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1.
Decay of the energy for the Cauchy problem of the wave equation of variable coeffcients with a dissipation is considered.It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coeffcients.In particular,some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given. 相似文献
2.
Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases. 相似文献
3.
A note on geometric conditions for boundary control of wave equations with variable coefficients 总被引:1,自引:0,他引:1
The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent. 相似文献
4.
Shigeki Aida 《Journal of Functional Analysis》2007,251(1):59-121
We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis. 相似文献
5.
Peng-Fei Yao 《Journal of Differential Equations》2007,241(1):62-93
We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback. 相似文献
6.
On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection
Neda Bokan 《Rendiconti del Circolo Matematico di Palermo》1990,39(3):331-380
We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated
with a symmetric connection of Riemannian manifold. We solve the problem under the action ofS0(n). The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined.
Several applications for Riemannian manifolds with symmetric connection are given. The group of projective transformations
of a Riemannian manifold and its subgroups are considered. 相似文献
7.
Keomkyo Seo 《Monatshefte für Mathematik》2012,97(3):525-542
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature. 相似文献
8.
Keomkyo Seo 《Monatshefte für Mathematik》2012,166(3-4):525-542
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature. 相似文献
9.
It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related
parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations
are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance
of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations
on manifolds and semigroups, we study C
∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions
of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a
unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional
separation of curvature.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006. 相似文献
10.
We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given. 相似文献
11.
D. A. Popov 《Functional Analysis and Its Applications》2012,46(2):133-146
Let the standard Riemannian metric of constant curvature K = −1 be given on a compact Riemannian surface of genus g > 1. Under this condition, for a class of strictly hyperbolic Fuchsian groups, we obtain an explicit expression for the spectral
counting function of the Laplace operator in the form of a series over the zeros of the Selberg zeta function. 相似文献
12.
Ye Li 《Advances in Mathematics》2010,223(6):1924-1957
We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L2-norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C1,α-close to the original metric on a smaller ball but still of definite size. 相似文献
13.
We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting. 相似文献
14.
风机叶片的Riemann几何研究 总被引:1,自引:0,他引:1
李根全 《数学的实践与认识》2004,34(3):73-78
用 Riemann几何方法研究了风机叶片的设计问题 .得到了螺旋线切曲面的 Riemann曲率张量 Rlkij=0 ,从而指出了切曲面为可展曲面 ,并给出了设计单片式连续曲面结构叶片的有关参数 . 相似文献
15.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.supported by the National Natural Science Foundation of China (10371138). 相似文献
16.
Boundary Controllability for the Quasilinear Wave Equation 总被引:1,自引:0,他引:1
Peng-Fei Yao 《Applied Mathematics and Optimization》2010,61(2):191-233
We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric
analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around
the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such
a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another
location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results
show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria
of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of
the Riemann metric. Some examples are presented to verify the global exact controllability. 相似文献
17.
We show that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle, equipped with the induced metric, onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented. 相似文献
18.
Plateau's problem (PP) is studied for surfaces of prescribed mean curvature spanned by a given contour in a 3-d Riemannian
manifold. We consider the local situation where a neighborhood of a given point on the manifold is described by a single normal
chart. Under certain conditions on and the contour, existence of a small -surface to (PP) is guaranteed by [HK]. The purpose of this paper is the investigation of large -surfaces. Our result states: For sufficiently large (constant) mean curvature and a sufficiently small contour depending
on the local geometry of the manifold, (PP) has at least two solutions, a small one and a large one. The proof is based on
mountain pass arguments and uses – in contrast to results in the 3-d Euclidean space and in order to derive conformality directly
– also a deformation constructed by variations of the independent variable.
Received November 8, 1995 / Accepted April 29, 1996 相似文献
19.
S. E. Kozlov 《Journal of Mathematical Sciences》2004,119(2):223-229
An infinitesimal criterion indicating when a two-dimensional submanifold of a Riemannian symmetric space is totally geodesic is given. As an application, the classification of two-dimensional totally geodesic submanifolds of the Grassmannian of bivectors is given in a new way, and it is proved that the sectional curvature takes stationary values on tangent spaces of such submanifolds. Bibliography: 9 titles. 相似文献
20.
Thilo Kuessner 《Central European Journal of Mathematics》2011,9(1):173-183
We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which
is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can
be approximated by isotopies of a given contact structure. 相似文献