Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
GRADIENT ESTIMATE AND HARNACK INEQUALITY ON NON-COMPACT RIEMANNIAN MANIFOLDS
 

Summary: GRADIENT ESTIMATE AND HARNACK INEQUALITY
ON NON-COMPACT RIEMANNIAN MANIFOLDS
MARC ARNAUDON, ANTON THALMAIER, AND FENG-YU WANG
Abstract. A gradient-entropy inequality is established for elliptic diffusion semi-
groups on arbitrary complete Riemannian manifolds. As applications, a global Har-
nack inequality with power and a heat kernel estimate are derived without curvature
conditions.
1. The main result
Let M be a non-compact complete connected Riemannian manifold, and Pt be the
Dirichlet diffusion semigroup generated by L = + V for some C2
function V . We
intend to establish reasonable gradient estimates and Harnack type inequalities for Pt.
In case that Ric - HessV is bounded below, a dimension-free Harnack inequality was
established in [15], which according to [17], is indeed equivalent to the corresponding
curvature condition. See e.g. [2] for equivalent statements on heat kernel functional
inequalities; see also [8, 3, 9] for a parabolic Harnack inequality using the dimension-
curvature condition by shifting time, which goes back to the classical local parabolic
Harnack inequality of Moser [10].
Recently, some sharp gradient estimates have been derived in [13, 19] for the Dirichlet
semigroup on relatively compact domains. More precisely, for V = 0 and a relatively

  

Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers

 

Collections: Mathematics