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Gradient estimates and Harnack inequalities on noncompact Riemannian manifolds, Stochastic Process
 Appl
"... Abstract. A new type of gradient estimate is established for diffusion semigroups on noncompact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1. The m ..."
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Cited by 34 (15 self)
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Abstract. A new type of gradient estimate is established for diffusion semigroups on noncompact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1. The main result Let M be a noncompact complete Riemannian manifold, and Pt be the Dirichlet diffusion semigroup generated by L = ∆ + ∇V for some C 2 function V. We intend to establish reasonable gradient estimates and Harnack type inequalities for Pt. In case that Ric − HessV is bounded below, a dimensionfree 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 dimensioncurvature 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 compact open C2 domain D, the Dirichlet heat semigroup P D t satisfies (1.1) ∇P D t f(x) ≤ C(x, t)P D t f(x), x ∈ D, t> 0, for some locally bounded function C: D ×]0, ∞ [ →]0, ∞ [ and all f ∈ B + b, the space of bounded nonnegative measurable functions on M. Obviously, this implies the Harnack inequality (1.2) P D t f(x) ≤ ˜ C(x, y, t)P D t f(y), t> 0, x, y ∈ D, f ∈ B+ b, for some function ˜ C: M 2 ×]0, ∞ [ →]0, ∞[. The purpose of this paper is to establish inequalities analogous to (1.1) and (1.2) globally on the whole manifold M. On the other hand however, both (1.1) and (1.2) are in general wrong for Pt in place of P D t. A simple counterexample is already the standard heat semigroup on R d. Hence, we turn to search for the following slightly weaker version of gradient
SOME GRADIENT ESTIMATES FOR THE HEAT EQUATION ON DOMAINS AND FOR AN EQUATION BY PERELMAN
, 2006
"... Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bo ..."
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Cited by 29 (2 self)
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Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound. Contents
GRADIENT ESTIMATES FOR THE DEGENERATE PARABOLIC EQUATION ut = ∆F(u) ON MANIFOLDS AND SOME LIOUVILLE THEOREMS OF POROUS MEDIA EQUATIONS
, 805
"... Abstract. In this paper, we first prove a gradient estimate for the positive solutions of the ecumenic degenerate parabolic equation: ut = ∆F(u), with F ′ (u)> 0, on a complete Riemannian manifold with Ricci curvature lower bound Ric(M) ≥ −k with k ≥ 0. The second part, we apply the gradient est ..."
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Abstract. In this paper, we first prove a gradient estimate for the positive solutions of the ecumenic degenerate parabolic equation: ut = ∆F(u), with F ′ (u)> 0, on a complete Riemannian manifold with Ricci curvature lower bound Ric(M) ≥ −k with k ≥ 0. The second part, we apply the gradient estimates to Porous Media Equations: F(u) = u p, p> 0, to obtain the gradient estimates in a larger range of p than the range of p for Harnack inequalities and Cauchy problems in the literature, and also prove some Liouville theorems for positive global solutions on noncompact complete manifolds with nonnegative Ricci curvature for the Porous Media Equations. 1.
doi:10.1112/blms/bdm089 ELLIPTICTYPE GRADIENT ESTIMATE FOR SCHRÖDINGER EQUATIONS ON NONCOMPACT MANIFOLDS
"... In this paper, the author discusses an elliptictype gradient estimate for the solution of the timedependent Schrödinger equations on noncompact manifolds. As an application, the dimensionfree Harnack inequality and a Liouvilletype theorem for the Schrödinger equation are proved. 1. ..."
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In this paper, the author discusses an elliptictype gradient estimate for the solution of the timedependent Schrödinger equations on noncompact manifolds. As an application, the dimensionfree Harnack inequality and a Liouvilletype theorem for the Schrödinger equation are proved. 1.