Skip to main content
Boundary Value Problems
Boundary Value Problems Cover Image
Download PDF

Harnack Inequalities: An Introduction

Boundary Value Problems volume 2007, Article number: 081415 (2007) Cite this article

Abstract

The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.

[123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128]

References

  1. 1.

    Voss A: Zur Erinnerung an Axel Harnack. Mathematische Annalen 1888,32(2):161-174. 10.1007/BF01444065

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Axel Harnack C-G: Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene. Teubner, Leipzig, Germany; 1887. see also The Cornell Library Historical Mathematics Monographs

    Google Scholar 

  3. 3.

    Poincaré H: Sur les Equations aux Derivees Partielles de la Physique Mathematique. American Journal of Mathematics 1890,12(3):211-294. 10.2307/2369620

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Lichtenstein L: Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Unendliche Folgen positiver Lösungen. Rendiconti del Circolo Matematico di Palermo 1912, 33: 201-211. 10.1007/BF03015300

    MATH  Google Scholar 

  5. 5.

    Lichtenstein L: Randwertaufgaben der Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. I. Journal für die Reine und Angewandte Mathematik 1913, 142: 1-40.

    MathSciNet  Google Scholar 

  6. 6.

    Feller W: Über die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Mathematische Annalen 1930,102(1):633-649. 10.1007/BF01782367

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Serrin J: On the Harnack inequality for linear elliptic equations. Journal d'Analyse Mathématique 1955/1956, 4: 292-308.

    MathSciNet  Google Scholar 

  8. 8.

    Bers L, Nirenberg L: On linear and non-linear elliptic boundary value problems in the plane. In Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954. Edizioni Cremonese, Rome, Italy; 1955:141-167.

    Google Scholar 

  9. 9.

    Lichtenstein L: Neuere Entwicklung der Potentialtheorie. Konforme Abbildung. Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen 1918, 2, T.3, H.1: 177-377.

    Google Scholar 

  10. 10.

    Kellogg OD: Foundations of Potential Theorie, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Bd. 31. Springer, Berlin, Germany; 1929.

    Google Scholar 

  11. 11.

    Riesz M: Intégrales de Riemann-Liouville et potentiels. Acta Scientiarum Mathematicarum (Szeged) 1938, 9: 1-42.

    Google Scholar 

  12. 12.

    Landkof NS: Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer, New York, NY, USA; 1972:x+424.

    Google Scholar 

  13. 13.

    Pini B: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rendiconti del Seminario Matematico della Università di Padova 1954, 23: 422-434.

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Hadamard J: Extension à l'équation de la chaleur d'un théorème de A. Harnack. Rendiconti del Circolo Matematico di Palermo. Serie II 1954, 3: 337-346 (1955). 10.1007/BF02849264

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Moser J: A Harnack inequality for parabolic differential equations. Communications on Pure and Applied Mathematics 1964, 17: 101-134. 10.1002/cpa.3160170106

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Auchmuty G, Bao D: Harnack-type inequalities for evolution equations. Proceedings of the American Mathematical Society 1994,122(1):117-129. 10.1090/S0002-9939-1994-1219716-X

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Li P, Yau Sh-T: On the parabolic kernel of the Schrödinger operator. Acta Mathematica 1986,156(3-4):153-201.

    MathSciNet  Google Scholar 

  18. 18.

    Moser J: On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics 1961, 14: 577-591. 10.1002/cpa.3160140329

    MATH  MathSciNet  Google Scholar 

  19. 19.

    De Giorgi E: Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali. Serie III 1957, 3: 25-43.

    MathSciNet  Google Scholar 

  20. 20.

    John F, Nirenberg L: On functions of bounded mean oscillation. Communications on Pure and Applied Mathematics 1961, 14: 415-426. 10.1002/cpa.3160140317

    MATH  MathSciNet  Google Scholar 

  21. 21.

    Bombieri E, Giusti E: Harnack's inequality for elliptic differential equations on minimal surfaces. Inventiones Mathematicae 1972,15(1):24-46. 10.1007/BF01418640

    MATH  MathSciNet  Google Scholar 

  22. 22.

    Han Q, Lin F: Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics. Volume 1. New York University Courant Institute of Mathematical Sciences, New York, NY, USA; 1997:x+144.

    Google Scholar 

  23. 23.

    Grüter M, Widman Kj-O: The Green function for uniformly elliptic equations. Manuscripta Mathematica 1982,37(3):303-342. 10.1007/BF01166225

    MATH  MathSciNet  Google Scholar 

  24. 24.

    Bensoussan A, Frehse J: Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences. Volume 151. Springer, Berlin, Germany; 2002:xii+441.

    Google Scholar 

  25. 25.

    Nash J: Continuity of solutions of parabolic and elliptic equations. American Journal of Mathematics 1958,80(4):931-954. 10.2307/2372841

    MATH  MathSciNet  Google Scholar 

  26. 26.

    DiBenedetto E, Trudinger NS: Harnack inequalities for quasiminima of variational integrals. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1984,1(4):295-308.

    MATH  MathSciNet  Google Scholar 

  27. 27.

    DiBenedetto E: Harnack estimates in certain function classes. Atti del Seminario Matematico e Fisico dell'Università di Modena 1989,37(1):173-182.

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Serrin J: Local behavior of solutions of quasi-linear equations. Acta Mathematica 1964,111(1):247-302. 10.1007/BF02391014

    MATH  MathSciNet  Google Scholar 

  29. 29.

    Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics 1967, 20: 721-747. 10.1002/cpa.3160200406

    MATH  MathSciNet  Google Scholar 

  30. 30.

    Trudinger NS: Harnack inequalities for nonuniformly elliptic divergence structure equations. Inventiones Mathematicae 1981,64(3):517-531. 10.1007/BF01389280

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York, NY, USA; 1968:xviii+495.

    Google Scholar 

  32. 32.

    Moser J: Correction to: "A Harnack inequality for parabolic differential equations". Communications on Pure and Applied Mathematics 1967, 20: 231-236. 10.1002/cpa.3160200107

    MATH  MathSciNet  Google Scholar 

  33. 33.

    Moser J: On a pointwise estimate for parabolic differential equations. Communications on Pure and Applied Mathematics 1971, 24: 727-740. 10.1002/cpa.3160240507

    MATH  MathSciNet  Google Scholar 

  34. 34.

    Fabes EB, Garofalo N: Parabolic B.M.O. and Harnack's inequality. Proceedings of the American Mathematical Society 1985,95(1):63-69.

    MATH  MathSciNet  Google Scholar 

  35. 35.

    Ferretti E, Safonov MV: Growth theorems and Harnack inequality for second order parabolic equations. In Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), Contemp. Math.. Volume 277. American Mathematical Society, Providence, RI, USA; 2001:87-112.

    Google Scholar 

  36. 36.

    Safonov MV: Mean value theorems and Harnack inequalities for second-order parabolic equations. In Nonlinear Problems in Mathematical Physics and Related Topics, II, Int. Math. Ser. (N. Y.). Volume 2. Kluwer/Plenum, New York, NY, USA; 2002:329-352. 10.1007/978-1-4615-0701-7_18

    Google Scholar 

  37. 37.

    Aronson DG: Bounds for the fundamental solution of a parabolic equation. Bulletin of the American Mathematical Society 1967, 73: 890-896. 10.1090/S0002-9904-1967-11830-5

    MATH  MathSciNet  Google Scholar 

  38. 38.

    Fabes EB, Stroock DW: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Archive for Rational Mechanics and Analysis 1986,96(4):327-338.

    MATH  MathSciNet  Google Scholar 

  39. 39.

    Fabes EB, Stroock DW:The-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations. Duke Mathematical Journal 1984,51(4):997-1016. 10.1215/S0012-7094-84-05145-7

    MATH  MathSciNet  Google Scholar 

  40. 40.

    Aronson DG, Serrin J: Local behavior of solutions of quasilinear parabolic equations. Archive for Rational Mechanics and Analysis 1967, 25: 81-122. 10.1007/BF00281291

    MATH  MathSciNet  Google Scholar 

  41. 41.

    Ivanov AV: The Harnack inequality for generalized solutions of second order quasilinear parabolic equations. Trudy Matematicheskogo Instituta imeni V. A. Steklova 1967, 102: 51-84.

    MATH  Google Scholar 

  42. 42.

    Trudinger NS: Pointwise estimates and quasilinear parabolic equations. Communications on Pure and Applied Mathematics 1968, 21: 205-226. 10.1002/cpa.3160210302

    MATH  MathSciNet  Google Scholar 

  43. 43.

    DiBenedetto E: Degenerate Parabolic Equations, Universitext. Springer, New York, NY, USA; 1993:xvi+387.

    Google Scholar 

  44. 44.

    Ivanov AV: Second-order quasilinear degenerate and nonuniformly elliptic and parabolic equations. Trudy Matematicheskogo Instituta imeni V. A. Steklova 1982, 160: 285.

    Google Scholar 

  45. 45.

    Porper FO, Èĭdel'man SD: Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them. Uspekhi Matematicheskikh Nauk 1984,39(3(237)):107-156.

    Google Scholar 

  46. 46.

    Ladyzhenskaya OA, Solonnikov VA, Ural'tseva NN: Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs. Volume 23. American Mathematical Society, Providence, RI, USA; 1968.

    Google Scholar 

  47. 47.

    Kruzhkov SN: A priori estimation of solutions of linear parabolic equations and of boundary value problems for a certain class of quasilinear parabolic equations. Soviet Mathematics. Doklady 1961, 2: 764-767.

    MATH  MathSciNet  Google Scholar 

  48. 48.

    Kruzhkov SN: A priori estimates and certain properties of the solutions of elliptic and parabolic equations. American Mathematical Society Translations. Series 2 1968, 68: 169-220.

    MATH  Google Scholar 

  49. 49.

    Kružkov SN: Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications. Mathematical Notes 1969, 6: 517-523. 10.1007/BF01450257

    Google Scholar 

  50. 50.

    Chiarenza FM, Serapioni RP: A Harnack inequality for degenerate parabolic equations. Communications in Partial Differential Equations 1984,9(8):719-749. 10.1080/03605308408820346

    MATH  MathSciNet  Google Scholar 

  51. 51.

    DiBenedetto E: Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Archive for Rational Mechanics and Analysis 1988,100(2):129-147. 10.1007/BF00282201

    MATH  MathSciNet  Google Scholar 

  52. 52.

    DiBenedetto E, Urbano JM, Vespri V: Current issues on singular and degenerate evolution equations. In Evolutionary Equations. Vol. I, Handb. Differ. Equ.. North-Holland, Amsterdam, The Netherlands; 2004:169-286.

    Google Scholar 

  53. 53.

    DiBenedetto E, Gianazza U, Vespri V: Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equatoins. Electronic Research Announcements of the American Mathematical Society 2006, 12: 95-99. 10.1090/S1079-6762-06-00166-1

    MATH  MathSciNet  Google Scholar 

  54. 54.

    Gianazza U, Vespri V: A Harnack inequality for a degenerate parabolic equation. Journal of Evolution Equations 2006,6(2):247-267. 10.1007/s00028-006-0242-2

    MATH  MathSciNet  Google Scholar 

  55. 55.

    Gianazza U, Vespri V:Parabolic De Giorgi classes of order and the Harnack inequality. Calculus of Variations and Partial Differential Equations 2006,26(3):379-399. 10.1007/s00526-006-0022-4

    MATH  MathSciNet  Google Scholar 

  56. 56.

    Kruzhkov SN: Certain properties of solutions to elliptic equations. Soviet Mathematics. Doklady 1963, 4: 686-690.

    MATH  Google Scholar 

  57. 57.

    Murthy MKV, Stampacchia G: Boundary value problems for some degenerate-elliptic operators. Annali di Matematica Pura ed Applicata 1968,80(1):1-122. 10.1007/BF02413623

    MATH  MathSciNet  Google Scholar 

  58. 58.

    Murthy MKV, Stampacchia G: Errata corrige: "Boundary value problems for some degenerate-elliptic operators". Annali di Matematica Pura ed Applicata 1971,90(1):413-414. 10.1007/BF02415055

    MATH  MathSciNet  Google Scholar 

  59. 59.

