Skip to main content
SpringerLink
Log in

A Note on Unique Continuation for Parabolic Operators with Singular Potentials

  • Chapter
  • First Online:
Studies in Phase Space Analysis with Applications to PDEs

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 84))

  • 1628 Accesses

Abstract

We consider a class of heat-type differential operators. The coefficients in the lower-order terms are allowed to have critical singularities. These operators can be viewed as a perturbation of a simple model operator by a subcritical one. Under some conditions on the vector and scalar potentials in the critical part, we establish strong unique continuation theorems for such operators. For proof, we use a two-stage Carleman method. Firstly we derive a Carleman inequality for the model operators with critical potentials through an analysis of spectrum of some Schrödinger operators with compact resolvent. The obtained Carleman inequality at the first stage guarantees us to choose a weight function with higher singularity in a Carleman inequality at the second stage for the perturbed operators.

2010 Mathematics Subject Classification: 35B60, 35K10, 35R45.

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

Access this chapter

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
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

Institutional subscriptions

References

  1. Alessandrini, G., Vessella, S.: Remark on the strong unique continuation property for parabolic operators. Proc. AMS 132, 499–501 (2003)

    Article  MathSciNet  Google Scholar 

  2. Alinhac, S., Baouendi, M.S.: A counterexample to strong uniqueness for partial differential equations of Schrödinger’s type. Comm. P.D.E. 19, 1727–1733 (1994)

    Google Scholar 

  3. Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pure. Appl. 36, 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  4. Carleman, T.: Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26,1–9 (1939)

    MathSciNet  MATH  Google Scholar 

  5. Chen, X.Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cordes, H.: Über die Besstimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Göttingen II a, 230–258 (1956)

    Google Scholar 

  7. Escauriaza, L.: Carleman inequalities and the heat operator. Duke Math. J. 104, 113–127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator II. Indiana U. Math. J. 50, 1149–1169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grammatico, C.: A result on strong unique continuation for the Laplace operator. Comm. P. D. E. 22, 1475–1491 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jerison, D., Kenig, C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. 121, 463–494 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kato, T.: Perturbation Theory for Linear Operators. Springer (1976)

    Google Scholar 

  12. Koch, H., Tataru, D.: Carleman estimates and unique continuation for second order parabolic equations with non-smooth coefficients. Comm. P. D. E. 34, 305–366 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Landis, E.M., Oleinik, O.A.: Generalized analyticity and some related properties of solutions of elliptic and parabolic equations. Uspekhi Mat. Nauk 29(176):2, 190–206 (1974) (Russian). English transl.: Russian Math. Surveys 29, 195–212 (1974)

    Article  MathSciNet  Google Scholar 

  14. Lees, M., Protter, M.H.: Unique continuation for parabolic differential equations and inequalities. Duke Math. J. 28, 369–382 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, F.H.: A uniqueness theorem for parabolic equations. CPAM 43, 127–136 (1990)

    MATH  Google Scholar 

  16. Mizohata, S.: Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques. Mem. Coll. Sci. Univ. Kyoto Ser. A, 31(3), 219–239 (1958)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Okaji, T.: Strong unique continuation property for time harmonic Maxwell equations. J. Math. Soc. Japan. 54-1, 87–120 (2002)

    MathSciNet  Google Scholar 

  19. Poon, C.C.: Unique continuation for parabolic operators. Comm. P. D. E. 21, 521–539 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Regbaoui, R.: Prolongement unique pour les opérateurs de Schrödinger. Thesis, University of Rennes (1993)

    Google Scholar 

  21. Regbaoui, R.: Strong uniqueness for second order differential operators. J. Differ. Equat. 141, 201–217 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 2. Academic, New York (1975)

    MATH  Google Scholar 

  23. Schechter, M.: Spectra of Partial Differential Operators, 2nd edn. Elsevier, Amsterdam (1986)

    MATH  Google Scholar 

  24. Sogge, C.D.: Strong uniqueness theorems for second order elliptic differential equations. Am. J. Math. 112, 943–984 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sogge, C.D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28, 159–182 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stummel, F.: Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann. 132, 150–176 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  27. Weidmann, J.: Linear Operators in Hilbert Spaces (English translation). Springer, Berlin (1980)

    Book  Google Scholar 

  28. Wolff, T.H.: Recent work on sharp estimates in second order elliptic unique continuation problems. In: Fourier Analysis and partial Differential Equations, pp. 99–128. Studies in Advanced Mathematics. CRC, Boca Raton (1995)

    Google Scholar 

  29. Yamabe, H.: A unique continuation theorem of a diffusion equation. Ann. Math. (2), 69, 462–466 (1959)

    Google Scholar 

Download references

Acknowledgements

This research was partially supported by Grants-in-Aid for Scientific Research (No. 22540185), Japan Society for the Promotion of Science.

Author information

Authors and Affiliations

  1. Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Takashi Okaji

Authors
  1. Takashi Okaji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takashi Okaji .

Editor information

Editors and Affiliations

  1. , Dipartimento di Matematica, UniversitĂ  di Bologna, Piazza di Porta San Donato, 5, Bologna, I-40126, Italy

    Massimo Cicognani

  2. , Dipartimento di Matematica, UniversitĂ  di Pisa, Largo Bruno Pontecorvo, 5, Pisa, I-56127, Italy

    Ferruccio Colombini

  3. , Dipartimento di Matematica e Geoscienze, UniversitĂ  di Trieste, Via Valerio, 12/1 12/1, Trieste, I-34127, Italy

    Daniele Del Santo

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Okaji, T. (2013). A Note on Unique Continuation for Parabolic Operators with Singular Potentials. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_13

Download citation

Publish with us

Policies and ethics

Search

Navigation