Advertisement

Parabolic Harnack inequality for divergence form second order differential operators

Chapter
  • 201 Downloads

Abstract

Old and recent results concerning Harnack inequalities for divergence form operators are reviewed. In particular, the characterization of the parabolic Harnack principle by simple geometric properties -Poincaré inequality and doubling property- is discussed at length. It is shown that these two properties suffice to apply Moser’s iterative technique.

Key words

Harnack inequality Poincaré inequality doubling property heat equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexopoulos G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canadian J. Math. 44, 1992, 691–727.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aronson D. G.: Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 1967, 890–896.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.a.
    Aronson D. G.: Non-negative solutions of linear parabolic equations Ann. Scu. Norm. Sup. Pisa. CI. Sci. 22, 1968, 607–694;MathSciNetzbMATHGoogle Scholar
  4. 3.b.
    Aronson D. G., Non-negative solutions of linear parabolic equations Addendum 25, 1971, 221–228.MathSciNetzbMATHGoogle Scholar
  5. 4.
    Aronson D. G. and Serrin J.: Local behavior of solutions of quasilinear parabolic equations. Arch. Rat Mech. Anal. 25, 1967, 81–122.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 5.
    Bakry D., Coulhon Th., Ledoux M. Saloff-Coste L.: Sobolev inequalities in disguise. Preprint 1994.Google Scholar
  7. 6.
    Biroli M. and Mosco U.: Formes de Dirichlet et estimations structurelles dans les milieux discontinus. C. R. Acad. Sci. Paris, 313, 1991, 593–598.MathSciNetzbMATHGoogle Scholar
  8. 7.
    Biroli M. and Mosco U.: A Saint-Venant Principle for Dirichlet forms on discontinuous media. Ann. di Mat. Pura e Appl.Google Scholar
  9. 8.
    Biroli M. Mosco U.: Sobolev and isoperimetric inequalities for Dirichlet forms on homoge- neous spaces. Atti Accad. Naz. Lincei CI. Sci. Fis. Mat. Natur., 1994Google Scholar
  10. 9.
    Bôcher M.: Singular points of functions which satisfy partial differential equations of elliptic type. Bull. Amer. Math. Soc. 9, 1903, 455–465.MathSciNetCrossRefGoogle Scholar
  11. 10.
    Bombieri E.: Theory of mininal surfaces and a counter-example to the Berstein conjecture in high dimensions. Mineographed Notes of Lectures held at Courant Institute, New-York University, 1970.Google Scholar
  12. 11.
    Bombieri E. and Giusti E.: Harnack’s inequality for elliptic differential equations on mininal surfaces. Invent. Math. 15, 1972, 24–46.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 12.
    Bony J-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier, 19, 1969, 277–304.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 13.
    Brelot M.: Elements de la théorie classique du potentiel. Centre de documentation universitaire, Paris, 1965.Google Scholar
  15. 14.
    Buser P.: A note on the isoperimetric constant. Ann. Sci. Ecole Norm. Sup. 15, 1982, 213–230.MathSciNetzbMATHGoogle Scholar
  16. 15.
    Cao H-D. and Yau S-T.: Gradients estimates, Harnack inequalities and estimates for heat kernels of sum of squares of vector fields. Math. Zeit. 211, 1992, 485–504.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 16.
    Chanillo S. and Wheeden R.: Haxnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations. Coram. Part. Diff. Equ. 11, 1986, 1111–1134.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 17.
    Charienza F. and Serapioni R.: A Harnack inequality for degenerate parabolic equations. Coram. Part. Diff. Equ. 9, 1984, 719–749.CrossRefGoogle Scholar
  19. 18.
    Cheeger J., Gromov M. and Taylor M.: Finite propagation speed, kernel estimates for func- tions of the Laplace operator and the geometry of complete Riemannian manifolds. J. Diff. Geo. 17, 1982, 15–23.MathSciNetzbMATHGoogle Scholar
  20. 19.
    Cheng S., Li P. and Yau S-T.: On the upper estimate of the heat kernel of a complete Riemannian manifold. Amer. J. Math. 103, 1981, 1021–1036.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 20.
    Cheng S. and Yau S-T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure. Appl. Math. 28, 1975, 333–354.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 21.
    Coulhon Th. and Saloff-Coste L. Variétés riemanniennes isométriques à l’infini. 1994.Google Scholar
  23. 22.
    Davies E.B.: Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math. 109, 319–333.Google Scholar
  24. 23.
    Davies E.B.: Heat kernels and spectral theory. Cambridge University Press, 1989.zbMATHCrossRefGoogle Scholar
