domains with boundaries. J. Math. Pures et Appl. 73 (1994), 355–387.

    MathSciNet  MATH  Google Scholar 

  26. Taylor, M.: Partial Differential Equations. 1, 2, Springer, Berlin etc. 1991.

    Google Scholar 

  27. Triggiani, R.: Carleman estimates and exact boundary controllability for a system of coupled non-conservative Schrödinger equations. Rend. Istit. Math. Univ. Trieste (Italy), Suppl. Vol. XXVIII (1997), 453–504; volume in memory of P. Grisvard.

    Google Scholar 

  28. Triggiani, R., Yao, P.F.: Inverse/observability estimates for Schrödinger equations with variable coefficients. Special volume on control of PDEs. Control and Cybernetics 28 (1999), 627–664.

    MathSciNet  MATH  Google Scholar 

  29. Wu, H., Shen, C.L., Yu, Y.L.: An introduction to Riemann geometry. University of Beijing, 1989 (in Chinese).

    Google Scholar 

  30. Yao, P.F.: On the observability inequalities for the exact controllability of wave equations with variable coefficients, SIAM Control and Optimiz. 37 (5) (1999), 1568–1599.

    Article  MATH  Google Scholar 

  31. Yao, P.F.: Observability inequalities for the Euler-Bernoulli plate with variable coefficients. Contemporary Mathematics, Amer. Math. Soc. 268 (2000), 383–406.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    I. Lasiecka & R. Triggiani

  2. Institute of Systems Science Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, PRC

    P. F. Yao

Authors
  1. I. Lasiecka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Triggiani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. F. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Heinrich G. W. Begehr

  2. University of Delaware, Newark, DE, USA

    Robert P. Gilbert

  3. York University, Toronto, Canada

    Man Wah Wong

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Lasiecka, I., Triggiani, R., Yao, P.F. (2003). Carleman Estimates for a Plate Equation on a Riemann Manifold with Energy Level Terms. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_15

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3741-7_15

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5247-9

  • Online ISBN: 978-1-4757-3741-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation