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1.
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
2.
In the space of functions B
a3+={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T3/2)=g(x, −t)}, we establish that if the condition aT
3
(2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u
u
−a
2
u
xx
=g(x, t), u(0, t)=u(π, t)=0, u(x, t+T
3
)=u(x, t), ℝ2, is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997 相似文献
3.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
4.
D. V. Maksimov 《Journal of Mathematical Sciences》2008,148(6):850-859
Consider functions u1, u2,..., un ∈ D(ℝk) and assume that we are given a certain set of linear combinations of the form ∑i, j a
ij
(l)
∂jui. Sufficient conditions in terms of coefficients a
ij
(l)
are indicated under which the norms
are controlled in terms of the L1-norms of these linear combinations. These conditions are mostly transparent if k = 2. The classical Gagliardo inequality
corresponds to a single function u1 = u and the collection of its partial derivatives ∂1u,..., ∂ku. Bibliography: 2 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 120–139. 相似文献
5.
Letp be any odd prime number. Letk be any positive integer such that
. LetS = (a
1,a
2,...,a
2p−k
) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows:
whereu ≥v,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. This extends a result in [13] to all primesp andk satisfying (p + 1)/4 + 3 ≤k ≤ (p + 1)/3 + 1. Also, we prove that ifg denotes the number of distinct residue classes modulop appearing in the sequenceS in ℤp of length 2p -k (2≤k ≤ [(p + 1)/4]+1), and
, then there exists a subsequence of S of lengthp whose sum is zero in ℤp. 相似文献
(1) |
6.
On weighted approximation by Bernstein-Durrmeyer operators 总被引:6,自引:0,他引:6
Zhang Zhenqiu 《分析论及其应用》1991,7(2):51-64
In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in Lp[0, 1] (1≤p≤∞), where the weight function w(x)=xα(1−x)β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals.
Supported by Zhejiang Provincial Science Foundation. 相似文献
7.
Rosa M. Migo-Roig 《manuscripta mathematica》1993,80(1):89-94
We show the following theorem of compensated compactness type: Ifu
n
⇁u weakly in the spaceH
1,p
(Ω, ℝ
k
) and if also
in the sense of distributions then ∂α(∣∇u∣
p-2∂α
u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S
k
−1. 相似文献
8.
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
9.
E. G. Goluzina 《Journal of Mathematical Sciences》1998,89(1):958-966
Let TR be the class of functions
that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z0), f(r)} for z0 ∈ ℝ, r∈(-1,1) is studied. The sets Dr={f(z0):f∈TR, f(r)=a} for −1≤r≤1 and Δr={(c2, c3): f ∈ TR, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)−2, r(1−r)−2) is an arbitrary fixed number. Bibliography: 11 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79. 相似文献
10.
Li Changpin 《应用数学学报(英文版)》2001,17(2):191-199
In this paper, we investigate the bifurcations of one class of steady-state reaction-diffusion equations of the formu″ + μu − u
k=0, subjectu(0)=u(π)=0, where μ is a parameter, 4≤kεZ
+. Using the singularity theory based on the Liapunov-Schmidt reduction, some satisfactory results are obtained.
This work is supported by the National Natural Science Foundation of China (No.19971057) and the Youth Science Foundation
of Shanghai Municipal Commission of Education (No.99QA66). 相似文献
11.
Jean Bourgain Jeff Kahn Gil Kalai Yitzhak Katznelson Nathan Linial 《Israel Journal of Mathematics》1992,77(1-2):55-64
LetX be a probability space and letf: X
n
→ {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI
f
(k), as follows: Foru=(u
1,u
2,…,u
n−1) ∈X
n−1 consider the setl
k
(u)={(u
1,u
2,...,u
k−1,t,u
k
,…,u
n−1):t ∈X}.
More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI
f
(S), be the probability that assigning values to the variables not inS at random, the value off is undetermined.
Theorem 1:There is an absolute constant c
1
so that for every function f: X
n
→ {0, 1},with Pr(f
−1(1))=p≤1/2,there is a variable k so that
Theorem 2:For every f: X
n
→ {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c
2(ε)n/logn so that I
f
(S)≥1−ε.
These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.
Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science. 相似文献
12.
L. I. Sazonov 《Mathematical Notes》1999,65(2):202-207
In the exterior domain Ω⊂ℝ2 we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral
and satisfies the condition lim|x|→∞
u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x|
−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the
body.
Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999. 相似文献
13.
Let ℤ2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ2N→ℝ is given by φf(k)=arg
for k ∈ ℤ2N, where
denotes the discrete Fourier transforms of f.
For a fixed s∈L let Ks denote the cone of all f:ℤ2N→ℝ which satisfy φf ≡ φs and let Ms be its linear span. The angle αs between Ms and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove
the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:
14.
Roy Meshulam 《Graphs and Combinatorics》1992,8(3):287-289
Let ℱ be a family ofn−k-dimensional faces of the discrete cube {0,1}
n
such that for allF ε ℱ, F ⊄ ∪ { F′: F ≠ F′ ∈ ℱ}. It is shown that ifn≥n
0
(k) then |ℱ| ≤
. This was conjectured by Aharoni and Holzman and is the casem=2 of a more general result on faces of {0,...,m−1}n. 相似文献
15.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
16.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
17.
We prove inequalities about the quermassintegralsV
k
(K) of a convex bodyK in ℝ
n
(here,V
k
(K) is the mixed volumeV((K, k), (B
n
,n − k)) whereB
n
is the Euclidean unit ball). (i) The inequality
18.
We generalize a result by H. Brezis, Y. Y. Li and I. Shafrir [6] and obtain an Harnack type inequality for solutions of −Δu = |x|2α Ve u in Ω for Ω ⊂ ℝ2 open, α ∈ (−1, 0) and V any Lipschitz continuous function satisfying 0 < a ≤ V ≤ b < ∞ and ‖∇V‖∞ ≤ A. 相似文献
19.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
20.
R. J. Cook 《Proceedings Mathematical Sciences》1989,99(2):147-153
Letf(x)=θ1
x
1
k
+...+θ
s
x
s
k
be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ
1,…,θ
s
, are algebraic ands = 4k then there are integersx
1,…,x
s
, satisfying l ≤x
1,≤ N and ∥f(x)∥ ≤ N
E
, withE = − 1 + 2/e.
Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ
1,…,θ
s
, be algebraic then the result holds for almost all values of θεℝ
s
. Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate. 相似文献
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