  25. 24.
    Doob J.L.: Classical potential theory and its probabilistic counterpart. New-York, Springer-Verlag, 1984.zbMATHCrossRefGoogle Scholar
  26. 25.
    Fabes E.: Gaussian upper bounds on fundamental solutions of parabolic equations: the method of Nash. In Dirichlet forms, Led. Not. Math. 1563, Springer-Verlag, 1993, 1–20.CrossRefGoogle Scholar
  27. 26.
    Fabes E. and Stroock D.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 1986, 327–338.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 27.
    Fefferman C. and Phong D.H: Subelliptic eigenvalue problems. In Proceedings of the conference in harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Ser., Wadsworth, Belmont, California, 1981, 590–606.Google Scholar
  29. 28.
    Fefferman C. and Sanchez-Calle A.: Fundamental solutions for second order subelliptic oper- ators. Ann. Math. 124, 1986, 247–272.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 29.
    Fernandes J.: Mean value and Harnack inequalities for certain class of degenerate parabolic equations. Rev. mat. Iberoamericana, 7 1991, 247–286.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 30.
    Franchi B.: Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Amer. Math. Soc. 327, 1991, 125–158.MathSciNetzbMATHGoogle Scholar
  32. 31.
    Franchi B. and Lanconelli E.: An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality. Commm. P.D.E. 9, 1984, 1237–1264.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 32.
    Franchi B. and Serapioni R.: Pointwise estimates for a class of strongly degenerate elliptic operators: a geometric approach. Ann. Scul. Norm. Sup. Pisa, 14, 1987, 527–568.MathSciNetzbMATHGoogle Scholar
  34. 33.
    Gilbarg D. and Trudinger N.: Elliptic partial differential equations of second order. Sec. Ed. Berlin, Heidelberg, Springer-Verlag 1983.zbMATHCrossRefGoogle Scholar
  35. 34.
    de Giorgi E.: Sulla differentiabilita el’analiticita delle estremali degli integrali multipli regolari Mem. Accad. Sci. Torino, CI. Sci. Fis. Mat. Nat, Ser 3, 3, 1957, 25–43.Google Scholar
  36. 35.
    Grigory’an A.: The heat equation on noncompact Riemannian manifold. Math. USSR Sbornik, 72, 1992, 47–76.MathSciNetCrossRefGoogle Scholar
  37. 36.
    Guivarc’h Y.: Croissance polynomiale et periode des functions harmoniques. Bull. Soc. Math. France, 101, 1973, 149–152.MathSciNetGoogle Scholar
  38. 37.
    Gutiérrez E. and Wheeden R.: Mean value and Harnack inequalties for degenerate parabolic equations. Coll. Math. 60, Volume dédié a M. Anton Zygmund, 1990, 157–194.Google Scholar
  39. 38.
    Hadamard J.: Extension à l’équation de la chaleur d’un théorème de A. Harnack. Rend. Circ. Mat. Palermo, Ser. 2, 3, 1954, 337–346.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 39.
    Harnack A.: Die Grundlagen der Theorie des logarthmischen Potentials und der eindeutigen Potentialfunktion. Leipzig, Teubner, 1887.Google Scholar
  41. 40.
    Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 1967, 147–171.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 41.
    Jerison D.: The Poincaré inequality for vector fields satisfying the Hörmander’s condition. Duke Math. J. 53, 1986, 503–523.MathSciNetzbMATHGoogle Scholar
  43. 42.
    Jerison D. and Sanchez-Calle A.: Subelliptic second order differential operators. In Complexe analysis III, Procedings, Lect. Not. Math. 1277, Springer-Verlag, 1986, 47–77.Google Scholar
  44. 43.
    Krylov N. and Safonov M.: A certain property of solutions of parabolic equations with mesurable coefficients. Izv. Akad. Nauk. SSSR, 44, 1980, 81–98.MathSciNetGoogle Scholar
  45. 44.
    Kusuoka S. and Stroock D.: Application of Malliavin calculus, part 3. J. Fac. Sci. Univ. Tokyo, Série IA, Math. 34, 1987, 391–442.MathSciNetzbMATHGoogle Scholar
  46. 45.
    Kusuoka S. and Stroock D.: Long time estimates for the heat kernel associated with uniformly subelliptic symmetric second order operator. Ann. Math. 127, 1988, 165–189. 391–442.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 46.
    Li P. and Yau S-T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 1986, 153–201.MathSciNetCrossRefGoogle Scholar
  48. 47.
    Lu G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Rev. Mat. Iberoamericana, 8, 1992, 367–439.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 48.
    Maheux P. and Saloff-Coste L.: Analyse sur les boules d’un opérateur sous-elliptique. Preprint 1994.Google Scholar
  50. 49.
    Moser J.: On Harnack’s Theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 1961, 577–591.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 50.a.
