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1.
M. S. Agranovich 《Functional Analysis and Its Applications》2009,43(3):165-183
We consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω+ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained
in the standard torus $
\mathbb{T}^n
$
\mathbb{T}^n
. In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces H
p
σ
and B
p
σ
without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface
potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω+ and the complementing domain Ω− are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular
operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission
problems. 相似文献
2.
An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed
convex proper cone inR
n and −Γ′ be the antipodes of the dual cone of Γ. Let
be a partial differential operator with constant coefficients inR
n, whereQ(ζ)≠0 onR
n−iΓ′ andP
i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R
n−iΓ′;P
j(ζ)=0, gradP
j(ζ)≠0} contains some real point on which gradP
j≠0 and gradP
j∉Γ∪(−Γ). LetC be an open cone inR
n−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in
{ξ∈R
n;P(ξ)=0}. Ifu∈ℒ′∩L
loc
2
(R
n−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition
implies that the support ofu is contained in Γ. 相似文献
3.
Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C
ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ3 of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC
1 cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes
based on finite elements of this type. 相似文献
4.
S. A. Nazarov 《Journal of Applied and Industrial Mathematics》2009,3(3):377-390
Taking various viewpoints, we study the selfadjoint extensions $
\mathcal{A}
$
\mathcal{A}
of the operator A of the Dirichlet problem in a 3-dimensional region Ω with an edge Γ. We identify the infinite dimensional nullspace def A with the Sobolev space H
−ϰ(Γ) on Γ with variable smoothness exponent −ϰ ∈ (−1, 0); while the selfadjoint extensions, with selfadjoint operators $
\mathcal{T}
$
\mathcal{T}
on the subspaces of H
−ϰ(Γ). To the boundary value problem in the region with a “smoothed” edge we associate a concrete extension, which yields a
more precise approximate solution to the singularly perturbed problem. 相似文献
5.
Let Ω be a bounded Lipschitz domain. Define B
0,1
1,
r
(Ω) = {f∈L
1 (Ω): there is an F∈B
0,1
1 (ℝ
n
) such that F|Ω = f} and B
0,1
1
z
(Ω) = {f∈B
0,1
1 (ℝ
n
) : f = 0 on ℝ
n
\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the
regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ
n
+.
Received June 8, 2000, Accepted October 24, 2000 相似文献
6.
Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W
2
2
(Ω) of the equation Δ
x
2
u = f with the boundary conditions u = Δxu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained
by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported
polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If
the variable opening angle α ∈ C∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the
iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case
α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel
and co-kernel of the operators and determining the solution u ∈ W
2
2
(Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a
nonnegative function f ∈ L2(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to
introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical
situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property
remain open. Bibliography: 46 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198. 相似文献
7.
Byron L. Walden 《Journal d'Analyse Mathématique》1994,63(1):231-253
In 1981, Hayman and Wu proved that for any simply connected domain Ω and any Riemann mappingF: Ω →D,F′ ∈ L1 (L ∩ Ω), whereL is any line in the complex plane. Several years later, Fernández, Heinonen and Martio showed that there is anε > 0 such thatF′ ∈ L1+∈(L ∩ Ω). The question arises as to which curves other than lines satisfy such a statement. A curve Γ is said to be Ahlfors-David
regular if there is a constantA such that for any B(x, r) (the disk of radiusr centered atx), l(Γ ∩ B(x, r))≤ Ar. The major result of the paper is the following theorem: Let Γ be an Ahlfors-David regular curve with constantA. Then there exists an∈ > 0, depending only onA, such thatF′ ∈ L1+∈(Γ ∩ Ω). This result is the synthesis of the extension of Fernández, Heinonen and Martio, and the result of Bishop and Jones
showing thatF′ ∈ L1(Γ ∩ Ω). The proof of the results uses a stopping-time argument which seeks out places in the curve where small pieces may
be added in order to control the portions of the curve where |F′ | is large. This is accomplished with an estimate on the
vanishing of the harmonic measure of the curve in such places. The paper also includes simpler arguments for the special cases
where Γ = ∂Ω and Γ ⊂Ω. 相似文献
8.
In this paper, we first consider a delay difference equation of neutral type of the form:
Δ(y
n
+ py
n−k
+ q
n
y
n−l
= 0 for n∈ℤ+(0) (1*)
and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form:
Δ1(u
n,m
+ pu
n−k,m
) + q
n,m
u
n−l,m
= a
2Δ2
2
u
n
+1,
m−1
for (n,m) ∈ℤ+ (0) ×Ω, (2*)
study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses.
Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000 相似文献
9.
We consider the problem −Δu=|u|
p−1u+λu in Ω with
on δΩ, where Ω is a bounded domain inR
N
,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR
N
, then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR
3, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial
solution.
This work was supported by the Paris VI-Leiden exchange program
Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016. 相似文献
10.
A rate of convergence for the set compound estimation in a family of certain retracted distributions
Yoshiko Nogami 《Annals of the Institute of Statistical Mathematics》1982,34(1):241-257
Summary This paper is concerned with the set compound squared-error loss estimation problem. Here, the author obtains Lévy consistent
estimate
of the empiric distributionG
n of the parameters θ1,...,θn for a more general family of retracted distributions on the interval [θ, θ+1) than the uniform on [θ, θ+1) as in R. Fox (1970,Ann. Math. Statist.,41, 1845–1852; 1978,Ann. Statist.,6, 846–853) and exhibits a decision procedure based on
with a convergence rateO((n
−1 logn)1/4) for the mofified regret uniformly in (θ1, θ2, ..., θn ∈ Ωn with bounded Ω. The author also gives a counterexample to the convergence of the modified regret for Ω=(−∞, ∞).
