A Note on Krylov-Tso's Parabolic Inequality

1992 ◽  
Vol 115 (4) ◽  
pp. 1053
Author(s):  
Luis Escauriaza
Keyword(s):  
Parabolic Inequality

Strongly nonlinear parabolic inequality in orlicz spaces via a sequence of penalized equations

2015 ◽  
Vol 26 (7-8) ◽  
pp. 1669-1695 ◽  
Author(s):  
Y. Akdim ◽  
J. Bennouna ◽  
M. Mekkour ◽  
H. Redwane
Keyword(s):  
Orlicz Spaces ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
Parabolic Inequality

scholarly journals A note on Krylov-Tso’s parabolic inequality

1992 ◽  
Vol 115 (4) ◽  
pp. 1053-1053
Author(s):  
Luis Escauriaza
Keyword(s):  
Parabolic Inequality

scholarly journals A logarithmic Sobolev form of the Li-Yau parabolic inequality

2006 ◽  
pp. 683-702 ◽  
Author(s):  
Dominique Bakry ◽  
Michel Ledoux
Keyword(s):  
Parabolic Inequality

Curvature Pinching Based on Integral Norms of the Curvature

1993 ◽  
Vol 45 (3) ◽  
pp. 599-611 ◽  
Author(s):  
Miroslav Lovrić
Keyword(s):  
Curvature Tensor ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Negative Number ◽  
Crucial Step ◽  
Nonlinear Parabolic ◽  
Sobolev Constant ◽  
Parabolic Inequality ◽  
Curvature Pinching ◽  
Negative Sectional Curvature

AbstractA compact Riemannian manifold (M, g) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in LP norm for p > n/2, and the metric is almost spherical or almost hyperbolic in the LP sense. The idea of the proof is to obtain stronger (i.e. L∞) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case rg < 0 and of Grove, Karcher and Ruh in the case rg > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.

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