Skip to main content
SpringerLink
Log in

Interpolation and Extrapolation Spaces in Evolution Equations

  • Conference paper
Partial Differential Equations and Functional Analysis

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 22))

Summary

In the first part of the paper we show how some methods of functional analysis (i.e. semigroup theory and abstract interpolation) have been used to obtain known and new results in the field of parabolic differential equations of many different types.

In the second part we indicate a new method based on abstract extrapolation theory to study non homogeneous evolution equations and its applications to hyperbolic partial differential equations: in particular to linear (and nonlinear) Volterra integrodifferential equations. These results will be applied to a wave equation with memory effects.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. P. Acquistapace, Abstract linear non autonomous parabolic equations: a survey, Dekker Lecture Notes in Pure and Appl. Math. 148 (1993), 1–19.

    MathSciNet  Google Scholar 

  2. P. Acquistapace, B. Terreni, H00F6lder classes with boundary conditions as interpolation spaces, Math. Zeit. 195 (1987) 451–471.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.B. Baillon, Charactère borné de certains générateurs de semigroupes linéaires dans les espaces de Banach, C.R. Acad. Sci. Paris 290 (1980), 757–760.

    MathSciNet  MATH  Google Scholar 

  4. H. Berens, P.L. Butzer, Approximation theorems for semigroup operators in intermediate spaces, Bull. Amer. Math. Soc. 70 (1964), 689–692.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, 1972.

    Google Scholar 

  6. P.L. Butzer, H. Berens, Semigroups of Operators and Approximation, Springer, 1967.

    Google Scholar 

  7. P. Clement, H.J.A.M. Heijmans, S. Angenent, C.J. Van Duijn, and B. De Pagter, One-Parameter Semigroups, North-Holland, 1987.

    MATH  Google Scholar 

  8. D. Daners, P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Longman, 1992.

    MATH  Google Scholar 

  9. G. Da Prato, P. Grisvard, Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures et Appl. 54 (1975), 305–387.

    MATH  Google Scholar 

  10. G. Da Prato, P. Grisvard, Equations d’évolution abstraites non linéaires de type parabolique, Annali Mat. Pura e Appl. 120 (1979), 329–396.

    Article  MATH  Google Scholar 

  11. G. Da Prato, P. Grisvard, On extrapolation spaces, Rend. Accad. Naz. Lincei 72 (1982), 330–332.

    MATH  Google Scholar 

  12. G. Da Prato, P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal. 58 (1984), 107–124.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Da Prato, E. Sinestrari, Hölder regularity for non autonomous abstract parabolic equations, Israel J. Math. 42 (1982) 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Da Prato, E. Sinestrari, Differential operators with non dense domain, Annali Sc. Norm. Sup. Pisa 14 (1987), 285–344.

    MATH  Google Scholar 

  15. L. De Simon, Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova 34 (1964), 205–232.

    MATH  Google Scholar 

  16. W. Desch, W. Schappacher, On relatively bounded perturbations of linear c 0-semigroups, Annali Sc. Norm. Sup. Pisa 11, (1984) 327–341.

    MathSciNet  MATH  Google Scholar 

  17. G. Di Blasio, Linear parabolic evolution equations in L p-spaces. Annali Mat. Pura Appl. 138 (1984), 55–104.

    Article  MATH  Google Scholar 

  18. G. Dore, A. Venni, On the closedness of the sum of two closed operators, Math. Zeit. 196 (1987), 189–201.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Grisvard, Commutativité de deux foncteurs d’interpolation et applications, J. Math. Pures et Appl. 45 (1966), 143–206.

    MathSciNet  MATH  Google Scholar 

  20. P. Grisvard, Equations différentielles abstraites, Ann. Scient. Ec. Norm. Sup. 2 (1969), 311–395.

    MathSciNet  MATH  Google Scholar 

  21. T. Kato, Perturbations Theory for Linear Operators,. Springer, 1966.

    Google Scholar 

  22. O. Ladyzhenskaya,V. Solonnikov, N. Uralceva, Linear and quasilinear equations of parabolic type, Am. Math. Soc. 1968.

    Google Scholar 

  23. J. L. Lions, Théorèmes de trace et d’interpolation, Ann. Sc. Norm. Sup. Pisa Z 13 (1959), 389–403.

    Google Scholar 

  24. J. L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968.

    MATH  Google Scholar 

  25. J. L. Lions, J. Peetre, Sur une classe d’espaces d’interpolation, IHES, 1964.

    Google Scholar 

  26. W. Littman, The wave operator and L p norms, J. Math. Mech. 12 (1963), 55–68.

    MathSciNet  MATH  Google Scholar 

  27. A. Lunardi, Characterization of interpolation spaces between domains of elliptic operators and spaces of continuous functions, Math. Nachr. 121 (1985), 295–318.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Lunardi, Maximal space regularity in non homogeneous initial boundary value parabolic problem, Numer. Fund. Anal. Optim. 10 (1989), 323–339.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, preprint 1994.

    Google Scholar 

  30. A. Lunardi, E. Sinestrari, W. Von Wahl, A semigroup approach to the time-dependent parabolic initial boundary value problem, Diff. Int. Eq. 5 (1992), 1275–1306.

    MATH  Google Scholar 

  31. R. Nagel, Sobolev spaces and semigroups, Semesterbericht Funktion-alanalysis Tübingen, (Sommersemester 1983) 1–19.

    Google Scholar 

  32. R. Nagel, E. Sinestrari, Inhomogeneous volterra integrodifferential equations for Hille-Yosida operators, Dekker Lecture Notes Pure Appl. Math. no. 150 (1994), 51–70.

    MathSciNet  Google Scholar 

  33. J. Neerven, The adjoint of a semigroup of linear operators, Lecture Notes in Mathematics no. 1529, Springer, (1992).

    MATH  Google Scholar 

  34. E. Sinestrari, Continuous interpolation spaces and spatial regularity in non linear Volterra integrodifferential equations, J. Int. Eq. 5 (1983), 287–308.

    MathSciNet  MATH  Google Scholar 

  35. E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16–66.

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Sinestrari, Optimal Regularity for Parabolic Equations and its Applications, in: Semigroups, theory and applications. Pitman Research Notes in Mathematics, H. Brezis, M. G. Crandall, F. Kappel, eds: 152 (1986), 217–233.

    Google Scholar 

  37. E. Sinestrari, P. Vernole, Semilinear evolution equations in interpolation spaces, Nonlin. Anal. 1 (1977), 249–261.

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Sinestrari, W. Von Wahl, On the solution of the first boundary value problem for linear parabolic equations, Proc. Roy. Soc. Edinburgh 108A (1988), 339–355.

    Google Scholar 

  39. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 299–310.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, P. le Aldo Moro 2, 00185, Roma, Italy

    E. Sinestrari

Authors
  1. E. Sinestrari
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques, University of Nice, F-06108, Nice Cedex, France

    Jean Cea  & Denise Chenais  & 

  2. Laboratoire de Mathématiques, Ecole Normale Supérieure, 94235, Cachan Cedex, France

    Giuseppe Geymonat

  3. Collège de France, 3, rue d’Ulm, 75005, Paris, France

    Jacques Louis Lions

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Birkhäuser Boston

About this paper

Cite this paper

Sinestrari, E. (1996). Interpolation and Extrapolation Spaces in Evolution Equations. In: Cea, J., Chenais, D., Geymonat, G., Lions, J.L. (eds) Partial Differential Equations and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 22. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2436-5_16

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2436-5_16

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7536-7

  • Online ISBN: 978-1-4612-2436-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation