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.
This paper is concerned with the large time behavior of disturbed planar fronts in the buffered bistable system inℝn(n≥2). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up tot=+∞. And then we prove that the planar front is asymptotically stable inL∞(ℝn)under ergodic perturbations, which include quasiperiodic and almost periodic ones as special cases.
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.
Abstract
In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.
Abstract
Let
F_{n}
: (Σ,
h_{n}
)
\to
\mathbb{C}^{2}
be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics
\{
h_{n}
\}
converges smoothly to a Riemannian metric h. We show that a subsequence of
\{
F_{n}
\}
converges smoothly to a branched conformally immersed Lagrangian self-shrinker
F_{\infty}
: (Σ, h)
\to
\mathbb{C}^{2}
. When the area bound is less than 16π, the limit
{F_{\infty}}
is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence
h_{n}
\to
h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.
Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs