Skip to main content
SpringerLink
Log in
Book cover

Semigroups, Boundary Value Problems and Markov Processes pp 591–604Cite as

Concluding Remarks

  • Chapter
  • First Online:

  • 1893 Accesses

Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

In this final chapter we summarize the contents of the first edition of the present monograph “Semigroups, boundary value problems and Markov processes” which was published in 2004. In Sect. 13.1 we study a class of degenerate boundary value problems for second-order elliptic differential operators which includes as particular cases the Dirichlet and Robin problems. We state existence and uniqueness theorems for this class of degenerate elliptic boundary value problems (Theorems 13.1 and 13.2). The crucial point is how to define modified boundary Besov and Hölder spaces in which our boundary value problems are uniquely solvable.

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   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   139.99
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

References

  1. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/New York (1976)

    Book  MATH  Google Scholar 

  2. Bony, J.-M., Courrège, P., Priouret, P.: Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier (Grenoble) 18, 369–521 (1968)

    Article  MATH  Google Scholar 

  3. Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  4. Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston Inc., New York/Montreal/London (1969)

    MATH  Google Scholar 

  5. Galakhov, E.I., Skubachevskiĭ, A.L.: On Feller semigroups generated by elliptic operators with integro-differential boundary conditions. J. Differ. Equ. 176, 315–355 (2001)

    Article  MATH  Google Scholar 

  6. Garroni, M.G., Menaldi, J.-L.: Green Functions for Second Order Parabolic Integro-Differential Problems. Pitman Research Notes in Mathematics Series, vol. 275. Longman Scientific & Technical, Harlow (1992)

    Google Scholar 

  7. Garroni, M.G., Menaldi, J.-L.: Second Order Elliptic Integro-Differential Problems. Chapman & Hall/CRC Research Notes in Mathematics, vol. 430. Chapman & Hall/CRC, Boca Raton (2002)

    Google Scholar 

  8. Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. 83, 129–209 (1966)

    Article  MATH  Google Scholar 

  9. Komatsu, T.: Markov processes associated with certain integro-differential operators. Osaka J. Math. 10, 271–303 (1973)

    MATH  MathSciNet  Google Scholar 

  10. Masuda, K.: Evolution Equations (in Japanese). Kinokuniya Shoten, Tokyo (1975)

    Google Scholar 

  11. Runst, T., Youssfi, A.: Boundary value problems for Waldenfels operators. Indiana Univ. Math. J. 54, 237–255 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Seeley, R.T.: Refinement of the functional calculus of Calderón and Zygmund. Nederl. Akad. Wetensch. Proc. Ser. A 68, 521–531 (1965)

    MATH  MathSciNet  Google Scholar 

  13. Seeley, R.T.: Singular integrals and boundary value problems. Am. J. Math. 88, 781–809 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  14. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  15. Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 209–244 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  16. Taibleson, M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-space I. Principal properties. J. Math. Mech. 13, 407–479 (1964)

    MathSciNet  Google Scholar 

  17. Taira, K.: On some degenerate oblique derivative problems. J. Fac. Sci. Univ. Tokyo Sect. IA 23, 259–287 (1976)

    MATH  MathSciNet  Google Scholar 

  18. Taira, K.: Diffusion Processes and Partial Differential Equations. Academic, Boston (1988)

    MATH  Google Scholar 

  19. Taira, K.: On the existence of Feller semigroups with boundary conditions. Mem. Am. Math. Soc. 99(475) (1992). American Mathematical Society, Providence, vii+65

    Google Scholar 

  20. Taira, K.: Analytic Semigroups and Semilinear Initial-Boundary Value Problems. London Mathematical Society Lecture Note Series, vol. 223. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  21. Taira, K.: Boundary value problems for elliptic integro-differential operators. Math. Z. 222, 305–327 (1996)

    Article  MathSciNet  Google Scholar 

  22. Taira, K.: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics, vol. 1499, 2nd edn. Springer, Berlin (2009)

    Google Scholar 

  23. Taira, K.: On the existence of Feller semigroups with discontinuous coefficients. Acta Math. Sin. (English Series), 22, 595–606 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Taira, K.: On the existence of Feller semigroups with discontinuous coefficients II. Acta Math. Sin. (English Series), 25, 715–740 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tanabe, H.: Equations of Evolution. Iwanami Shoten, Tokyo (1975, in Japanese); English translation: Monographs and Studies in Mathematics, vol. 6. Pitman, Boston/London (1979)

    Google Scholar 

  26. Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser Verlag, Basel (1983)

    Google Scholar 

  27. von Waldenfels, W.: Positive Halbgruppen auf einem n-dimensionalen Torus. Archiv der Math. 15, 191–203 (1964)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Kazuaki Taira

Authors
  1. Kazuaki Taira
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Taira, K. (2014). Concluding Remarks. In: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43696-7_13

Download citation

Publish with us

Policies and ethics

Search

Navigation