8,605 research outputs found
Local gradient estimate for harmonic functions on Finsler manifolds
In this paper, we prove the local gradient estimate for harmonic functions on
complete, noncompact Finsler measure spaces under the condition that the
weighted Ricci curvature has a lower bound. As applications, we obtain
Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.Comment: Calc. Var. to appea
A Minkowski type inequality in space forms
In this note we apply the general Reilly formula established in \cite{QX} to
the solution of a Neumann boundary value problem to prove an optimal Minkowski
type inequality in space forms.Comment: 8 pages, an equivalent statement of the main theorem is adde
Inverse anisotropic mean curvature flow and a Minkowski type inequality
In this paper, we show that the inverse anisotropic mean curvature flow in
, initiating from a star-shaped, strictly -mean convex
hypersurface, exists for all time and after rescaling the flow converges
exponentially fast to a rescaled Wulff shape in the topology. As an
application, we prove a Minkowski type inequality for star-shaped, -mean
convex hypersurfaces.Comment: final version, to appear in Adv. Mat
On an anisotropic Minkowski problem
In this paper, we study the anisotropic Minkowski problem. It is a problem of
prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly
convex hypersurface in Euclidean space as a function on its anisotropic normals
in relative or Minkowski geometry. We first formulate such problem to a
Monge-Amp\'ere type equation on the anisotropic support function and then prove
the existence and uniqueness of the admissible solution to such equation. In
conclusion, we give an affirmative answer to the anisotropic Minkowski problem.Comment: 28 page
Locally constrained inverse curvature flows
We consider inverse curvature flows in warped product manifolds, which are
constrained subject to local terms of lower order, namely the radial coordinate
and the generalized support function. Under various assumptions we prove
longtime existence and smooth convergence to a coordinate slice. We apply this
result to deduce a new Minkowski type inequality in the anti-de-Sitter
Schwarzschild manifolds and a weighted isoperimetric type inequality in the
hyperbolic space.Comment: The proof of Proposition 7.6 has been minor revise
On Rigidity of hypersurfaces with constant curvature functions in warped product manifolds
In this paper, we first investigate several rigidity problems for
hypersurfaces in the warped product manifolds with constant linear combinations
of higher order mean curvatures as well as "weighted'' mean curvatures, which
extend the work \cite{Mon, Brendle,BE} considering constant mean curvature
functions. Secondly, we obtain the rigidity results for hypersurfaces in the
space forms with constant linear combinations of intrinsic Gauss-Bonnet
curvatures . To achieve this, we develop some new kind of Newton-Maclaurin
type inequalities on which may have independent interest.Comment: 24 pages, Ann. Glob. Anal. Geom. to appea
Renormalized solutions for the fractional p(x)-Laplacian equation with L^1 data
In this paper, we prove the existence and uniqueness of nonnegative
renormalized solutions for the fractional p(x)-Laplacian problem with L1 data.
Our results are new even in the constant exponent fractional p-Laplacian
equation case.Comment: 22 page
An optimal anisotropic Poincar\'e inequality for convex domains
In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue
of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we
prove an optimal anisotropic Poincar\'e inequality for convex domains, which
generalizes the result of Payne-Weinberger. A lower bound of the first
(nonzero) eigenvalue of Finsler-Laplacian with the Dirichlet boundary condition
is also proved.Comment: 18 page
A note on local gradient estimate on Alexandrov spaces
In this note, we prove Cheng-Yau type local gradient estimate for harmonic
functions on Alexandrov spaces with Ricci curvature bounded below. We adopt a
refined version of Moser's iteration which is based on Zhang-Zhu's Bochner type
formula. Our result improves the previous one of Zhang-Zhu in the case of
negative Ricci lower bound.Comment: 10 pages, a revision, to appear in Tohoku Math. J. (2
Christoffel-Minkowski problem: the case
We consider a fully nonlinear partial differential equation associated to the
intermediate Christoffel-Minkowski problem in the case . We
establish the existence of convex body with prescribed -th even -area
measure on , under an appropriate assumption on the prescribed
function. We construct examples to indicate certain geometric condition on the
prescribed function is needed for the existence of smooth strictly convex body.
We also obtain regularity estimates for admissible solutions of the
equation when .Comment: 23 page
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