    Edmunds DE, Peletier LA: A Harnack inequality for weak solutions of degenerate quasilinear elliptic equations. Journal of the London Mathematical Society. Second Series 1972, 5: 21-31. 10.1112/jlms/s2-5.1.21

    MATH  MathSciNet  Google Scholar 

  60. 60.

    Fabes EB, Kenig CE, Serapioni RP: The local regularity of solutions of degenerate elliptic equations. Communications in Partial Differential Equations 1982,7(1):77-116. 10.1080/03605308208820218

    MATH  MathSciNet  Google Scholar 

  61. 61.

    Trudinger NS: On the regularity of generalized solutions of linear, non-uniformly elliptic equations. Archive for Rational Mechanics and Analysis 1971,42(1):50-62.

    MATH  MathSciNet  Google Scholar 

  62. 62.

    Franchi B, Serapioni R, Serra Cassano F: Irregular solutions of linear degenerate elliptic equations. Potential Analysis 1998,9(3):201-216. 10.1023/A:1008684127989

    MATH  MathSciNet  Google Scholar 

  63. 63.

    Chiarenza F, Serapioni R: Pointwise estimates for degenerate parabolic equations. Applicable Analysis 1987,23(4):287-299. 10.1080/00036818708839648

    MATH  MathSciNet  Google Scholar 

  64. 64.

    Kružkov SN, Kolodīĭ ĪM: A priori estimates and Harnack's inequality for generalized solutions of degenerate quasilinear parabolic equations. Sibirskiĭ Matematičeskiĭ Žurnal 1977,18(3):608-628, 718.

    Google Scholar 

  65. 65.

    Chanillo S, Wheeden RL: Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations. Communications in Partial Differential Equations 1986,11(10):1111-1134. 10.1080/03605308608820458

    MATH  MathSciNet  Google Scholar 

  66. 66.

    Chanillo S, Wheeden RL: Existence and estimates of Green's function for degenerate elliptic equations. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1988,15(2):309-340 (1989).

    MATH  MathSciNet  Google Scholar 

  67. 67.

    Chiarenza F, Rustichini A, Serapioni R: De Giorgi-Moser theorem for a class of degenerate non-uniformly elliptic equations. Communications in Partial Differential Equations 1989,14(5):635-662. 10.1080/03605308908820623

    MATH  MathSciNet  Google Scholar 

  68. 68.

    Salinas O: Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. Revista Matemática Iberoamericana 1991,7(3):313-349.

    MATH  MathSciNet  Google Scholar 

  69. 69.

    Franchi B, Gutiérrez CrE, Wheeden RL: Weighted Sobolev-Poincaré inequalities for Grushin type operators. Communications in Partial Differential Equations 1994,19(3-4):523-604. 10.1080/03605309408821025

    MATH  MathSciNet  Google Scholar 

  70. 70.

    De Cicco V, Vivaldi MA: Harnack inequalities for Fuchsian type weighted elliptic equations. Communications in Partial Differential Equations 1996,21(9-10):1321-1347. 10.1080/03605309608821229

    MATH  MathSciNet  Google Scholar 

  71. 71.

    Gutiérrez CrE, Lanconelli E:Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for-elliptic operators. Communications in Partial Differential Equations 2003,28(11-12):1833-1862. 10.1081/PDE-120025487

    MATH  MathSciNet  Google Scholar 

  72. 72.

    Mohammed A: Harnack's inequality for solutions of some degenerate elliptic equations. Revista Matemática Iberoamericana 2002,18(2):325-354.

    MATH  MathSciNet  Google Scholar 

  73. 73.

    Trudinger NS, Wang X-J: On the weak continuity of elliptic operators and applications to potential theory. American Journal of Mathematics 2002,124(2):369-410. 10.1353/ajm.2002.0012

    MATH  MathSciNet  Google Scholar 

  74. 74.

    Zamboni P: Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions. Journal of Differential Equations 2002,182(1):121-140. 10.1006/jdeq.2001.4094

    MATH  MathSciNet  Google Scholar 

  75. 75.

    Fernandes JD, Groisman J, Melo ST: Harnack inequality for a class of degenerate elliptic operators. Zeitschrift für Analysis und ihre Anwendungen 2003,22(1):129-146.

