[1] Schaefer P W, Sperb B P. Maximum principles and bounds in some inhomogeneous elliptic boundary value problems. SIAM J Math Anal, 1977, 8: 871--878
[2] Hopf E. A remark on linear elliptic differential equations of second order. Proc Amer Math Soc, 1952, 3: 791--793
[3] Payne L E. Bounds for the maximum stress in the Saint-Venant torsion problem. Indian J Mech Math, Special issue, 1968, 51--59
[4] Webb J R L. Maximum principles for functionals associated with the solution of semilinear elliptic boundary value problems, Zeitschrift fr Angewandte Mathematik und Physik, 1989, 40:330-338.
[5] Sperb R P. Maximum Principles and Their Applications.New York: Academic Press, 1981
[6] Lott J. Some geometric properties of the Bakry-Emery-Ricci tensor. Comm Math Helv, 2003, 78: 865--883
[7] Wei G F, Wylie W. Comparison geometry for the Bakry-Emery Ricci tensor. J Different Geometry, 2009, 83(2): 377--405
[8] Qian Z. Estimates for weighted volumes and applications. Quart J Math, 1997, 48(2): 235--242
[9] Ruan Q H. Bakry-Emery curvature operator and Ricci flow. Potential Analysis, 2006, 25(4): 399--406
[10] Ruan Q H. Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature. Proc Japan Acad Ser A Math Sci, 2009, 85(6): 71--74
[11] Sturm K T. On the geometry of metric measure spaces. I. Acta Math, 2006, 196(1): 65--131
[12] Sturm K T. On the geometry of metric measure spaces. II. Acta Math, 2006, 196(1): 133--177
[13] Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal mass transport. Ann of Math, 2009, 169: 903--991
[14] Ohta S, Sturm K T. Heat flow on Finsler manifolds.Comm Pure Appl Math, 2009, 62: 1386--1433
[15] Ohta S. Finsler interpolation inequalities. Calc Var Partial Differential Equations, 2009, 36: 211--249
[16] Li X D. Liouville theorem for symmeric diffusion operators on complete Riemannian manifolds. J Math Pure Appl, 2005, 84: 1295--1361
[17] Wu J Y. Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Emery Ricci curvature. J Mathematical Analysis and Applications, 2010, 361(1): 10--18
[18] Huang G Y, Ma B Q. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Archiv der Mathematik , 2010, 94(3): 265--275
[19] Qian B. Elliptic gradient estimates for diffusion operators on complete Riemannian manifolds.Acta Math Sci, 2010, 30B(5): 1555--1560
[20] Ruan Q H. Elliptic-type gradient estimate schr\"{o}dinger equations on noncompact manifolds. Bull London Math Soc, 2007,39: 982--988
[21] Ruan Q H. Harnack inequality and Liouville properties for L-harmonic operator. J Math Anal Appl, 2007, 329: 1430--1439
[22] Payne L E, Stakgold I.On the mean value of the fundamental mode in the fixed membrane problem. Applicable Anal, 1973, 3: 295--306 |