Interior gradient estimates for mean curvature type equations and related flows

2021 ◽  
pp. 421-442
Keyword(s):  
Mean Curvature ◽  
Gradient Estimates ◽  
Curvature Type

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Gradient estimates of mean curvature equation with Neumann boundary condition in domains of Riemannian manifold

2017 ◽  
Vol 28 (08) ◽  
pp. 1750065
Author(s):  
Jinju Xu ◽  
Dekai Zhang
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Existence Result ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
The Maximum Principle ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation

We study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

scholarly journals On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

2020 ◽  
Vol 59 (4) ◽  
Author(s):  
Leonardo Bonorino ◽  
Jean-Baptiste Casteras ◽  
Patricia Klaser ◽  
Jaime Ripoll ◽  
Miriam Telichevesky
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mean Curvature ◽  
Partial Differential ◽  
Curvature Type

Gradient Estimates for Spacelike Mean Curvature Flow with Boundary Conditions

2018 ◽  
Vol 62 (2) ◽  
pp. 459-469
Author(s):  
Ben Lambert
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Mean Curvature Flow ◽  
Neumann Boundary ◽  
Curvature Flow ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
The Mean

AbstractWe prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

scholarly journals Counterexamples to C2 Boundary Estimates for a Fully Nonlinear Yamabe Problem on Manifolds with Boundary

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
YanYan Li ◽  
Luc Nguyen
Keyword(s):  
Mean Curvature ◽  
A Priori Estimates ◽  
A Priori ◽  
Gradient Estimates ◽  
Manifolds With Boundary ◽  
Fully Nonlinear ◽  
Priori Estimates ◽  
Hessian Estimates ◽  
The Mean ◽  
Boundary Estimates

AbstractOn closed manifolds, gradient and Hessian a priori estimates for fully nonlinear Yamabe problems are known to hold. On manifolds with boundary, gradient estimates are known to hold, while Hessian estimates hold if the prescribed mean curvature is positive. Examples are given here which show that Hessian estimates can fail when the mean curvature is negative.

scholarly journals Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Fangcheng Guo ◽  
Guanghan Li ◽  
Chuanxi Wu
Keyword(s):  
Boundary Condition ◽  
Mean Curvature ◽  
Convex Cone ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Type Flow ◽  
Maximal Principle ◽  
Curvature Type

We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially asttends to infinity.

Global Gradient Estimate of the Mean Curvature Type Equation

2021 ◽  
Vol 11 (06) ◽  
pp. 1130-1136
Author(s):  
玉苏普 阿迪莱•
Keyword(s):  
Mean Curvature ◽  
Type Equation ◽  
Gradient Estimate ◽  
The Mean ◽  
Curvature Type
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