Potential Analysis

, Volume 19, Issue 1, pp 69–87 | Cite as

On Regularity for Beurling–Deny Type Dirichlet Forms

  • Moritz Kassmann
Article

Abstract

Characteristic examples of Beurling–Deny type Dirichlet forms are considered. The forms are identified with bilinear forms of integro-differential operators that arise as generators of jump-diffusion processes. The aim of this article is to prove Harnack inequalities for these operators and consequently Hölder regularity of weak H1-solutions. Moser's iteration technique is used.

Dirichlet forms Harnack inequality Beurling–Deny formula Hölder regularity integro-differential operators 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albeverio, S., Rüdiger, B. and Wu, J.-L.: 'Invariant measures and symmetry property of Lévy type operators', Potential Anal. 13(2) (2000), 147-168.Google Scholar
  2. 2.
    Aronson, D.G. and Serrin, J.: 'Local behavior of solutions of quasilinear parabolic equations', Arch. Rat. Mech. Anal. 25 (1967), 81-122.Google Scholar
  3. 3.
    Bass, R. and Levin, D.A.: 'Harnack inequalities for jump processes', Potential Anal., to appear.Google Scholar
  4. 4.
    Bensoussan, A. and Lions, J.-L.: Impulse Control and Quasi-variational Inequalities, Gauthier-Villars, Paris, 1984.Google Scholar
  5. 5.
    Biroli, M. and Mosco, U.: 'A Saint-Venant type principle for Dirichlet forms on discontinuous media',Ann. Mat. Pura Appl. (4) 169 (1995), 125-181.Google Scholar
  6. 6.
    De Giorgi, E.: 'FrSulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari', Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3) 3 (1957), 25-43.Google Scholar
  7. 7.
    Elliott, J.: 'Dirichlet spaces associated with integro-differential operators, Part I', Illinois J. Math. 9 (1965),87-98.Google Scholar
  8. 8.
    Elliott, J.: 'Dirichlet spaces associated with integro-differential operators, Part II', Illinois J. Math. 10 (1965), 66-89.Google Scholar
  9. 9.
    Fabes, E.B., Kenig, C.E. and Serapioni, R.P.: 'The local regularity of solutions of degenerate elliptic equations', Comm. Partial Differential Equations 7(1) (1982), 77-116.Google Scholar
  10. 10.
    Fabes, E.B. and Stroock, D.W.: 'A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash',Arch. Rational Mech. Anal. 96(4) (1986), 327-338.Google Scholar
  11. 11.
    Farkas, W., Jacob, N. and Schilling, R.L.: 'Feller semigroups, Lp-sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols', Forum Math. 13(1) (2001), 51-90.Google Scholar
  12. 12.
    Fukushima, M.: 'On an Lp estimate of resolvents of Markov processes', Publ. Res. Inst. Math. Sci. 13 (1977), 277-284.Google Scholar
  13. 13.
    Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Berlin, 1994.Google Scholar
  14. 14.
    Garroni, M. and Menaldi, J.: Green Functions for Second Order Parabolic Integro-Differential Problems,Longman, Essex, 1992.Google Scholar
  15. 15.
    Giaquinta, M. and Modica, G.: 'Nonlinear systems of the type of the stationary Navier—Stokes system',J. Reine Angew. Math. 330 (1982), 173-214.Google Scholar
  16. 16.
    Gilbarg, D. and Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.Google Scholar
  17. 17.
    Grüter, M. and Widman, K.-O.: 'The Green function for uniformly elliptic equations', Manuscripta Math. 37(3) (1982), 303-342.Google Scholar
  18. 18.
    Hoh, W.: 'A symbolic calculus for pseudo-differential operators generating Feller semigroups', Osaka J. Math. 35(4) (1998), 789-820.Google Scholar
  19. 19.
    Hoh, W. and Jacob, N.: 'On the Dirichlet problem for pseudodifferential operators generating Feller semigroups',J. Funct. Anal. 137(1) (1996), 19-48.Google Scholar
  20. 20.
    Jacob, N.: Pseudo-Differential Operators and Markov Processes, Part I: Fourier Analysis and Semigroups, Imperial College Press, London, 2001.Google Scholar
  21. 21.
    Kassmann, M.: 'Harnack-Ungleichungen für nichtlokale Differential operatoren und Dirichlet-Formen', Bonner Math. Schriften 336 (2001), 1-91.Google Scholar
  22. 22.
    Komatsu, T.: 'Markov processes associated with a certain integro-differential operator', Osaka J. Math. 10 (1973), 71-303.Google Scholar
  23. 23.
    Komatsu, T.: 'Uniform estimates for fundamental solutions associated with non-local Dirichlet forms', Osaka J. Math. 32 (1995), 833-860.Google Scholar
  24. 24.
    Krylov, N.V. and Safonov, M.V.: 'A property of the solutions of parabolic equations with measurable coefficients', Izv. Akad. Nauk SSSR Ser. Mat. 44(1) (1980), 161-175, 239Google Scholar
  25. 25.
    Mikulevicius, R. and Pragarauskas, H.: 'On Hölder continuity of solutions of certain integrodifferential equations',Ann. Acad. Sci. Fenn. 13(2) (1988), 231-238.Google Scholar
  26. 26.
    Moser, J.: 'On Harnack's theorem for elliptic differential equations',Comm. Pure Appl. Math. 14 (1961), 577-591.Google Scholar
  27. 27.
    Moser, J.: 'A Harnack inequality for parabolic differential equations',Comm. Pure Appl. Math. 17 (1964), 101-134Google Scholar
  28. 28.
    Murthy, M.K.V. and Stampacchia, G.: 'Boundary value problems for some degenerate-elliptic operators',Ann. Mat. Pura Appl. (4) 80 (1968), 1-122.Google Scholar
  29. 29.
    Nash, J.: 'Continuity of solutions of parabolic and elliptic equations',Amer. J. Math. 80 (1958), 931-954.Google Scholar
  30. 30.
    Serrin, J.: 'A Harnack inequality for nonlinear equations', Bull. Amer. Math. Soc. 69 (1963), 481-486.Google Scholar
  31. 31.
    Song, R. and Vondraček, Z.: 'Harnack inequality for some classes of Markov processes', Preprint, 2002.Google Scholar
  32. 32.
    Stroock, D.: 'Diffusion processes associated with Levý generators', Z. Wahr. Verw. Gebiete 39 (1975), 209-244.Google Scholar
  33. 33.
    Tomisaki, M.: 'Some estimates for solutions of equations related to non-local Dirichlet forms', Rep. Fac. Sci. Engrg. Saga Univ. 6 (1978), 1-7.Google Scholar
  34. 34.
    Trudinger, N.S.: 'On Harnack type inequalities and their application to quasilinear elliptic equations', Comm. Pure Appl. Math. 20 (1967), 721-747.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Moritz Kassmann
    • 1
  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations