Regularizing effects of homogeneous evolution equations: the case of homogeneity order zero

  • Daniel HauerEmail author
  • José M. Mazón


In this paper, we develop a functional analytical theory for establishing that mild solutions of first-order Cauchy problems involving homogeneous operators of order zero are strong solutions; in particular, the first-order time derivative satisfies a global regularity estimate depending only on the initial value and the positive time. We apply those results to the Cauchy problem associated with the total variational flow operator and the nonlocal fractional 1-Laplace operator.


Nonlinear semigroups Local and nonlocal operators 1-Laplace operator Regularity Homogenous operators 

Mathematics Subject Classification

47H20 47H06 47J35 


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  1. 1.
    F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), pp. 347–403.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), pp. 321–360.MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Andreu-Vaillo, V. Caselles, and J. M. Mazón, Parabolic quasilinear equations minimizing linear growth functionals, vol. 223 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2004.Google Scholar
  4. 4.
    W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 1–85.Google Scholar
  5. 5.
    W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, vol. 96 of Monographs in Mathematics, Birkhäuser/Springer Basel AG, Basel, second ed., 2011.Google Scholar
  6. 6.
    V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Monographs in Mathematics, Springer, New York, 2010.CrossRefzbMATHGoogle Scholar
  7. 7.
    P. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, in Contributions to analysis and geometry (Baltimore, Md., 1980), Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 23–39.Google Scholar
  8. 8.
    P. Bénilan and M. G. Crandall, Completely accretive operators, in Semigroup theory and evolution equations (Delft, 1989), vol. 135 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1991, pp. 41–75.Google Scholar
  9. 9.
    P. Bénilan, M. G. Crandall, and A. Pazy, Evolution problems governed by accretive operators, book in preparation.Google Scholar
  10. 10.
    C. Bennett and R. Sharpley, Interpolation of operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988.Google Scholar
  11. 11.
    H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).Google Scholar
  12. 12.
    T. Coulhon and D. Hauer, Regularisation effects of nonlinear semigroups - Theory and Applications. to appear in BCAM Springer Briefs, 2017.Google Scholar
  13. 13.
    E. B. Davies, Heat kernels and spectral theory, vol. 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1990.Google Scholar
  14. 14.
    A. Grigor’yan, Heat kernel and analysis on manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.Google Scholar
  15. 15.
    G. Leoni, A first course in Sobolev spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009.Google Scholar
  16. 16.
    J. M. Mazón, J. D. Rossi, and J. Toledo, Fractional \(p\) -Laplacian evolution equations, J. Math. Pures Appl. (9), 105 (2016), pp. 810–844.Google Scholar
  17. 17.
    A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.Google Scholar
  18. 18.
    M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, vol. 146 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1991.Google Scholar
  19. 19.
    N. T. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of SydneySydneyAustralia
  2. 2.Departament d’Anàlisi MatemàticaUniversitat de ValènciaValenciaSpain

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