共查询到20条相似文献,搜索用时 15 毫秒
1.
A logarithmic Gauss curvature flow and the Minkowski problem 总被引:1,自引:0,他引:1
Kai-Seng Chou Xu-Jia Wang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2000,17(6):733
Let X0 be a smooth uniformly convex hypersurface and f a postive smooth function in Sn. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX0 along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ*>0 such that if θ<θ*, they shrink to a point in finite time and, if θ>θ*, they expand to an asymptotic sphere. Finally, when θ=θ*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x). 相似文献
2.
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc. 相似文献
3.
Let M^n be a smooth, compact manifold without boundary, and F0 : M^n→ R^n+1 a smooth immersion which is convex. The one-parameter families F(·, t) : M^n× [0, T) → R^n+1 of hypersurfaces Mt^n= F(·,t)(M^n) satisfy an initial value problem dF/dt (·,t) = -H^k(· ,t)v(· ,t), F(· ,0) = F0(· ), where H is the mean curvature and u(·,t) is the outer unit normal at F(·, t), such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton. 相似文献
4.
In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study
the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short).
Requiring that, for each time t ≥ 0, the evolving hypersurface M
t
meets such tgh orthogonally, we prove that: (a) the flow exists while M
t
does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M
t
remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set
(if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature. 相似文献
5.
Thomas Marquardt 《Mathematische Zeitschrift》2010,264(3):507-511
In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on ${x \in \varOmega \subset \mathbb {R}^n}In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on
x ? \varOmega ì \mathbb Rn{x \in \varOmega \subset \mathbb {R}^n}, the height u of the hypersurface M = graph u over
\varOmega{\varOmega} and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of
?\varOmega{\partial \varOmega} that guarantees the existence of smooth solutions even for not necessarily convex domains. 相似文献
6.
Bernhard Hein 《Calculus of Variations and Partial Differential Equations》2007,28(2):249-273
In
, n < 7, we treat the quasilinear, degenerate parabolic initial and boundary value problem which is the natural parabolic extension of Huisken and Ilmanen’s weak inverse mean curvature flow (IMCF). We prove long time existence and partial uniqueness of Lipschitz continuous weak solutions u(x,t) and show C
1,α-regularity for the sets ∂{x| u(x,t) < z }. Our approach offers a new approximation for weak solutions of the IMCF starting from a class of interesting and easily obtainable initial values; for these, the above sets are shown to converge against corresponding surfaces of the IMCF as t → ∞ globally in Hausdorff distance and locally uniformly with respect to the C
1,α-norm.Research partially supported by the DFG, SFB 382 at Tübingen University 相似文献
7.
A smooth, compact and strictly convex hypersurface evolving in ℝ
n+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We
show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink
to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough.
The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper.
Long-time existence and convergence of normalization of the flow are also investigated. 相似文献
8.
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following
situation. Letc(t) be a curve in a space formM
λ
n
of sectional curvature λ. LetP
0 be a totally geodesic hypersurface ofM
λ
n
throughc(0) and orthogonal toc(t). LetC
0 be a hypersurface ofP
0. LetC be the hypersurface ofM
λ
n
obtained by a motion ofC
0 alongc(t). We shall denote it byC
PorC
Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC
0, then volume(C) ≥ volume(C),P),and the equality holds whenC
0 is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion).
Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052. 相似文献
9.
10.
Luis Silvestre 《纯数学与应用数学通讯》2007,60(1):67-112
Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:
- u ≥ φ in ?n,
- (??)su ≥ 0 in ?n,
- (??)su(x) = 0 for those x such that u(x) > φ(x),
- lim|x| → + ∞ u(x) = 0.
11.
Let (M,g) be an n-dimensional Riemannian manifold and T*M be its cotangent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compatible paracomplex structures on T*M. Second, we construct locally decomposable Golden Riemannian structures on T*M. Finally we investigate curvature properties of T*M. 相似文献
12.
The Blow-up Locus of Heat Flows for Harmonic Maps 总被引:5,自引:0,他引:5
Abstract
Let M and N be two compact Riemannian manifolds. Let u
k
(x, t) be a sequence of strong stationary weak heat flows from M×R
+ to N with bounded energies. Assume that u
k→u weakly in H
1, 2(M×R
+, N) and that Σt is the blow-up set for a fixed t > 0. In this paper we first prove Σt is an H
m−2-rectifiable set for almost all t∈R
+. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the
limiting map u is also a strong stationary weak heat flow, Σt is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly
quasi-harmonic sphere and a blow-up set ∪t<0Σt× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.
This work is supported by NSF grant 相似文献
13.
Gabjin Yun 《Geometriae Dedicata》2002,89(1):133-139
Let M
n
, n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R
n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C
s
, where C
s
is the Sobolev constant for M, then there are no L
2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane. 相似文献
14.
Simeon M. Berman 《纯数学与应用数学通讯》2007,60(8):1238-1259
Let f(x), x ∈ ?M, M ≥ 1, be a density function on ?M, and X1, …., Xn a sample of independent random vectors with this common density. For a rectangle B in ?M, suppose that the X's are censored outside B, that is, the value Xk is observed only if Xk ∈ B. The restriction of f(x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {(u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let Km, n(x) be the classical kernel estimator of f(x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|Km, n(x) ? f(x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc. 相似文献
15.
Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons’ cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C0 sense to a minimal hypersurface which is tangent to Simons’ cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. 相似文献
16.
In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u t (x, t) = (?1) n?1 (J * Id ? 1) n (u(x, t)), x ∈ ? N , which is the nonlocal analogous to the higher order local evolution equation v t = (?1) n?1(Δ) n v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. 相似文献
17.
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
18.
Horatio Quadjovie 《Bulletin des Sciences Mathématiques》2004,128(6):447-466
A rotationally symmetric, compact, oriented, connected, uniformly convex hypersurface M0 of , with boundary ∂M0 in a rotationally symmetric cone S, is evolving under volume-preserving mean curvature flow. Then for n?2, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a part of a round sphere. 相似文献
19.
We consider an inverse boundary value problem for the heat equation ? t u = div (γ? x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| t=0 = u 0, in a bounded domain Ω ? ? n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k 2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?ν u(t, x)|(0, T)×?Ω. The knowledge of the initial data u 0 is not used in the proof. If we know min0≤t≤T (sup x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking. 相似文献
20.
Makoto Tsukada 《Journal of Approximation Theory》1984,40(4):301-309
Let X be a reflexive, strictly convex Banach space such that both X and X* have Fréchet differentiable norms, and let {Cn} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ Cn)} of any x ε X converges if and only if lim Cn exists and is not empty. We also discuss measurability of closed convex set valued functions. 相似文献