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1.
In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ1 of (−Δ,H
0
1
(Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical
point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions. 相似文献
2.
Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities
In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on
a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization
method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ 1 and λ ≥ λ 1, where λ1 is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality. 相似文献
3.
M. Vanninathan 《Proceedings Mathematical Sciences》1981,90(3):239-271
In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of
the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic
expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we
denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet:
λε = ε−2 λ + λ0 +O (ε), Stekloff: λε = ελ1 +O (ε2), Neumann: λε = λ0 + ελ1 +O (ε2).
Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in
the case of Neumann eigenvalue problem. 相似文献
4.
Marcello Lucia 《Calculus of Variations and Partial Differential Equations》2006,26(3):313-330
We consider the equation
If Ω is of class C
2, we show that this problem has a non-trivial solution u
λ for each λ ∊ (8 π, λ*). The value λ* depends on the domain and is bounded from below by 2 j
0
2 π, where j
0 is the first zero of the Bessel function of the first kind of order zero (λ*≥ 2 j
0
2 π > 8 π). Moreover, the family of solution u
λ blows-up as λ → 8 π. 相似文献
5.
6.
We explore connections between Krein's spectral shift function ζ(λ,H
0, H) associated with the pair of self-adjoint operators (H
0, H),H=H
0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K
*(H
0−λ−i0)−1
K) associated with the operator-valued Herglotz functionJ+K
*(H
0−z)−1
K, Im(z)>0 inH, whereV=KJK
* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H
0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E
J+A(λ)+tB(λ)(−∞, 0)),E
J((−∞, 0))), ℝ, whereA(λ)=Re(K
*(H
0−λ−i0−1
K),B(λ)=Im(K
*(H
0−λ-i0)−1
K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H
0, H) coincides with the trindex associated with the pair (Ξ(J+K
*(H
0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm
determinant of the abstract scattering matrix.
We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting
functions by appropriate spectral shift functions. 相似文献
7.
In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that
this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact.
Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C*-algebras, of which at least one is exact. In this paper we prove that the Effros–Ruan conjecture holds for general C*-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear
form on a pair of C*-algebras A and B, there exist states f
1, f
2 on A and g
1, g
2 on B such that for all a∈A and b∈B,
While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses more classical operator
algebra theory, namely, Tomita–Takesaki theory and special properties of the Powers factors of type IIIλ, 0<λ<1.
Mathematics Subject Classification (2000) 46L10, 47L25 相似文献
8.
Franz Kinzl 《Semigroup Forum》1989,38(1):105-118
Let S be a locally compact semigroup. We study the sequence (λn) of the convolution powers of a probability measure λ on S and their shifts by a probability measure η on S. We shall give
sufficient conditions for lim ‖λn−η*λn‖ = 0 (where ‖.‖ denotes the norm). In particular we consider the case the η is a point measure and we study the subsemigroup
LO(λ) = {x ∈ S : lim ‖λn−δX*λn‖ = 0}. We shall give necessary and sufficient conditions for Lo(λ)=S. In this case we want to treat the problem of the convergence of the sequence (λn). 相似文献
9.
We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in RN (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution uλ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of uλ with the exact second term and the ‘optimal’ estimate of the third term. 相似文献
10.
Dorin Bucur Ioan R. Ionescu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(6):1042-1056
We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent
friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions
of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as
their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale
collinear faults periodically disposed. If β0* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic
problem behaves as β0*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result
is that the macroscopic critical slip Dc scales with Dc∈/∈ (here Dc∈ is the small scale critical slip). 相似文献
11.
Ching-Shui Cheng 《Annals of the Institute of Statistical Mathematics》1981,33(1):155-164
A method to compare two-associate-class PBIB designs is discussed. As an application, it is shown that ifd
* is a group-divisible design withλ
2=λ1+1, a group divisible design with group size two andλ
2=λ1+1>1, a design based on a triangular scheme andv=10 andλ
1=λ2+1, a design with anL
2 scheme andλ
2=λ1+1, a design with anL
s scheme,v=(s+1)
2, andλ
2=λ1+1, wheres is a positive integer, or a design with a cyclic schemev=5, andλ
1=λ2±1, thend
* is optimum with respect to a very general class of criteria over all the two-associate-class PBIB designs with the same values
ofv, b andk asd
*. The best two-associate-class PBIB design, however, is not necessarily optimal over all designs.
This paper was prepared with the support of Office of Naval Research Contract No. N00014-75-C-0444/NR 042-036 and National
Science Foundation Grant No. MCS-79-09502. 相似文献
12.
Julián Fernández Bonder Pablo Groisman Julio D. Rossi 《Annali di Matematica Pura ed Applicata》2007,186(2):341-358
The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the
Rayleigh quotient ‖u‖2
H
1
(Ω)
2/‖u‖2
L
2
(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed
volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence
of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations
that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal
configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes.
Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10 相似文献
13.
B. M. Hambly 《Probability Theory and Related Fields》2000,117(2):221-247
We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove
an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ
ds/2
as λ→∞ where we can explicitly compute the spectral dimension d
s
. Moreover the limit N(λ) λ
−ds/2
will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable
is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume
of the fractal.
Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000 相似文献
14.
Summary We study the asymptotic behaviour of the solutions of the equation ut=Au+λu−|u|αu. Denoting by λ0 the principal eigenvalue of the second-order differential operator A, we shall prove that if λ ⩽ λ0 the only equilibrium solution, namely zero, is asymptotically stable, whereas, if λ>λ0, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attractivity and stability are proved
both in the L2-norm and in the H
0
1
-norm.
Entrata in Redazione il 15 ottobre 1976. 相似文献
15.
Xie Siqing 《分析论及其应用》1996,12(3):52-58
The object of this paper is to show regularity of (0,1, …, r−2, r)* interpolation on the set obtained by projecting vertically the zeros of (1−x2)p
n
(λ)
(x) (λ≥1/2) onto the unit circle, where P
n
(λ)
(x) stands for the nth ultraspherical polynomial. 相似文献
16.
Dan Mangoubi 《Mathematische Annalen》2008,341(1):1-13
We consider Riemannian metrics compatible with the natural symplectic structure on T
2 × M, where T
2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive
eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is
that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic
question. 相似文献
17.
Jun Wu 《Monatshefte für Mathematik》2006,54(4):259-264
For
log\frac1+?52 £ l* £ l* < ¥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty
, let E(λ*, λ*) be the set
{x ? [0,1): liminfn ? ¥\fraclogqn(x)n=l*, limsupn ? ¥\fraclogqn(x)n=l*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}.
It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that
dimE(l*, l*) 3 \fracl* -log\frac1+?522l*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}} 相似文献
18.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》1995,66(1):277-294
The nonlinear two-parameter Sturm-Liouville problemu
"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show
the existence of ann-th variational eigenvalue λ=λn(μ). Furthermore, for specialf andg, the asymptotic formula of λ1(μ)) as μ→∞ is established. 相似文献
19.
Mihail N. Kolountzakis 《Journal of Fourier Analysis and Applications》2012,18(1):21-26
A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential
functions e
λ
(x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L
2(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of
a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan. 相似文献
20.
Abstract. It is proved that the semilinear elliptic problem with zero boundary value 相似文献