Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T22:07:33.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Gradient Estimates on Rd

Published online by Cambridge University Press:  20 November 2018

Feng-Yu Wang
Feng-Yu Wang*
Affiliation:
Department of Mathematics, Beijing Normal University Beijing, 100875 People's Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper uses both the maximum principle and coupling method to study gradient estimates of positive solutions to Lu = 0 on Rd, where

with (aij ) uniformly positive definite and aij,bi € C1(Rd). We obtain some upper bounds of |∇u |/u and ∥∇u∥∞/∥u∥∞, which imply a Harnack inequality and improve the corresponding results proved in Cranston [4]. Besides, two examples show that our estimates can be sharp.

Keywords

58G3258C50gradient estimatecouplingdiffusion process

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 4 , 01 December 1994, pp. 560 - 570
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Brandt, A., Interior estimates for second-order differential equations via the maximum principle, Israel J. Math. 7(1969), 95–121.Google Scholar
2. Chen, M. F. and Li, S. F, Coupling methods for multidimensional diffussion processes, Ann. Probab. 17 (1989), 151–177.Google Scholar
3. Cranston, M., Gradient estimates on manifolds using coupling, J. Funct. Anal. 99(1991), 110–124.Google Scholar
4. Cranston, M., A probabilistic approach to gradient estimates, Canad. Math. Bull. (1) 35(1992), 46–55.Google Scholar
5. Cranston, M. and Zhao, Z. X., Some regularity results and eigenfunction estimates for the heat equation, Diffusions and Related Problems in Analysis, Vol. I, (ed. Mark Pinsky), Birkhauser, 1990.Google Scholar
6. Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes (Second Edition), North-Holland, 1989.Google Scholar
7. Li, P. and Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156(1986), 153–201.Google Scholar
8. Lindvall, T. and G, L. C.. Rogers, Coupling of multidimensional diffusions by reflection, Ann. Probab. 14(1986), 860–827.Google Scholar
9. Wang, F. Y, Gradient estimates for generalized harmonic functions on manifolds, Chinese Sci. Bull. (6) 39(1994), 492–495.Google Scholar
10. Yau, S. T., Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28(1975), 201–228.Google Scholar