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Global existence of solutions to Schrödinger equations on compact riemannian manifolds below H 1

Sijia Zhong (2010)

Bulletin de la Société Mathématique de France

In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. s < 1 , under some bilinear Strichartz assumption. We will find some s ˜ < 1 , such that the solution is global for s > s ˜ .

Gradient estimates for a nonlinear equation Δ f u + c u - α = 0 on complete noncompact manifolds

Jing Zhang, Bingqing Ma (2011)

Communications in Mathematics

Let ( M , g ) be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation Δ f u + c u - α = 0 in M , where α , c are two real constants and α > 0 , f is a smooth real valued function on M and Δ f = Δ - ∇ f ∇ . When N is finite and the N -Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that ∞ -Bakry-Emery Ricci tensor is bounded from below and | ∇ f | is bounded from above,...

Gradient estimates for inverse curvature flows in hyperbolic space

Julian Scheuer (2015)

Geometric Flows

We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p > 1 of F-1 and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface....

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds ( M , g ) for the following general heat equation u t = Δ V u + a u log u + b u where a is a constant and b is a differentiable function defined on M × [ 0 , ∞ ) . We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.

Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman, Stéphane Nonnenmacher (2007)

Annales de l’institut Fourier

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

Currently displaying 561 – 580 of 1746