Skip to main content
SpringerLink
Log in

Lower Bounds at Infinity for Solutions of Differential Equations with Constant Coefficients in Unbounded Domains

  • Conference paper
Singularities in Boundary Value Problems

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 65))

  • 199 Accesses

Abstract

Let Q be an unbounded domain in Rn and P be a differential operator. In this note, for the equation:

$$Pu\; = \,\,0\;in\;\Omega ,$$

we consider the following problems in the case that P is an operator with constant coefficients.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agmon, S., Lower hounds for solutions of Schrödinger-type equations in unbounded domains, Proc. Int. Conf. Functional Analysis and Related Topics, Tokyo, (1969), 216–224.

    Google Scholar 

  2. Agmon, S., Lover bounds for solutions of Schrödinger equations, J. d’Anal. Math., 23 (1970), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  3. Goodman, R. W., One sided invariant subspaces and domains of uniqueness for hyperbolic equations, Proc A.M.S., 15 (1964), 653–660.

    Article  MATH  Google Scholar 

  4. k. Hormander, L., Lover bounds at infinity for solutions of differential equations with constant coefficients, Israel J. Math., 16 (1973), 103–116.

    Article  MathSciNet  Google Scholar 

  5. Ikebe, T. and Uchiyama, J., On the asymptotic behavior of eigenfunctions of second order elliptic operators, J. Math Kyoto Univ., 11 (1971), 425–48.

    MathSciNet  MATH  Google Scholar 

  6. Kato, T., Grovth properties of solutions of the reduced wave equation with a variable coefficients, Comm. Pure Appl. Math., 12 (1959), 403–425.

    Article  MathSciNet  MATH  Google Scholar 

  7. Konno, R., Non-existence of positive eigenvalues of Schrödinger operators in infinite domains, J. Fac. Sci Univ. Tokyo Sec. IA, 19 (1972), 393–402.

    MathSciNet  MATH  Google Scholar 

  8. Landis, E.M., Some problems of the qualitative theory of second order elliptic equations (case of several independent variables), Russian Math. Surveys, 18: 1 (1963), 1–62.

    Article  MathSciNet  MATH  Google Scholar 

  9. Landis, E.M., On the behavior of solutions of higher order elliptic equations in unbounded domains, Trans. Moscov Math. Soc., 31 (1974), 30–54.

    MathSciNet  Google Scholar 

  10. Littman, W., Decay at infinity of solutions to partial differential equations with constant coefficients, Trans. Amer. Math. Soc., 123 (1966), 449–459.

    Article  MathSciNet  MATH  Google Scholar 

  11. Littman, W., Maximal rates of decay of solutions of partial differential equations, Arch. Rat. Mech. Anal., 37 (1970), 11–20.

    Article  MathSciNet  MATH  Google Scholar 

  12. Littman, W., Decay at infinity of solutions to partial differential equations; removal of the curvature assumption, Israel J., Math., 8 (1970), 403–407.

    Article  MathSciNet  MATH  Google Scholar 

  13. Littman, W., Décroissance a l’infini des solutions, a I’exterieur d’un cône, d’équations aux derivees partielles a coefficients constants, C.R.Acad. Se. Paris, t., 287 (3 juillet 1978) Serie A 15–17.

    Google Scholar 

  14. Mochizuki, K. and Uchiyama, J., On eigenvalues in the continuum of 2-body or many body Schrodinger operators, Nagoya Math. J., 70 (1974), 125–141.

    MathSciNet  Google Scholar 

  15. Murata, M., A theorem of Liouville type for partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sec. IA, 21 (1974), 395–404.

    MathSciNet  MATH  Google Scholar 

  16. Murata, M., Asymptotic behaviors at infinity of solutions of certain linear partial differential equations, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 107–148.

    MathSciNet  Google Scholar 

  17. Murata, M. and Shibata, Y., Lower bounds at infinity of soltions of partial differential equations in the exterior of a proper cone, Israel J. Math., 31 (1978), 193–203.

