共查询到20条相似文献,搜索用时 15 毫秒
1.
In this work we study an asymptotic behaviour of solutions to the Laplace–Beltrami operator on a rotation surface near a cuspidal point. To this end we use the WKB-approximation. This approach describes the asymptotic behaviour of the solution more explicitly than abstract theory for operators with operator-valued coefficients. 相似文献
2.
Let u+u=v+v= 0, where isthe Laplace–Beltrami operator on a compact connected smoothmanifold M and > 0. If H1(M) = 0then there exists pM such that u(p)=v(p) = 0 For homogeneous M,H1(M) 0 implies the existence of a pair u,v as above that has no common zero. 相似文献
3.
We study the spectrum of the Laplace–Beltrami operator on noncompact Riemannian manifolds of a special form, in particular on model manifolds. We obtain a discreteness criterion for the spectrum in terms of the volume and capacity of some domains on a manifold. 相似文献
4.
《Mathematical Methods in the Applied Sciences》2018,41(1):270-280
The homogenization of kinetic laminates in the framework of time‐dependent linearized elasticity is studied from a variational point of view through the Γ‐convergence of the associated energies. The characterization of the effective coefficients is achieved by means of a finite dimensional minimization problem. 相似文献
5.
For a parabolic equation with drift on a Riemannian manifold of positive curvature we obtain a representation for the logarithmic gradient in the form of the sum of two vector fields one of which is known and the other is bounded. The drift field is assumed to be of sufficiently rapid decay at infinity. 相似文献
6.
In [14] Fernández, Heinonen and Llorente extend the Hornblower's results, about boundary behaviour of subharmonic functions in the unit disc of the complex plane, to subharmonic functions in the unit ball or the upper half space in higher dimensions. In this paper we establish that those results are also valid in the much more general setting of linear axiomatic potential theory. The interest of our general formulation relies on the applications to differential operators. We apply our result to Laplace–Beltrami operator and some uniformly elliptic second order operators in divergence form. 相似文献
7.
We construct a fundamental solution for a parabolic equation with drift on a Riemannian manifold of nonpositive curvature. We obtain some estimates for this fundamental solution that depend on the conditions on the drift field. 相似文献
8.
Andrii Khrabustovskyi 《Mathematical Methods in the Applied Sciences》2009,32(16):2123-2137
We study the asymptotic behavior of the eigenvalues and the eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold Mε depending on a small parameter ε>0 and whose structure becomes complicated as ε→0. Under a few assumptions on scales of Mε we obtain the homogenized eigenvalue problem. In addition we study the behavior of the heat equation on Mε and investigate the large time behavior of the homogenized equation. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
9.
M. Negri 《Mathematical Methods in the Applied Sciences》2011,34(4):384-396
Considering anti‐plane elasticity we provide an existence result for the energy release rate along a piecewise C1, 1 path that admits a kink. We provide two representations: an asymptotic one in terms of the stress intensity factor and an integral one in terms of the Eshelby tensor. Both the formulas make use of an implicit coefficient, depending on the kink angle and obtained by a minimum problem. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
10.
Recently, there has been much interest in the solution of differential equations on surfaces and manifolds, driven by many applications whose dynamics take place on such domains. Although increasingly powerful algorithms have been developed in this field, many straightforward questions remain, particularly in the area of coupling advanced discretizations with efficient linear solvers. In this paper, we develop a structured refinement algorithm for octahedral triangulations of the surface of the sphere. We explain the composite‐grid finite‐element discretization of the Laplace–Beltrami operator on such triangulations and extend the fast adaptive composite‐grid scheme to provide an efficient solution of the resulting linear system. Supporting numerical examples are presented, including the recovery of second‐order accuracy in the case of a nonsmooth solution. 相似文献
11.
Tong Kang Ran Wang Huai Zhang 《Numerical Methods for Partial Differential Equations》2021,37(1):546-582
We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system. 相似文献
12.
Let e(x, y, ) be the spectral function and the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, ) by Hörmander (Acta Math.88 (1968), 341–370) to that of xye(x,y,)|x=y for any multiindices , in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L2,Lp) (2 p) estimates of by Sogge (J. Funct. Anal.77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L2, Sobolev Lp) estimates of . 相似文献
13.
B. Amaziane M. Goncharenko L. Pankratov 《Mathematical Methods in the Applied Sciences》2005,28(15):1847-1865
We study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem posed in a domain Ω(ε) of asymptotically degenerating measure, i.e. meas Ω(ε) → 0 as ε → 0, where ε is the parameter that characterizes the scale of the microstructure. We obtain the convergence of the solution and the homogenized model of the problem is constructed using the notion of convergence in domains of degenerating measure. Proofs are given using the method of local characteristics of the medium Ω(ε) associated with our problem in a variational form. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
14.
István Heckenberger 《Compositio Mathematica》2000,123(3):329-354
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra
(SL
q
(N)) are computed. 相似文献
15.
16.
By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed‐point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation, which extend and improve significantly existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
17.
For domains with concave corners, the solutions to elliptic boundary values have the typical rα‐singularity. The so‐called singularity exponents α are the eigenvalues of an eigenvalue problem which is associated with the given boundary value problem. This paper is aimed at deriving the mentioned eigenvalue problems for two examples, the Laplace equation and the linear elasticity problem. We will show interesting properties of these eigenvalue problems. For the linear elasticity problem, we explain in addition why the classical symmetry and positivity assumptions of the material tensor have to be used with care. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
18.
Davood Domiri Ganji Mehdi Safari Reza Ghayor 《Numerical Methods for Partial Differential Equations》2011,27(4):887-897
In this article, the Sawada–Kotera–Ito seventh‐order equation is studied. He's variational iteration method and Adomian's decomposition method (ADM) are applied to obtain solution of this equation. We compare these methods together. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 887–897, 2011 相似文献
19.
20.
Ya‐nan Zhang Zhi‐zhong Sun Ting‐chun Wang 《Numerical Methods for Partial Differential Equations》2013,29(5):1487-1503
A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L2 ‐norm by the discrete energy method. The convergence order is begin{align*}mathcal{O}(tau^2+h_1^2+h^2_2)end{align*}, where τ is the temporal grid size and h1,h2 are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献