×

Stability results for Harnack inequalities. (English) Zbl 1115.58024

Summary: We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically non-negative sectional curvature.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML Link

References:

[1] Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73, 890-896, (1967) · Zbl 0153.42002
[2] Brownian motion and harmonic analysis on sierpinski carpets, Canad. J. Math., 54, 673-744, (1999) · Zbl 0945.60071
[3] Random walks on graphical sierpinski carpets, Random walks and discrete potential theory (Cortona, Italy, 1997), 39, 26-55, (1999), Cambridge Univ. Press, Cambridge · Zbl 0958.60045
[4] A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., 15, 213-230, (1982) · Zbl 0501.53030
[5] Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc., 24, 371-377, (1991) · Zbl 0728.53026
[6] Eigenvalues in Riemannian geometry, (1984), Academic Press, New York · Zbl 0551.53001
[7] Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., 17, 15-53, (1982) · Zbl 0493.53035
[8] A lower bound for the heat kernel, Comm. Pure Appl. Math., 34, 465-480, (1981) · Zbl 0481.35003
[9] Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28, 333-354, (1975) · Zbl 0312.53031
[10] Spectral Graph Theory, 92, (1996), Amer. Math. Soc. Publications · Zbl 0867.05046
[11] Variétés riemanniennes isométriques à l’infini, Revista Matematica Iberoamericana, 11, 3, 687-726, (1995) · Zbl 0845.58054
[12] Graphs between elliptic and parabolic Harnack inequalities, Potential Analysis, 16, 2, 151-168, (2000) · Zbl 1081.39012
[13] What do we know about the metropolis algorithm?, J. Computer and System Sciences, 57, 20-36, (1998) · Zbl 0920.68054
[14] Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., 32, 5, 703-716, (1983) · Zbl 0526.58047
[15] A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash, Arch. Rat. Mech. Anal., 96, 327-338, (1986) · Zbl 0652.35052
[16] Function theory of manifolds which possess a pole, 699, (1979), Springer · Zbl 0414.53043
[17] The heat equation on non-compact riemannian manifolds (Russian), Mat. Sbornik, 182, 1, 55-87, (1991) · Zbl 0743.58031
[18] Heat kernel upper bounds on a complete non-compact manifold, Revista Matematica Iberoamericana, 10, 2, 395-452, (1994) · Zbl 0810.58040
[19] Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36, 135-249, (1999) · Zbl 0927.58019
[20] Surgery of Faber-Krahn inequalities and applications to heat kernel upper bounds on manifolds with ends, (2000)
[21] Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math., 55, 93-133, (2002) · Zbl 1037.58018
[22] Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures et Appl., 81, 115-142, (2002) · Zbl 1042.58022
[23] Structures métriques pour les variétés Riemannienes, (1981), Cedic/Ferdnand Nathan, Paris · Zbl 0509.53034
[24] Sobolev Met Poincaré, 688, (2000), Memoirs of the AMS · Zbl 0954.46022
[25] On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier, 51, 5, 1437-1481, (2001) · Zbl 0988.58007
[26] The Poincaré inequality for vector fields satisfying Hörmander condition, Duke Math. J., 53, 503-523, (1986) · Zbl 0614.35066
[27] Analytic inequalities, and rough isometries between non-compact Riemannian manifolds, 1201, 122-137, (1986), Springer · Zbl 0593.53026
[28] Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I., Geometry and Analysis on Manifolds (Katata/Kyoto, 1987), 158-181, (1988), Springer · Zbl 0685.31004
[29] Prescribing curvatures, Proceedings of Symposia in Pure Mathematics, 27, 2, 309-319, (1975) · Zbl 0313.53017
[30] Application of Malliavin calculus, III, J. Fac. Sci. Tokyo Univ., Sect. 1A, Math., 34, 391-442, (1987) · Zbl 0633.60078
[31] The second order equations of elliptic and parabolic type (Russian), (1971), Nauka, Moscow · Zbl 0226.35001
[32] Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. Math., 125, 171-207, (1987) · Zbl 0622.58001
[33] Green’s function, harmonic functions and volume comparison, J. Diff. Geom., 41, 277-318, (1995) · Zbl 0827.53033
[34] On the parabolic kernel of the Schrödinger operator, Acta Math., 156, 3-4, 153-201, (1986) · Zbl 0611.58045
[35] Ball covering property and nonnegative Ricci curvature outside a compact set, Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), 54, Part 3, 459–464 pp., (1993), Amer. Math. Soc., Providence, RI · Zbl 0788.53028
[36] Some Liouville theorems on riemannian manifolds of a special type (Russian), Izv. Vyssh. Uchebn. Zaved. Matematika, 12, 15-24, (1991) · Zbl 0764.58035
[37] Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Revista Matematica Iberoamericana, 8, 3, 367-439, (1992) · Zbl 0804.35015
[38] Brownian motion and harmonic functions on rotationally symmetric manifolds, Ann. Prob., 14, 3, 793-801, (1986) · Zbl 0593.60078
[39] On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, 577-591, (1961) · Zbl 0111.09302
[40] A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17, 101-134, (1964) · Zbl 0149.06902
[41] Positive harmonic functions on rotationary symmetric Riemannian manifolds, Potential Theory, 251-259, (1992), Walter de Gruyter, Berlin · Zbl 0777.53039
[42] Two-side estimates of fundamental solutions of second-order parabolic equations and some applications (Russian), Uspekhi Matem. Nauk, 39, 3, 101-156, (1984) · Zbl 0582.35052
[43] A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, 2, 27-38, (1992) · Zbl 0769.58054
[44] Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, 4, 429-467, (1995) · Zbl 0840.31006
[45] Lectures on finite Markov chains,, (1997), Springer · Zbl 0885.60061
[46] Aspects of Sobolev inequalities, 289, (2002), Cambridge Univ. Press · Zbl 0991.35002
[47] Sharp estimates for capacities and applications to symmetrical diffusions, Probability theory and related fields, 103, 1, 73-89, (1995) · Zbl 0828.60062
[48] Spaces of harmonic functions, J. London Math. Soc., 2, 3, 789-806, (2000) · Zbl 0963.31004
[49] Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28, 201-228, (1975) · Zbl 0291.31002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.