Progress in the Problem of the Lp-Contractivity of Semigroups for Partial Differential Operators

  • Alberto Cialdea
Part of the International Mathematical Series book series (IMAT, volume 13)


This is a survey of results concerning the L p -dissipativity of partial differential operators and the L p -contractivity of the generated semigroups, mostly obtained by V. Maz’ya, G. Kresin, M. Langer, and A. Cialdea. Necessary and sufficient conditions are discussed for scalar second order operators with complex coefficients and some systems, including the two-dimensional elasticity. The case of higher order operators is reviewed as well.


Cauchy Problem Dirichlet Problem Heat Kernel Elliptic Operator Erential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Amann, H.: Dual semigroups and second order elliptic boundary value problems. Israel J. Math. 45, 225–254 (1983)MATHMathSciNetGoogle Scholar
  2. 2.
    Auscher, P., Barthélemy, L., Bénilan, P., Ouhabaz, El M.: Absence de la L -contractivité pour les semi-groupes associés auz opérateurs elliptiques complexes sous forme divergence. Poten. Anal. 12, 169–189 (2000)MATHCrossRefGoogle Scholar
  3. 3.
    Brezis, H., Strauss, W.A.: Semi-linear second order elliptic equations in L 1. J. Math. Soc. Japan 25, 565–590 (1973)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cialdea, A., Maz'ya, V.: Criterion for the L p-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. 84, 1067–1100 (2005)MATHMathSciNetGoogle Scholar
  5. 5.
    Cialdea, A., Maz'ya, V.: Criteria for the L p-dissipativity of systems of second order differential equations. Ric. Mat. 55, 233–265 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Daners, D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Davies, E.B.: One-Parameter Semigroups. Academic Press, London etc. (1980)MATHGoogle Scholar
  8. 8.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)MATHGoogle Scholar
  9. 9.
    Davies, E.B.: L p spectral independence and L 1 analyticity. J. London Math. Soc. (2) 52, 177–184 (1995)MATHMathSciNetGoogle Scholar
  10. 10.
    Davies, E.B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132, 141–169 (1995)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fattorini, H.O.: The Cauchy Problem. In: Encyclopedia Math. Appl. 18, Addison-Wesley, Reading, Mass. (1983)Google Scholar
  12. 12.
    Fattorini, H.O.: On the angle of dissipativity of ordinary and partial differential operators. Functional Analysis, Holomorphy and Approximation Theory II. Math. Stud. 86, 85–111 (1984)MathSciNetGoogle Scholar
  13. 13.
    Karrmann, S.: Gaussian estimates for second order operators with unbounded coefficients. J. Math. Anal. Appl. 258, 320–348 (2001)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kovalenko, V., Semenov, Y.: C 0-semigroups in L p(Rd) and Ĉ(Rd) spaces generated by the differential expression d+b·N. Theor. Probab. Appl. 35, 443–453 (1990)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kresin, G.: Sharp constants and maximum principles for elliptic and parabolic systems with continuous boundary data. In: The Maz'ya Anniversary Collection 1m pp. 249–306. Birkhäuser, Basel (1999)Google Scholar
  16. 16.
    Kresin, G., Maz'ya, V.: Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems. Ark. Mat. 32, 121–155 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Langer, M.: L p-contractivity of semigroups generated by parabolic matrix differential operators. In: The Maz'ya Anniversary Collection, 1, pp. 307–330. Birkhäuser, Basel (1999)Google Scholar
  18. 18.
    Langer, M., Maz'ya, V.: On L p-contractivity of semigroups generated by linear partial differential operators. J. Funct. Anal. 164, 73–109 (1999)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Liskevich, V.: On C 0-semigroups generated by elliptic second order differential expressions on L p-spaces. Differ. Integral Equ. 9, 811–826 (1996)MATHMathSciNetGoogle Scholar
  20. 20.
    Liskevich, V.A., Semenov, Yu.A.: Some problems on Markov semigroups, In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras. Math. Top. 11, pp. 163–217. Akademie-Verlag, Berlin (1996)Google Scholar
  21. 21.
    Liskevich, V., Sobol, Z., Vogt, H.: On the L p-theory of C 0 semigroups associated with second order elliptic operators. II. J. Funct. Anal. 193, 55–76 (2002)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lumer, G., Phillips, R.S.: Dissipative operators in a Banach space. Pacific J. Math. 11, 679–698 (1961)MATHMathSciNetGoogle Scholar
  23. 23.
    Maz'ya, V.G.: The negative spectrum of the higher-dimensional Schrödinger operator (Russian). Dokl. Akad. Nauk SSSR 144, 721–722 (1962); English transl.: Sov. Math. Dokl. 3, 808–810 (1962)MATHGoogle Scholar
  24. 24.
    Maz'ya, V.: On the theory of the higher-dimensional Schrödinger operator (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 28, 1145–1172 (1964)MATHMathSciNetGoogle Scholar
  25. 25.
    Maz'ya, V.: Analytic criteria in the qualitative spectral analysis of the Schrödinger operator. Proc. Sympos. Pure Math. 76, no. 1, 257–288 (2007)MathSciNetGoogle Scholar
  26. 26.
    Maz'ya, V., Sobolevskii. P.: On the generating operators of semigroups (Russian). Usp. Mat. Nauk 17, 151–154 (1962)MATHGoogle Scholar
  27. 27.
    Metafune, G., Pallara, D., Prüss, J., Schnaubelt, R.: L p-theory for elliptic operators on Rd with singular coefficients. Z. Anal. Anwe. 24, 497–521 (2005)MATHGoogle Scholar
  28. 28.
    Okazawa, N.: Sectorialness of second order elliptic operators in divergence form. Proc. Am. Math. Soc. 113, 701–706 (1991)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ouhabaz, El M.: Gaussian upper bounds for heat kernels of second order elliptic operators with complex coefficients on arbitrary domains. J. Operator Theory 51, 335–360 (2004)MathSciNetGoogle Scholar
  30. 30.
    Ouhabaz, El M.: Analysis of Heat Equations on Domains. Princeton Univ. Press, Princeton, NJ (2005)Google Scholar
  31. 31.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)MATHGoogle Scholar
  32. 32.
    Robinson, D.W.: Elliptic Operators on Lie Groups. Oxford Univ. Press, Oxford (1991)Google Scholar
  33. 33.
    Sobol, Z., Vogt, H.: On the L p-theory of C 0 semigroups associated with second order elliptic operators. I. J. Funct. Anal. 193, 24–54 (2002)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Strichartz: R.S.: L p contractive projections and the heat semigroup for differential forms. J. Funct. Anal. 65, 348–357 (1986)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità della BasilicataViale dell’AteneoItaly

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