    MATH  MathSciNet  Google Scholar 

  76. 76.

    Ferrari F:Harnack inequality for two-weight subelliptic-Laplace operators. Mathematische Nachrichten 2006,279(8):815-830. 10.1002/mana.200410396

    MATH  MathSciNet  Google Scholar 

  77. 77.

    Gutiérrez CrE, Wheeden RL: Mean value and Harnack inequalities for degenerate parabolic equations. Colloquium Mathematicum 1990,60/61(1):157-194.

    Google Scholar 

  78. 78.

    Gutiérrez CrE, Wheeden RL: Harnack's inequality for degenerate parabolic equations. Communications in Partial Differential Equations 1991,16(4-5):745-770. 10.1080/03605309108820776

    MATH  MathSciNet  Google Scholar 

  79. 79.

    Gutiérrez CrE, Wheeden RL: Bounds for the fundamental solution of degenerate parabolic equations. Communications in Partial Differential Equations 1992,17(7-8):1287-1307.

    MATH  MathSciNet  Google Scholar 

  80. 80.

    Ishige K: On the behavior of the solutions of degenerate parabolic equations. Nagoya Mathematical Journal 1999, 155: 1-26.

    MATH  MathSciNet  Google Scholar 

  81. 81.

    Pascucci A, Polidoro S: On the Harnack inequality for a class of hypoelliptic evolution equations. Transactions of the American Mathematical Society 2004,356(11):4383-4394. 10.1090/S0002-9947-04-03407-5

    MATH  MathSciNet  Google Scholar 

  82. 82.

    Krylov NV, Safonov MV: An estimate for the probability of a diffusion process hitting a set of positive measure. Doklady Akademii Nauk SSSR 1979,245(1):18-20.

    MathSciNet  Google Scholar 

  83. 83.

    Krylov NV, Safonov MV: A property of the solutions of parabolic equations with measurable coefficients. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 1980,44(1):161-175, 239.

    MATH  MathSciNet  Google Scholar 

  84. 84.

    Safonov MV: Harnack's inequality for elliptic equations and Hölder property of their solutions. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI) 1980, 96: 272-287, 312. Boundary value problems of mathematical physics and related questions in the theory of functions, 12

    MATH  MathSciNet  Google Scholar 

  85. 85.

    Nirenberg L: On nonlinear elliptic partial differential equations and Hölder continuity. Communications on Pure and Applied Mathematics 1953, 6: 103-156; addendum, 395. 10.1002/cpa.3160060105

    MATH  MathSciNet  Google Scholar 

  86. 86.

    Cordes HO: Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. Mathematische Annalen 1956,131(3):278-312. 10.1007/BF01342965

    MATH  MathSciNet  Google Scholar 

  87. 87.

    Landis EM: Harnack's inequality for second order elliptic equations of Cordes type. Doklady Akademii Nauk SSSR 1968, 179: 1272-1275.

    MathSciNet  Google Scholar 

  88. 88.

    Nirenberg L: On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations. In Contributions to the Theory of Partial Differential Equations, Annals of Mathematics Studies, no. 33. Princeton University Press, Princeton, NJ, USA; 1954:95-100.

    Google Scholar 

  89. 89.

    Landis EM: Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov. Izdat. "Nauka", Moscow, Russia; 1971:287.

    Google Scholar 

  90. 90.

    Landis EM: Second Order Equations of Elliptic and Parabolic Type, Translations of Mathematical Monographs. Volume 171. American Mathematical Society, Providence, RI, USA; 1998:xii+203.

    Google Scholar 

  91. 91.

    Evans LC: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Communications on Pure and Applied Mathematics 1982,35(3):333-363. 10.1002/cpa.3160350303

    MATH  MathSciNet  Google Scholar 

  92. 92.

    Krylov NV: Boundedly inhomogeneous elliptic and parabolic equations. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 1982,46(3):487-523, 670.

    MATH  MathSciNet  Google Scholar 

  93. 93.

    Krylov NV: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 1983,47(1):75-108.

    MathSciNet  Google Scholar 

  94. 94.

    Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin, Germany; 2001:xiv+517.

    Google Scholar 

  95. 95.

    Caffarelli LA: Interior a priori estimates for solutions of fully nonlinear equations. Annals of Mathematics. Second Series 1989,130(1):189-213. 10.2307/1971480

    MATH  MathSciNet  Google Scholar 

  96. 96.

    Caffarelli LA, Cabré X: Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications. Volume 43. American Mathematical Society, Providence, RI, USA; 1995:vi+104.

    Google Scholar 

  97. 97.

    Yau Sh-T: Harmonic functions on complete Riemannian manifolds. Communications on Pure and Applied Mathematics 1975, 28: 201-228. 10.1002/cpa.3160280203

    MATH  MathSciNet  Google Scholar 

  98. 98.

    Hamilton RS: The Ricci flow on surfaces. In Mathematics and General Relativity (Santa Cruz, CA, 1986), Contemp. Math.. Volume 71. American Mathematical Society, Providence, RI, USA; 1988:237-262.

    Google Scholar 

  99. 99.

    Chow B:The Ricci flow on the-sphere. Journal of Differential Geometry 1991,33(2):325-334.

    MATH  MathSciNet  Google Scholar 

  100. 100.

    Hamilton RS: The Harnack estimate for the Ricci flow. Journal of Differential Geometry 1993,37(1):225-243.

    MATH  MathSciNet  Google Scholar 

  101. 101.

    Andrews B: Harnack inequalities for evolving hypersurfaces. Mathematische Zeitschrift 1994,217(2):179-197.

    MATH  MathSciNet  Google Scholar 

  102. 102.

    Yau Sh-T: On the Harnack inequalities of partial differential equations. Communications in Analysis and Geometry 1994,2(3):431-450.

    MATH  MathSciNet  Google Scholar 

  103. 103.

    Chow B, Chu S-C: A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow. Mathematical Research Letters 1995,2(6):701-718.

    MATH  MathSciNet  Google Scholar 

  104. 104.

    Hamilton RS: Harnack estimate for the mean curvature flow. Journal of Differential Geometry 1995,41(1):215-226.

    MATH  MathSciNet  Google Scholar 

  105. 105.

    Hamilton RS, Yau Sh-T: The Harnack estimate for the Ricci flow on a surface—revisited. Asian Journal of Mathematics 1997,1(3):418-421.

    MATH  MathSciNet  Google Scholar 

  106. 106.

    Chow B, Hamilton RS: Constrained and linear Harnack inequalities for parabolic equations. Inventiones Mathematicae 1997,129(2):213-238. 10.1007/s002220050162

    MATH  MathSciNet  Google Scholar 

  107. 107.

    Cao HD, Chow B, Chu SC, Yau Sh-T (Eds): Collected Papers on Ricci Flow, Series in Geometry and Topology. Volume 37. International Press, Somerville, Mass, USA; 2003:viii+539.

    Google Scholar 

  108. 108.

    Müller R: Differential Harnack inequalities and the Ricci flow. EMS publishing house 2006., 2006, 100 pages:

    Google Scholar 

  109. 109.

    Saloff-Coste L: A note on Poincaré, Sobolev, and Harnack inequalities. International Mathematics Research Notices 1992,1992(2):27-38. 10.1155/S1073792892000047

    MATH  MathSciNet  Google Scholar 

  110. 110.

    Grigor'yan AA: The heat equation on noncompact Riemannian manifolds. Matematicheskiĭ Sbornik 1991,182(1):55-87.

    MATH  Google Scholar 

  111. 111.

    Saloff-Coste L: Parabolic Harnack inequality for divergence-form second-order differential operators. Potential Analysis 1995,4(4):429-467. 10.1007/BF01053457

    MATH  MathSciNet  Google Scholar 

  112. 112.

    Delmotte T: Parabolic Harnack inequality and estimates of Markov chains on graphs. Revista Matemática Iberoamericana 1999,15(1):181-232.

    MATH  MathSciNet  Google Scholar 

  113. 113.

    Sturm KT: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. Journal de Mathématiques Pures et Appliquées. Neuvième Série 1996,75(3):273-297.

    MATH  MathSciNet  Google Scholar 

  114. 114.

    Barlow MT, Bass RF: Coupling and Harnack inequalities for Sierpiński carpets. Bulletin of the American Mathematical Society 1993,29(2):208-212. 10.1090/S0273-0979-1993-00424-5

    MATH  MathSciNet  Google Scholar 

  115. 115.

    Barlow MT, Bass RF: Brownian motion and harmonic analysis on Sierpinski carpets. Canadian Journal of Mathematics 1999,51(4):673-744. 10.4153/CJM-1999-031-4

    MATH  MathSciNet  Google Scholar 

  116. 116.

    Barlow MT, Bass RF: Divergence form operators on fractal-like domains. Journal of Functional Analysis 2000,175(1):214-247. 10.1006/jfan.2000.3597

    MATH  MathSciNet  Google Scholar 

  117. 117.

    Barlow MT, Bass RF: Stability of parabolic Harnack inequalities. Transactions of the American Mathematical Society 2004,356(4):1501-1533. 10.1090/S0002-9947-03-03414-7

    MATH  MathSciNet  Google Scholar 

  118. 118.

    Delmotte T: Graphs between the elliptic and parabolic Harnack inequalities. Potential Analysis 2002,16(2):151-168. 10.1023/A:1012632229879

    MATH  MathSciNet  Google Scholar 

  119. 119.

    Hebisch W, Saloff-Coste L: On the relation between elliptic and parabolic Harnack inequalities. Annales de l'Institut Fourier (Grenoble) 2001,51(5):1437-1481. 10.5802/aif.1861

    MATH  MathSciNet  Google Scholar 

  120. 120.

    Barlow MT: Some remarks on the elliptic Harnack inequality. Bulletin of the London Mathematical Society 2005,37(2):200-208. 10.1112/S0024609304003893

    MATH  MathSciNet  Google Scholar 

  121. 121.

    Bogdan K, Sztonyk P: Estimates of potential Kernel and Harnack's inequality for anisotropic fractional Laplacian. preprint, published as math.PR/0507579 in www.arxiv.org, 2005

  122. 122.

    Bass RF, Kassmann M: Harnack inequalities for non-local operators of variable order. Transactions of the American Mathematical Society 2005,357(2):837-850. 10.1090/S0002-9947-04-03549-4

    MATH  MathSciNet  Google Scholar 

  123. 123.

    Alexopoulos GK: Random walks on discrete groups of polynomial volume growth. Annals of Probability 2002,30(2):723-801.

    MATH  MathSciNet  Google Scholar 

  124. 124.

    Diaconis P, Saloff-Coste L: An application of Harnack inequalities to random walk on nilpotent quotients. Journal of Fourier Analysis and Applications 1995, 189-207.

    Google Scholar 

  125. 125.

    Dodziuk J: Difference equations, isoperimetric inequality and transience of certain random walks. Transactions of the American Mathematical Society 1984,284(2):787-794. 10.1090/S0002-9947-1984-0743744-X

    MATH  MathSciNet  Google Scholar 

  126. 126.

    Grigor'yan A, Telcs A: Harnack inequalities and sub-Gaussian estimates for random walks. Mathematische Annalen 2002,324(3):521-556. 10.1007/s00208-002-0351-3

    MATH  MathSciNet  Google Scholar 

  127. 127.

    Lawler GrF: Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments. Proceedings of the London Mathematical Society. Third Series 1991,63(3):552-568. 10.1112/plms/s3-63.3.552

    MATH  MathSciNet  Google Scholar 

  128. 128.

    Lawler GrF, Polaski ThW: Harnack inequalities and difference estimates for random walks with infinite range. Journal of Theoretical Probability 1993,6(4):781-802. 10.1007/BF01049175

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

  1. Institute of Applied Mathematics, University of Bonn, Beringstrasse 6, Bonn, 53115, Germany

    Moritz Kassmann

Authors
  1. Moritz Kassmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moritz Kassmann.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Kassmann, M. Harnack Inequalities: An Introduction. Bound Value Probl 2007, 081415 (2007). https://doi.org/10.1155/2007/81415

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • General Theory
  • Functional Equation