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1.
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
2.
Yu. N. Podil'chuk V. F. Tkachenko E. E. Shevchenko 《Journal of Mathematical Sciences》1995,77(6):3480-3485
By solving the stationary thermoelastic problem for a transversally isotropic hyperboloid of revolution we conduct an analytic
study of the thermal stress state of a finite solid of revolution with a meridional section of complicated shape and propose
a way to solve the problem of the reliability of information obtained using the numerical methods developed and algorithms
implemented on a computer.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 7–15. 相似文献
3.
Roman G. Novikov 《Arkiv f?r Matematik》1999,37(1):141-169
We consider the Newton equation
for |j|≤2 and some α>1.
We give estimates and asymptotics for scattering solutions and scattering data for the equation (*) for the case of small
angle scattering. We show that scattering data at high energies uniquely determine theX-ray transformsPF andPv. Applying results on inversion of theX-ray transformP we obtain that ford≥2 scattering data at high energies uniquely determineF andv. For the case of potentials with compact support we give a connection between boundary value data and scattering data and
ford≥2 we obtain, using known results, a uniqueness theorem in the inverse scattering problem at fixed energy. 相似文献
((*)) |
4.
In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ
n
which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal
vector fields in ℍ
n
.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean
curvature (CMC) hypersurface. Our definition coincides with previous ones.
Our main result describes which are the CMC hypersurfaces of revolution in ℍ
n
.The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential
equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart
in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean
space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ
n
. 相似文献
5.
We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method. 相似文献
6.
In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ?X. In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry. 相似文献
7.
In [2, 3] a nonreflecting boundary condition(NBC) for time-dependent multiple scattering was derived, which is local in time but nonlocal in space. Here, based on a high-order local nonreflecting boundary condition (NBC) for single scattering [4], we seek a local NBC for time-dependent multiple scattering, which is completely local both in space and time. To do so, we first develop a high order representation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the artificial boundary condition. By combining that representation formula with a decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NBC for time-dependent multiple scattering. The accuracy and stability of this local NBC is evaluated by coupling it to standard finite element and finite difference methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
10.
Andreas Knauf 《Journal of the European Mathematical Society》2002,4(1):1-114
We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological
entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and
differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential
which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres.
Finally we show that different, non-universal, phenomena occur for collinear configurations.
Received October 16, 2000 / final version received June 18, 2001?Published online August 15, 2001 相似文献
11.
David Borthwick 《偏微分方程通讯》2013,38(8):1507-1539
For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r n+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an r n+1 lower bound on the counting function for scattering poles. 相似文献
12.
This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L 2 affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L 2 does not suffer from our L 2 setting, compared to schemes in higher order Sobolev spaces. 相似文献
13.
High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application
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We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L2(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
14.
We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves
by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering
problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The
latter are typically bounded on the space of tangential vector fields of mixed regularity
T H-\frac12(divG,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard
Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the
electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization
of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic
scattering problem. 相似文献
15.
16.
Hirobumi Mizuno 《Linear and Multilinear Algebra》2013,61(7):927-940
We give a decomposition formula for the determinant det(I ? U(λ)) of the weighted bond scattering matrix U(λ) of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express some determinant of the weighted bond scattering matrix of a regular covering of G by means of its L-functions. 相似文献
17.
By using the binary Darboux transformations, we construct scattering operators for a Dirac system with special potential depending
on 2n arbitrary functions of a single variable. It is shown that one of the operators coincides with the scattering operator obtained
by Nyzhnyk in the case of degenerate scattering data. It is also demonstrated that the scattering operator for the Dirac system
is either obtained as a composition of three Darboux self-transformations or factorized by two operators of binary transformations
of special form. We also consider several cases of reduction of these operators.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1097–1115, August, 2006. 相似文献
18.
Salvatore Milici 《Geometriae Dedicata》1993,47(1):49-56
For each admissiblev we exhibit a path designP(v, 3, 1) with a spanning set of minimum cardinality and aP(v, 3, 1) with a scattering set of maximum cardinality. Moreover, we study maximal independent sets (or complete arcs in the geometric terminology) having the minimum number of secants, i.e. sets which are both spanning and scattering, and complete arcs with the maximum number of secants.The research for this paper was supported by MURST and GNSAGA-CNR. 相似文献
19.
We formulate quantum scattering theory in terms of a discrete L
2-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators. 相似文献
20.
Rafael López 《Monatshefte für Mathematik》2008,154(4):289-302
In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of
type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution,
one of the Riemann minimal examples, or a generalized cone.
Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain 相似文献