共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
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
3.
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
4.
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
5.
Consider the system with perturbation g
k
∈ ℝ
n
and output z
k
= Cx
k
. Here, A
k
,A
k
(s) ∈ ℝ
n × n
, B
k
(1) ∈ ℝ
n × p
, B
k
(2) ∈ ℝ
n × m
, C ∈ ℝ
p × n
. We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u
k
(1) and u
k
(2) for which the following properties are satisfied:
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
相似文献
6.
Sergio Vessella 《数学学报(英文版)》2005,21(2):351-380
Let Γ be a portion of a C
1,α boundary of an n-dimensional domain D. Let u be a solution
to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We
prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous
result we prove that if u (x, t) = O (|x|k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically
equal to zero.
This work is partially supported by MURST, Grant No. MM01111258 相似文献
7.
F. Redig 《Bulletin of the Brazilian Mathematical Society》2002,33(3):427-446
We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}ℤ. For N ∈ ℕ let l
N
denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l
c
(N) of l
N
converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ,
lim
N
→∞ ℙ(l
c
(N) ≤ α log N + βx + γ) = e
–e
–x
.
Received: 2 September 2002 相似文献
8.
Given a∈L
1(ℝ) and A the generator of an L
1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫
−∞
t
a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L
1(ℝ) positive, nonincreasing and log-convex is already sufficient. 相似文献
9.
Let Zjt, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential
equations
where the matrix A(x)=(Aij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem
method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let
λ2 > λ1 > 0. We show that for any two vectors a, b∈ ℝd with |a|, |b| ∈ (λ1, λ2) and p sufficiently large, the Lp-norm of the operator whose Fourier multiplier is (|u · a|α - |u · b|α) / ∑j=1d |ui|α is bounded by a constant multiple of |a−b|θ for some θ > 0, where u=(u1 , . . . , ud) ∈ ℝd. We deduce from this the Lp-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|α / ∑j=1d |ui|α, where u=(u1,...,ud) and a is a continuous function on ℝd with |a(x)|∈ (λ1, λ2) for all x∈ ℝd.
Research partially supported by NSF grant DMS-0244737.
Research partially supported by NSF grant DMS-0303310. 相似文献
10.
Djemaïa Bensikaddour Sadek Gala Amina Lahmar-Benbernou 《Periodica Mathematica Hungarica》2008,57(1):1-22
The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ1(ℝ
d
) → (Ḣ−1(ℝ
d
)) is a complex-valued distribution on ℝ
d
, satisfy the regularity property D
k
u ∈ (Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.
相似文献
11.
Kernel regression estimation for continuous spatial processes 总被引:1,自引:0,他引:1
We investigate here a kernel estimate of the spatial regression function r(x) = E(Y
u | X
u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z
u = (X
u, Y
u), u ∈ ℝ
N
}. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the
bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal
rate is preserved by using a suitable spatial sampling scheme.
相似文献
12.
Sadek Gala 《Applications of Mathematics》2008,53(6):561-582
Consider the Navier-Stokes equation with the initial data a ∈ L
σ
2(ℝ
d
). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if ∇v ∈ L
2((0, T); (ℝ
d
)
d
) where (ℝ
d
) is the multiplier space, then we have u = v. 相似文献
13.
Sándor Csörgő 《Acta Appl Math》2007,96(1-3):159-174
We consider the generalized convolution powers G
α
*u
(x) of an arbitrary semistable distribution function G
α
(x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G
α
(k,j)(x;u)=∂
k+j
G
α
*u
(x)/∂
x
k
∂
u
j
, x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in
merge theorems from the domain of geometric partial attraction of semistable laws.
An erratum to this article can be found at 相似文献
14.
Shi Xia LUAN An Min MAO 《数学学报(英文版)》2005,21(4):685-690
In this paper, we develop the local linking theorem given by Li and Willein by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems (H) ü+A(t)u+∨V(t, u)=0, u∈R^N, t∈R. We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem. 相似文献
15.
W. Ishizuka C. Y. Wang 《Calculus of Variations and Partial Differential Equations》2008,32(3):387-405
For a bounded domain Ω ⊂ R
n
endowed with L
∞-metric g, and a C
5-Riemannian manifold (N, h) ⊂ R
k
without boundary, let u ∈ W
1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C
α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H
n-2(Σ) = 0, such that for some α ∈ (0, 1).
C. Y. Wang Partially supported by NSF. 相似文献
16.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
17.
Let ℤ denote the set of all integers and ℕ the set of all positive integers. Let A be a set of integers. For every integer u, we denote by d
A
(u) and s
A
(u) the number of solutions of u=a − a′ with a,a′ ∈ A and u=a+a′ with a,a′ ∈ A and a≤a′, respectively. 相似文献
18.
M. A. Nalbandyan 《Russian Mathematics (Iz VUZ)》2009,53(10):45-56
For any sequence {ω(n)}
n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n
2 + o(ω(n))}
n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $
\mathop \sum \limits_{k \in \Lambda }
$
\mathop \sum \limits_{k \in \Lambda }
α
k
ω
k
(x) that converges a. e. to this function, where {w
k
(x)}
k∈ℕ is the Walsh system. We find representation spectra in the form {n
l
+ o(n
l
)}
n∈ℕ, where l ∈ {2
k
}
k∈ℕ. 相似文献
19.
Masahiro Yasumoto 《manuscripta mathematica》1990,66(1):227-235
LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈Z[X], we denote byE(φ) the set of all integersa with φ
m
(a) =φ
n
(a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈Z[X] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ
m
(a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ
k
(T))=0 . 相似文献
20.
We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝ
N
with compact boundary, assuming thatf ∈ (L
loc
1
(Ω))+ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC
1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition,
we discuss the question of uniqueness of large solutions.
This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274. 相似文献
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