共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)<p≤1. 相似文献
3.
Zhu Xuexian 《分析论及其应用》1989,5(3):83-92
We show that if K(x,y)=Ω(x,y)/|x|n|y|m is a Calder n-Zygmund kerned on Rn×Rm, where Ω∈L2(Sn−1×Sm−1) and b(x,y) is any bounded function which is radial with x∈Rn and y∈Rm respectively, then b(x,y)K(x,y) is the kernel of a convolution operator which is bounded on Lp(Rn×Rm) for 1<p<∞ and n≧2, m≧2.
Project supported by NSFC 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.
Guido Cortesani 《Annali dell'Universita di Ferrara》1997,43(1):27-49
Let Ω be an open and bounded subset ofR
n
with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R
m
) whose jump setS
vis essentially closed and polyhedral and which are of classW
k, ∞ (S
v,R
m) for every integerk are strongly dense inGSBV
p(Ω,R
m
), in the sense that every functionu inGSBV
p(Ω,R
m
) is approximated inL
p(Ω,R
m
) by a sequence of functions {v
k{j∈N with the described regularity such that the approximate gradients ∇v
jconverge inL
p(Ω,R
nm
) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS
v
j converges to the (n−1)-dimensional measure ofS
u. The structure ofS
v can be further improved in casep≤2.
Sunto Sia Ω un aperto limitato diR n con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R m ) con insieme di saltoS v essenzialmente chiuso e poliedrale che sono di classeW k, ∞ (S v,R m ) per ogni interok sono fortemente dense inGSBV p(Ω,R m ), nel senso che ogni funzioneu∈GSBV p(Ω,R m ) è approssimata inL p(Ω,R m ) da una successione di funzioni {v j}j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v jconvergono inL p(Ω,R nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS v jconverge alla misura (n−1)-dimensionale diS u. La struttura diS vpuó essere migliorata nel caso in cuip≤2.相似文献
6.
In this paper, we consider the global existence, uniqueness and L
∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u
t
− div(|∇u|
m−2∇u) = u|u|
β−1 ∫Ω |u|
α
dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L
∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u
0 ∈ L
q
(Ω) (q > 1), and the case α + β < m − 1. 相似文献
7.
The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide. 相似文献
8.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
9.
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
10.
References: 《高校应用数学学报(英文版)》2007,22(1):29-36
In this note, the regularity of Poisson equation -△u = f with f lying in logarithmic function space Lp(LogL)a(Ω)(1<p <∞, a ∈ R) is studied. The result of the note generalizes the W2,p estimate of Poisson equation in Lp(Ω). 相似文献
11.
Jorge García-Melián Julio D. Rossi José C. Sabina de Lis 《Calculus of Variations and Partial Differential Equations》2008,31(2):187-204
In this work we consider the behaviour for large values of p of the unique positive weak solution u
p
to Δ
p
u = u
q
in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u
p
as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit. 相似文献
12.
Shu-Yu Hsu 《Mathematische Annalen》2006,334(1):153-197
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
13.
Harri Ojanen 《Journal of Fourier Analysis and Applications》2000,6(4):427-436
Weighted Lp estimates (1<p<∞) are shown for oscillatory singular integral operators with polynomial phase and a rough kernel of the form
eiP(x,y)Ω(x−y)h(|x−y|)|x−y|−n. We assume that Ω∈L logL(Sn−1) is homogeneous of degree zero and ∫Sn-1Ω=0. The radial factor h has bounded variation. The necessary condition on the weight is similar to the Ap condition but involves rectangles (instead of cubes) arising from a covering of a star-shaped set related to Ω. 相似文献
14.
Zhou Songping 《分析论及其应用》1989,5(1):11-14
In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f0(x)∈C
[−1,1]
2N
, for N+1≤k≤2N, such that p
n
(k)
(f0,1)→f
0
(k)
(1), n→∞, where pn(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture
is given. 相似文献
15.
An interpretation is given to point interactions of the form −Δ+d inL
p
(ℝ
N
), where Δ is the Laplacian operator andd is a pseudopotential related to the ‘Dirac measure at 0', depending on the dimension. They are described as extensions of
−Δ, defined on the space {u∈C
0
∞
(ℝ
N
)|u(0)=0} that are negative generators of analytic semigroups. This is done forN=1,2 and 1<p<∞ and forN=3 and 3/2<p<3. 相似文献
16.
Tetsutaro Shibata 《Annali di Matematica Pura ed Applicata》2003,182(2):211-229
We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u
p
+ f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as
μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics
of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort
of criticality for two-parameter problems seems to be new.
Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003 相似文献
17.
Abstract. It is proved that the semilinear elliptic problem with zero boundary value 相似文献
18.
Let f ∈ L
w
1
[−1, 1], let r
n,m(f) be the best rational L
w
1
-approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim
n → ∞ (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r
n,m
(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006. 相似文献
19.
Suppose thatx=|x(n)|n∈ℤ is a sequence of real numbers. For eachp∈ℕ,x
p
=|x
p
(n)|n∈ℤis the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx
(2p) andx(2
p}-1) are convergent. Thus the problem of convergence of the median filters of infinite-length sequences is completely solved.
Project supported by the National Natural Science Foundation of China (Grant No. 16971047). 相似文献
20.
In this paper, we study the initial-boundary value problem of the porous medium equation u
t
= Δu
m
+ V(x)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)
σ
. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u
0 ≥ 0. 相似文献