    Article  MathSciNet  MATH  Google Scholar 

  18. Oleinik, O.A. and Radkevič, E.V., Analiticity and theorems of Liouville and Phragmén-Lindelof type for general elliptic systems of differential equations, Math, USSR Sbornik, 24: l (1974), 127–143.

    Google Scholar 

  19. Pavlov, A.L., On general boundary value problems for differential equations with constant coefficients in a half-space, Math. USSR. Sbornik, 32: 3, (1977), 313–334.

    Article  MATH  Google Scholar 

  20. Rellich, F., Uber das asymptotische Verhalten der Lôsungen von ∆u + λu = 0 in unendlichen Gebieten, Jber. Deutsch. Math. Verein., 53 (1943), 57–65.

    MathSciNet  MATH  Google Scholar 

  21. Shibata, Y., Liouville type theorem for a system {P(D), Bj(D), j = l,.,.,p} of differential operators with constant coefficients in a half-space, Publ. RIMS Kyoto, Univ., l6 (1980), 61–104.

    Article  Google Scholar 

  22. Shibata, Y., Lower bounds of solutions of general boundary value problems for differential equations with constant coefficients in a half space, to appear.

    Google Scholar 

  23. Tayoshi, T., The asymptotic behavior of the solutions of (∆+λ)u = 0 in a domain with unbounded boundary, Publ.RIMS Kyoto Univ., 8 (1972), 375–391.

    Article  MathSciNet  MATH  Google Scholar 

  24. Trèves, F., Differential polynomials and decay at infinity, Bull. Amer. Math. Soc., 66 (1960), 183–186.

    Article  Google Scholar 

  25. Agmon, S. and Hormander, L., Asymptotic properties of solutions of differential equations with simple characteristics, J. d’Anal. Math., 30 (1976), 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  26. Grusin, V.V., On Sommerfeld-type conditions for a certain class of partial differential equations, Mat. Sbornik, 6l:2 (1963), 147–174; A.M.S. Transl. Ser. 2, 51 (1966), 82–112.

    Google Scholar 

  27. Vainberg, B.R., Principle of radiation, limit absorption and limit amplitude in the general theory of partial differential equations, Russian Math. Surveys, 21: 3 (1966), 115–193.

    Article  MathSciNet  Google Scholar 

  28. Hörmander, L., Linear partial differential operators, Springer verlag, Berlin-Göttingen-Heidelberg, 1963.

    MATH  Google Scholar 

  29. Hormander, L., Fourier integral operators I, Acta. Math., 127 (1971), 79–183.

    Article  MathSciNet  Google Scholar 

  30. Lax, P.D. and Phillips, R.S., Scattering theory, Pure and Applied Math., 26 Academic Press, New York (1967).

    Google Scholar 

  31. Rudin, W., Lectures on the edge-of-wedge theorem, CBMS Regional Conf., 6 (1970).

    Google Scholar 

  32. Trevès, F., Lectures on partial differential equations with constant coefficients, Notas de Math., 27, Instituto de Matemàtica Pura e Aplicada, Rio de Janeiro, (1961).

    Google Scholar 

  33. Vladimirov, V.S., Methods of the theory of functions of several complex variables, “nauka”, Moscow, ( 1964 ); English transl., M.I.T. Press, Cambridge, Mass., (1966).

    Google Scholar 

  34. Whitney, H., Elementary structure of real algebraic variety, Ann. Math., 66 (1957), 545–556.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tsukuba, Ibaraki, Japan

    Y. Shibata

Authors
  1. Y. Shibata
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Inst. de Math., University of Liège, 15, avenue des Tilleuls, B-4000, Liege, Belgium

    H. G. Garnir

Rights and permissions

Reprints and permissions

Copyright information

© 1981 D. Reidel Publishing Company

About this paper

Cite this paper

Shibata, Y. (1981). Lower Bounds at Infinity for Solutions of Differential Equations with Constant Coefficients in Unbounded Domains. In: Garnir, H.G. (eds) Singularities in Boundary Value Problems. NATO Advanced Study Institutes Series, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8434-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-8434-9_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-8436-3

  • Online ISBN: 978-94-009-8434-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation