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1.
The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR 2 with mixed boundary conditions, from the complete knowledge of the eigenvalues { j } j=1 for the Laplace operator, using the asymptotic expansion of the spectral function (t)= j=1 exp(–t j ) ast0.  相似文献
2.
Summary The trace function where { m} m=1 are the eigenvalues of the Laplacian is studied for a variety of domains. The dependence of(t) on the connectivity of a domain and the boundary conditions is analysed. Particular attention is given to annular domains.
Résumé Pour divers domaines, on étudie la fonction trace , où 1, 2, 3, sont les valeurs propres du laplacien. On analyse comment(t) dépend du domaine et des conditions aux limites. On considère notamment le cas de couronnes circulaires.
  相似文献
3.
The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) > 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.  相似文献
4.
this paper studies the influence of a finite container on an ideal gas,The trace of the heat kernel Θ(t)=∑(μ=1)^∞ exp(-tλμ),where{λμ}(μ=1)^∞ are the eigenvalues of the negative Laplacian-△n=-∑(p=1)^n (a/axp)^2 in R^n(n=2 or 3) ,is studied for a general mult-connected bounded drum Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries aΩi(i=1,……,m) where the Dirichlet ,Neumann and Robin boundary Conditions on aΩi(i=1,……,m) are considered.Some geometrical properties of Ω are determined ,The theremodynamic quantities for an ideal gas encolosed in Ω are examined by using the asymptotic expansions of Θ(t) for short-sime t.It is shown that the ideal gas can not feel the shape of its container Ω,althought it can feel some geometrical properties of it.  相似文献
5.
The trace of the wave kernel μ(t) =∑ω=1^∞ exp(-itEω^1/2), where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 (δ/δxk)^2 in the (x^1, x^2, x^3)-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j (j = 1,……, n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^* (i = 1 + kj-1,……, kj) of the bounding surfaces S j are considered, such that S j = Ui-1+kj-1^kj Si^*, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion ofexpansion of μ(t) for small │t│.  相似文献
6.
This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions, where the impedances are assumed to be discontinuous functions. Moreover, asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived. Further results may be obtained.  相似文献
7.
In this paper, we have studied the separation for the following biharmonic differential operator:
8.
In this paper we have studied the separation for the Laplace-Beltrami differential operator of the form
9.
The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.  相似文献
10.
The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.  相似文献
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