This is part of the author's Ph. D. Thesis at Michigan State University. 相似文献
11.
We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X
1,..., Xn−1) (codimension 1 differential systems) defined locally on ℝ
n
or ℂ
n
, and extend the standard duality(X
1,..., Xn−1)↦(ω), ω=Ω(X1,...,Xn−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX
i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension
1 differential systemsV and
are equivalent if and only if so are the corresponding Pfaffian equations (ω) and
provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V
⊥)⊥=V; (c)V
⊥=(ω); (d) (ω)⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator.
Supported by Polish KBN grant No 2 1090 91 01.
Partially supported by the fund for the promotion of research at the Technion, 100–942. 相似文献
12.
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Ω in a given complete
(not compact a priori) Riemannian manifold (M,g). For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general
metric way, interesting for itself. As applications, we prove that if the Ricci curvature of (M,g) is bounded below Ric
g
≥−(n−1)a
2, a≥0, then there exist constants A
n
>0,B
n
>0 only depending on the dimension, such that
where λ
k
(Ω) (k∈ℕ*) denotes the k-th eigenvalue of the Neumann problem on any bounded domain Ω⊂M of volume V=Vol (Ω,g). Furthermore, this upper bound is clearly in agreement with the Weyl law. As a corollary, we get also an estimate which
is analogous to Buser’s upper bounds of the spectrum of a compact Riemannian manifold with lower bound on the Ricci curvature.
相似文献
13.
Aissa Guesmia 《Israel Journal of Mathematics》2001,125(1):83-92
We consider in this paper the evolution systemy″−Ay=0, whereA =∂
i(aij∂j) anda
ij ∈C
1 (ℝ+;W
1,∞ (Ω)) ∩W
1,∞ (Ω × ℝ+), with initial data given by (y
0,y
1) ∈L
2(Ω) ×H
−1 (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT. 相似文献
14.
Jiří Neustupa 《Annali dell'Universita di Ferrara》2009,55(2):353-365
We prove the existence of a weak solution to the steady Navier–Stokes problem in a 2D domain Ω, whose boundary ∂Ω consists of two unbounded components Γ
− and Γ
+. We impose an inhomogeneous Dirichlet—type boundary condition on ∂Ω. The condition implies no restriction on fluxes of the solution through the components Γ
− and Γ
+. 相似文献
15.
We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ
0, A
0) ∈ L
2(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L
3(Ω)) using the Lorentz gauge.
相似文献
16.
Let Ω⊂R
n
be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W
1,p
(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W
0
1,p
(Ω). 相似文献
17.
Let Ω be an open, simply connected, and bounded region in ℝ
d
, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable,
and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate
exponential convergence. 相似文献
18.
Andreas Fr?hlich 《Annali dell'Universita di Ferrara》2000,46(1):11-19
We consider weights of Muckenhoupt classA
q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝn we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve
the weak Neumann problem for the Laplace equation in weighted spaces on ℝn, ℝn
+, on bounded and on exterior domains Ω with boundary of classC
1, which will yield the Helmholtz decomposition ofL
ω
q(Ω)n for general ω∈A
q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of
Farwig and Sohr [2] where the Helmholtz decomposition ofL
ω
p(Ω)n is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood
of ϖΩ.
Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝn, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝn, ℝn + in domini limitati ed in domini esterni con frontiera di classeC 1, che conduce alla decomposizione di Helmholtz diL ω q(Ω)n per un qualsiasi ω∈A q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL ω q(Ω)n è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.相似文献
19.
SoitM(Ω, η, ξ,g) une variété à (2m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ2
M est de rang 2m et Ω
m
Λη≠0). On définitM comme étant une variété semi-cosymplectique si en termes ded
ω-cohomologie la paire (Ω, η) satisfait àdη=0,d
−cη Ω=Ψ∈Λ3
M,c=constant. Dans ce cas le champ vectoriel de structure ξ=b
−1(η) est un champ conforme horizontal et siM est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés
et le cas oùM est une variété para Sasakienne dans le sens large est discuté. 相似文献
20.
In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” uf (·, T) is studied, where uf (x,t) is the solution of the initial boundary-value problem utt−Δu=0 in Ω×(0,T), u|t<0=0, u|∂Ω×(0,T)=f, and the (singular) control f runs over the class L2((0,T); H−m (∂Ω)) (m>0). The following result is established. Let ΩT={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝn (diam Ω<∞) filled with waves by a final instant of time t=T, let T*=inf{T : ΩT=Ω} be the time of filling the whole domain Ω. We introduce the notation Dm=Dom((−Δ)m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H2(Ω)∩H
0
1
(Ω);D−m=(Dm)′;D−m(ΩT)={y∈D−m:supp y ⋐ ΩT. If T<T., then the reachable set R
m
T
={ut(·, T): f ∈ L2((0,T), H−m (∂Ω))} (∀m>0), which is dense in D−m(ΩT), does not contain the class C
0
∞
(ΩT). Examples of a ∈ C
0
∞
, a ∈ R
m
T
, are presented.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21.
Translated by T. N. Surkova. 相似文献