Advertisement

Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds

  • Tim BinzEmail author
Article
  • 2 Downloads

Abstract

We study strictly elliptic differential operators with Dirichlet boundary conditions on the space \(\mathrm {C}(\overline{M})\) of continuous functions on a compact Riemannian manifold \(\overline{M}\) with boundary and prove sectoriality with optimal angle \(\frac{\pi }{2}\).

Keywords

Dirichlet boundary conditions Analytic semigroup Riemmanian manifolds 

Mathematics Subject Classification

47D06 34G10 47E05 47F05 

Notes

Acknowledgements

The author wishes to thank Professor Simon Brendle and Professor Klaus Engel for important suggestions and fruitful discussions. Moreover the author thanks the referee for his many helpful comments.

References

  1. 1.
    W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96. Birkhäuser (2001).CrossRefGoogle Scholar
  2. 2.
    R. A. Adams. Sobolev Spaces, Academic Press, New York-London (1975).zbMATHGoogle Scholar
  3. 3.
    S. Agmon. On th eigenfunctions and the eigenvalues of general boundary value problems. Comm. Pure Appl. Math. 25 (1962).Google Scholar
  4. 4.
    H. Amann. Linear and Quasilinear Parabolic Problems, vol. 1. Birkhäuser (2001).Google Scholar
  5. 5.
    W. Arendt. Resolvent positive operators and inhomogeneous boundary value problems. Ann. Scuola Norm. Sup. Pisa 24.70 (2000), 639–670.zbMATHGoogle Scholar
  6. 6.
    T. Binz and K. Engel Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann Operator. Math. Nachr. (to appear 2018).Google Scholar
  7. 7.
    T. Binz Strictly elliptic operators with Wentzell boundary conditions on spaces of continuous functions on manifolds. (preprint 2018).Google Scholar
  8. 8.
    F. Browder. On the spectral theory of elliptic differential operators I. Math. Ann. 142.1 (1961), 22–130.MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Campiti and G. Metafune. Ventcel’s boundary conditions and analytic semigroups. Arch. Math. 70 (1998), 377–390.MathSciNetCrossRefGoogle Scholar
  10. 10.
    K.-J. Engel and G. Fragnelli. Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions. Adv. Differential Equations 10 (2005), 1301–1320.MathSciNetzbMATHGoogle Scholar
  11. 11.
    K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194. Springer (2000).Google Scholar
  12. 12.
    K.-J. Engel. The Laplacian on \(C(\overline{\Omega })\) with generalized Wentzell boundary conditions. Arch. Math. 81 (2003), 548–558.MathSciNetCrossRefGoogle Scholar
  13. 13.
    L. C. Evans. Partial Differential Equations, Graduate Studies in Mathematics., vol. 19. Amer. Math. Soc. (1998).Google Scholar
  14. 14.
    A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht, and S. Romanelli. Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem. Math. Nachr. 283 (2010), 504–521.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2 (2002), 1–19.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order, Classics in Mathematics. Springer (2001).Google Scholar
  17. 17.
    E. Hebey. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes. Amer. Math. Soc. (2000).Google Scholar
  18. 18.
    A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser (1995).Google Scholar
  19. 19.
    M. Rudin. Real and Complex Analysis, Higher Mathematics Series, vol 3. McGraw-Hill (1986).Google Scholar
  20. 20.
    M. Renardy and R. C. Rogers. An Introduction to Partial Differential Equations, Texts in Appl. Math., vol 13. Springer (1993).Google Scholar
  21. 21.
    R. T. Seeley. Extenstion of \({\rm C}^\infty \) functions defined in a half space. Proc. Amer. Math. Soc. 15 (1964), 625–626.MathSciNetzbMATHGoogle Scholar
  22. 22.
    B. Stewart. Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974), 141–161.MathSciNetCrossRefGoogle Scholar
  23. 23.
    G. N. Watson A Treatise on the Theory of Bessel Functions, Cambridge University Press (1995).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TübingenTübingenGermany

Personalised recommendations