    Moser J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 16, 1964, 101–134CrossRefGoogle Scholar
  52. 50.b.
    Moser J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 20, 1967, 231–236.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 51.
    Moser J.: On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24, 1971, 727–440.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 52.
    Nagel A., Stein E. and Wainger S.: Balls and metrics defined by vector fields. Acta Math. 155, 1985, 103–147.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 53.
    Nash J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 1958, 931–953.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 54.
    Oleinik O. and Radkevic E.: Second order equations with nonnegative characteristic form. American Math. Soc, Providence, 1973.CrossRefGoogle Scholar
  57. 55.
    Pini B.: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Sem. Mat. Padova, 23, 1954, 422–434.MathSciNetzbMATHGoogle Scholar
  58. 56.
    Porper F. O. and Eidel’man S.D.: Two-sided estimates of fundamental solutions of second- order parabolic equations, and some applications. Russian Math. Surveys, 39, 1984, 119–178.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 57.
    Safonov M.N.: Harnack’s inequality for elliptic equations and Holder property of their solutions. J. Soviet Math. 21, 1983, 851–863.zbMATHCrossRefGoogle Scholar
  60. 58.
    Saloff-Coste L.: Analyse sur les groupes de Lie à croissance polynômiale. Ark. För Mat. 28, 1990, 315–331.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 59.
    Saloff-Coste L.: Opérateurs uniformèment elliptiques sur les variétés riemanniennes. C. R. Acad. Sci. Pans, Série I, Math. 312, 1991, 25–30.MathSciNetzbMATHGoogle Scholar
  62. 60.
    Saloff-Coste L.: Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geo. 36, 1992, 417–450.MathSciNetzbMATHGoogle Scholar
  63. 61.
    Saloff-Coste L.: A note on Poincaré, Sobolev, and Harnack inequality. Duke Math. J., I.M.R.N. 2, 1992, 27–38.MathSciNetGoogle Scholar
  64. 62.
    Saloff-Coste L. and Stroock D.: Opérateurs uniformèment sous-elliptiques sur les groupes de Lie. J. Funt. Anal. 98, 1991, 97–121.MathSciNetzbMATHCrossRefGoogle Scholar
  65. 63.
    Sturm K-T.: On the geometry defined by Dirichlet forms. Preprint, 1993.Google Scholar
  66. 64.
    Sturm K-T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativness and L p- Liouville properties. To appear in J. Reine Angew. Math. 1994.Google Scholar
  67. 65.
    Sturm K-T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for fundamental solutions of parabolic equations. To appear in Osaka J. Math. 1994.Google Scholar
  68. 66.
    Sturm K-T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. Preprint, 1994.Google Scholar
  69. 67.
    Serrin J.: On the Harnack inequality for linear elliptic differential equations. J. Anal. Math. 4, 1954–1956, 292–308.MathSciNetCrossRefGoogle Scholar
  70. 68.
    N. Trudinger: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math, 20, 1967, 721–747.MathSciNetzbMATHCrossRefGoogle Scholar
  71. 69.
    N. Trudinger: Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math, 21, 1968, 205–226.MathSciNetzbMATHCrossRefGoogle Scholar
  72. 70.
    N. Trudinger: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. sup. Pisa, 27, 1973, 265–308.MathSciNetzbMATHGoogle Scholar
  73. 71.
    Vaxopoulos N.: Fonctions harmoniques sur les groupes de Lie. C. R. Acad. Sci. Paris, Série I, Math. 309, 1987, 519–521.Google Scholar
  74. 72.
    Varopoulos N.: Analysis on Lie groups. J. Funt. Anal. 76, 1988, 346–410.MathSciNetzbMATHCrossRefGoogle Scholar
  75. 73.
    Varopoulos N.: Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semi- group technique. Bull Sci. Math. 113, 1989, 253–277.MathSciNetzbMATHGoogle Scholar
  76. 74.
    Varopoulos N.: Opérateurs sous-elliptique du second ordre. C. R. Acad. Sci. Paris, 308, Serie I, 1989, 437–440.MathSciNetzbMATHGoogle Scholar
  77. 75.
    Varopoulos N., Saloff-Coste L. and Coulhon Th.: Analysis and geometry on groups. Cambridge Universty Press, 1993.CrossRefGoogle Scholar
  78. 76.
    Yau S-T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure. Appl. Math. 28, 1975, 201–228.MathSciNetzbMATHCrossRefGoogle Scholar
  79. 77.
    Yau S-T. Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Sup. Paris, 8 1975, 487–507.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.CNRS, Statistique et ProbabilitésUniversité Paul SabatierToulouse cedexFrance

Personalised recommendations