共查询到20条相似文献,搜索用时 14 毫秒
1.
Albert Chau Jingyi Chen Weiyong He 《Calculus of Variations and Partial Differential Equations》2012,44(1-2):199-220
We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in ${{\mathbb R}^{2n}}$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t?=?0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L ∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow. 相似文献
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Rongli Huang Zhizhang Wang 《Calculus of Variations and Partial Differential Equations》2011,41(3-4):321-339
The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|2 D 2 u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? i are the eigenvalues of the Hessian D 2 u. 相似文献
4.
We observe that the comparison result of Barles–Biton–Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow. To cite this article: J. Chen, C. Pang, C. R. Acad. Sci. Paris, Ser. I 347 (2009). 相似文献
5.
Knut Smoczyk 《Mathematische Zeitschrift》2002,240(4):849-883
We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. .
Received: 11 April 2001 / Published online: 29 April 2002 相似文献
6.
Knut Smoczyk 《Calculus of Variations and Partial Differential Equations》1999,8(3):247-258
If is a Lagrangian manifold immersed into a K?hler-Einstein manifold, nothing is known about its behavior under the mean curvature
flow. As a first result we derive a Harnack inequality for the mean curvature potential of compact Lagrangian immersions immersed into .
Received March 16, 1997 / Accepted April 24, 1998 相似文献
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Vittorio Martino Annamaria Montanari 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(3-4):377-390
We prove interior gradient estimates of viscosity solutions of the prescribed Levi mean curvature equation.
The second author was partially supported by Indam, within the interdisciplinary project “Nonlinear subelliptic equations
of variational origin in contact geometry”. 相似文献
9.
Singularity of mean curvature flow of Lagrangian submanifolds 总被引:6,自引:0,他引:6
In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold 0 is Lagrangian and almost calibrated by Re in a Calabi-Yau n-fold (M,), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X0,T) is a stationary Lagrangian integer multiplicity current in R2n with volume density greater than one at X0. When n=2, the tangent cone is a finite union of at least two 2-planes in R4 which are complex in a complex structure on R4. 相似文献
10.
Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform
-bounds in space and C2-estimates in time for the underlying Monge-Ampére equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result.Received: 3 September 2002, Accepted: 12 June 2003, Published online: 4 September 2003Mathematics Subject Classification (2000):
53C44 相似文献
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André Neves 《Inventiones Mathematicae》2007,168(3):449-484
We study singularities of Lagrangian mean curvature flow in ℂ
n
when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are
unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian
isotopic to a plane that, nevertheless, develop finite time singularities under mean curvature flow.
We then prove two theorems regarding the tangent flow at a singularity when the initial condition is a zero-Maslov class Lagrangian.
The first one (Theorem A) states that that the rescaled flow at a singularity converges weakly to a finite union of area-minimizing
Lagrangian cones. The second theorem (Theorem B) states that, under the additional assumptions that the initial condition
is an almost-calibrated and rational Lagrangian, connected components of the rescaled flow converges to a single area-minimizing Lagrangian cone, as opposed to a possible non-area-minimizing union of area-minimizing Lagrangian cones.
The latter condition is dense for Lagrangians with finitely generated H
1(L,ℤ). 相似文献
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Marcos Dajczer Pedro A. Hinojosa Jorge Herbert de Lira 《Calculus of Variations and Partial Differential Equations》2008,33(2):231-248
It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian
manifolds.
M. Dajczer was partially supported by Procad, CNPq and Faperj. P. A. Hinojosa was partially supported by PADCT/CT-INFRA/CNPq/MCT
Grant #620120/2004-5. J. H. de Lira was partially supported by CNPq and Funcap. 相似文献
15.
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time. 相似文献
16.
In the Euclidean Space
\mathbb Rn+1{\mathbb {R}^{n+1}} with a density
ee\frac12 n m2 |x|2, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account
of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere)
depending on the relation between the bound of its principal curvatures and μ. 相似文献
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In this paper, we study the singularities of the mean curvature ?ow from a symplectic surface or from a Lagrangian surface in a K?hler-Einstein surface. We prove that the blow-up ?ow ∑ s ∞ at a singular point(X 0, T 0) of a symplectic mean curvature ?ow Σt or of a Lagrangian mean curvature ?ow Σt is a nontrivial minimal surface in ? 4 , if ∑ - ∞ ∞ is connected. 相似文献
19.
Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests. 相似文献
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We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established. 相似文献