scholarly journals On the Strong Maximum Principle and the Large Time Behavior of Generalized Mean Curvature Flow with the Neumann Boundary Condition

1999 ◽  
Vol 154 (1) ◽  
pp. 107-131 ◽  
Author(s):  
Giga Yoshikazu ◽  
Masaki Ohnuma ◽  
Moto-Hiko Sato
Keyword(s):  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Large Time ◽  
Mean Curvature Flow ◽  
Neumann Boundary ◽  
Curvature Flow ◽  
Large Time Behavior ◽  
Strong Maximum Principle ◽  
Time Behavior ◽  
Generalized Mean Curvature

scholarly journals Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

2020 ◽  
Author(s):  
Nils Dabrock ◽  
Martina Hofmanová ◽  
Matthias Röger
Keyword(s):  
Mean Curvature ◽  
Large Time ◽  
Mean Curvature Flow ◽  
Stochastic Perturbation ◽  
Curvature Flow ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Viscosity Sense ◽  
Stochastic Mean Curvature Flow ◽  
Martingale Solutions

Abstract We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $$L^{\infty }_{\omega ,x,t}$$ L ω , x , t ∞ estimate for the gradient and an $$L^{2}_{\omega ,x,t}$$ L ω , x , t 2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

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 On obstacle problem for mean curvature flow with driving force

2019 ◽  
Vol 4 (1) ◽  
pp. 9-29
Author(s):  
Yoshikazu Giga ◽  
Hung V. Tran ◽  
Longjie Zhang
Keyword(s):  
Mean Curvature ◽  
Driving Force ◽  
Obstacle Problem ◽  
Mean Curvature Flow ◽  
Boundary Regularity ◽  
Curvature Flow ◽  
Convergence Result ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Full Characterization

Abstract In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

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