Archiv der Mathematik

, Volume 98, Issue 4, pp 341–353 | Cite as

Projections onto convex sets and Lp-quasi-contractivity of semigroups

Article

Abstract

Based on a simple geometric description of orthogonal projections onto closed, convex sets we find an implicit formula for the L2-projection onto the Lp-unit ball. This allows us to to prove the Lp-quasi-contractivity of semigroups generated by linear elliptic operators and quasi-linear operators of q-Laplace-type in a simple way.

Mathematics Subject Classification

Primary 35K15 Secondary 35B45 

Keywords

Semigroup Lp-contraction Orthogonal projection Maximal regularity Quasi-linear equations p(x)-Laplace 

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References

  1. 1.
    W. Arendt and A.F.M. ter Elst, Sectorial forms and degenerate differential operators, arXiv:0812.3944v2, 2010.Google Scholar
  2. 2.
    Arendt W., Warma M.: The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal 19, 341–363 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cipriani F. Grillo G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angew. Math. 562, 201–235 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    CialdeaA. Maz’ya V.: Criterion for the L p -dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. 9(84), 1067–1100 (2005)MathSciNetGoogle Scholar
  5. 5.
    Daners D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41 (2000)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Daners D.: Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352, 4207–4236 (2000)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston Inc., Boston, MA, 1993.Google Scholar
  8. 8.
    Harjulehto P. et al.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72, 4551–4574 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Karrmann S.: Gaussian estimates for second-order operators with unbounded coefficients. J. Math. Anal. Appl. 258, 320–348 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.Google Scholar
  11. 11.
    P. C. Kunstmann and L. Weis, Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math. 1855, Springer, Berlin, 2004, pp. 65–311.Google Scholar
  12. 12.
    Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–426 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ming Ma Z., Röckner M.: Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext, Springer-Verlag, Berlin (1992)Google Scholar
  14. 14.
    Manavi A., Vogt H., Voigt J.: Domination of semigroups associated with sectorial forms. J. Operator Theory 54, 9–25 (2005)MathSciNetMATHGoogle Scholar
  15. 15.
    R. Nittka, Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains, Ph.D. thesis, University of Ulm, http://vts.uni-ulm.de/doc.asp?id=7304, 2010.
  16. 16.
    R. Nittka, Quasilinear elliptic and parabolic Robin problems on Lipschitz domains, arXiv:1104.5125v1, 2011.Google Scholar
  17. 17.
    Maati Ouhabaz E.: L -contractivity of semigroups generated by sectorial forms. J. London Math. Soc. 2(46), 529–542 (1992)CrossRefGoogle Scholar
  18. 18.
    Maati Ouhabaz E.: Gaussian upper bounds for heat kernels of second-order elliptic operators with complex coefficients on arbitrary domains. J. Operator Theory 51, 335–360 (2004)MathSciNetMATHGoogle Scholar
  19. 19.
    Maati Ouhabaz E.: Analysis of heat equations on domains, London Mathematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ (2005)Google Scholar
  20. 20.
    Serrin J.: On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101, 139–167 (1961)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Showalter R.E.: Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI (1997)Google Scholar
  22. 22.
    Weis L.: Operator-valued Fourier multiplier theorems and maximal L p-regularity. Math. Ann. 319, 735–758 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    W. P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation.Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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