Results 1  10
of
42
ON THE EQUIVALENCE OF THE ENTROPIC CURVATUREDIMENSION CONDITION AND BOCHNER’S INEQUALITY ON METRIC MEASURE SPACES
, 2013
"... ..."
BakryÉmery curvaturedimension condition and Riemannian Ricci curvature bounds
, 2014
"... ..."
(Show Context)
Yau’s gradient estimates on Alexandrov spaces
 University of Bonn, Institute for
"... ar ..."
(Show Context)
Eigenvalues of Laplacian and multiway isoperimetric constants on weighted Riemannian manifolds
"... Abstract. We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative BakryÉmery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantita ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract. We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative BakryÉmery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multiway isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multiway isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the BuserLedoux result to the kth eigenvalue and the kway isoperimetric constant. As a consequence the kth eigenvalue of the weighted Laplacian and the kway isoperimetric constant are equivalent up to polynomials of k on closed weighted manifolds of nonnegative BakryÉmery Ricci curvature.
On Harnack inequality and optimal transportation
, 2013
"... Abstract. – We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetrictype Harnack inequality is emphasized. Commutation properties between the heat ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. – We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetrictype Harnack inequality is emphasized. Commutation properties between the heat and HopfLax semigroups are developed consequently, providing direct access to heat flow contraction in Wasserstein spaces. 1.
TOPOLOGYPRESERVING DIFFUSION OF DIVERGENCEFREE VECTOR FIELDS AND MAGNETIC RELAXATION
, 2013
"... The usual heat equation is not suitable to preserve the topology of divergencefree vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topologypreserving diffusion equations for divergencefree vector fields. ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
The usual heat equation is not suitable to preserve the topology of divergencefree vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topologypreserving diffusion equations for divergencefree vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of ”dissipative solutions”, which shares common features with both P.L. Lions ’ dissipative solutions to the Euler equations and the concept of ”curves ofmaximal slopes”, à la DeGiorgi, recently used to study thescalar heat equation in very general metric spaces. We show that the initial value problem admits such global solutions, at least in the two space variable case, and they are unique whenever they are smooth.