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1.
A. P. Oskolkov 《Journal of Mathematical Sciences》1998,91(2):2840-2859
Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations
of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W
∞
r
(ℝ+;W
2
2+k
(Ω)), W
2
r
(ℝ+;W
2
2+k
(Ω)) and S
2
r
(ℝ+;W
2
2+k
(Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W
∞
r-1
(ℝ+;W
2
k
(Ω)), W
2
r-1
(ℝ+;W
2
k
(Ω)) and S
2
r-1
(ℝ+;W
2
k
(Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations
in the smoothness classes W
∞
r−1
(ℝ; W
2
2+k
(Ω)) and W
2
r−1
(0, T; W
2
2+k
(Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W
∞
r−1
(ℝ+; W
2
k
(Ω)) and W
2
r−1
(0,T; W
2
k
(Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {vɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight
fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems
for the equations of Kelvin-Voight fluids. Bibliography: 27 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242.
Translated by T. N. Surkova. 相似文献
2.
Let Ω ⊆ ℝn be a bounded convex domain with C
2 boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H
k
p,q,φ
(Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H
k
p,q,φ
) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained. 相似文献
3.
A. P. Oskolkov 《Journal of Mathematical Sciences》1997,83(2):320-326
In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations
(2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove
that these problems have global smooth solutions of the classW
∞
1
(ℝ+;W
2
2+k
(Ω)),k=1,2,...;Ω⊂ℝ3. Bibliography: 25 titles.
Dedicated to N. N. Uraltseva on her jubilee
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207.
Translated by N. A. Karazeeva. 相似文献
4.
Martin Kružík 《Applications of Mathematics》2007,52(6):529-543
We study convergence properties of {υ(∇u
k
)}k∈ℕ if υ ∈ C(ℝ
m×m
), |υ(s)| ⩽ C(1+|s|
p
), 1 < p < + ∞, has a finite quasiconvex envelope, u
k
→ u weakly in W
1,p
(Ω; ℝ
m
) and for some g ∈ C(Ω) it holds that ∫Ω
g(x)υ(∇u
k
(x))dx → ∫Ω
g(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L
1-weak convergence of {det ∇u
k
}
k∈ℕ to det ∇u if m = n = p.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday
This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR). 相似文献
5.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
6.
M. Langenbruch 《manuscripta mathematica》2000,103(2):241-263
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ
n
. Let L(P
m
) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P
m
. Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ
n
×(ℝ\{ 0}) for any Q∈L(P
m
) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P
m
must be locally hyperbolic.
Received: 24 January 2000 相似文献
7.
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
8.
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
9.
Bin Han 《Advances in Computational Mathematics》2006,24(1-4):375-403
In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented.
Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function
vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz
space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167)
in the univariate setting to the multivariate setting.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C20, 41A25, 39B12.
Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant
G121210654. 相似文献
10.
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
11.
Zeng Jian LOU Shou Zhi YANG Dao Jin SONG 《数学学报(英文版)》2005,21(4):949-954
We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2. This partially answers a well-known open problem. 相似文献
12.
Wu Huoxiong 《分析论及其应用》1994,10(2):92-104
In this paper we study the average δ-K width and the average δ-linear width of the unit ball of l
1
∞
(ℝ) in l
2
∞
(ℝ). The exact values of these widths are calculated and an optimal subspace with the optimal linear operator (for the δ-linear
width) are identified. 相似文献
13.
Flavia Giannetti Tadeusz Iwaniec Jani Onninen Anna Verde 《Journal of Geometric Analysis》2002,12(2):223-254
Let ƒ: Ω → ℝn be a mapping in the Sobolev space W1,n−1(Ω,ℝn), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D#ƒ satisfies
, where Lp(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L
loc
1
(Ω). Estimates above and below L
loc
1
(Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions
on the differential matrix. 相似文献
14.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
15.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
16.
Subhash J. Bhatt 《Proceedings Mathematical Sciences》2006,116(2):161-173
Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C
*-algebraE(S(ℝ, A∞, α)) of the smooth Schwartz crossed productS(ℝ,A
∞, α) of the Frechet algebra A∞ of C∞-elements ofA is isomorphic to the Σ-C
*-crossed productC
*(ℝ,E(A), α) of the enveloping Σ-C
*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK
*(S(ℝ, A∞, α)) =K
*(C
*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC
*-algebra defined by densely defined differential seminorms is given. 相似文献
17.
We investigate the Caucy problem for linear elliptic operators withC
∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at
the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH
4(Ω). 相似文献
18.
Lower semicontinuity for polyconvex functionals of the form ∫Ω
g(detDu)dx with respect to sequences of functions fromW
1,n
(Ω;ℝ
n
) which converge inL
1 (Ωℝ
n
) and are uniformly bounded inW
1,n−1 (Ω;ℝ
n
), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof
which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations
of these results.
Supported by MURST, Gruppo Nazionale 40%
Partially supported by Australian Research Council 相似文献
19.
We show that the Fréchet-Sobolev spaces C(ℝ) ∩ L
p
(ℝ) and C
k
(ℝ) ∩ L
p
(ℝ) are not isomorphic for p ≠ 2 and k ∈ ℕ.
Research supported by the Italian MURST. 相似文献
20.
Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>
David Swanson 《Arkiv f?r Matematik》2007,45(2):381-399
Let Ω⊂ℝ
n
be an arbitrary open set. We characterize the space W
1,1
loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation
and trace theorems for functions in the space W
1,1(ℝ
n
). 相似文献