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1.
Let Ω be a bounded open subset of ℝ
n
, n > 2. In Ω we deduce the global differentiability result for the solutions u ∈ H
1 (Ω, ℝ
n
) of the Dirichlet problem with controlled growth and nonlinearity q = 2.
The result was obtained by first extending the interior differentiability result near the boundary and then proving the global
differentiability result making use of a covering procedure. 相似文献
2.
This paper deals with the existence of weak solutions in W
01(Ω) to a class of elliptic problems of the form in a bounded domain Ω of ℝ
N
. Here a satisfies for all ξ∈ℝ
N
, a.e. x∈Ω,
, h
1∈ L
loc
1(Ω), h
1( x) ≧1 for a.e. x in Ω; λ
1 is the first eigenvalue for −Δ
p
on Ω with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.
相似文献
3.
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0. 相似文献
4.
Let where m
1 < m
2 < … < m
t
≦ , δ
x
→ 0, p runs over the primes p ≧ ≦ 1, | X
p
| ≦ 1. It is assumed that m
v
, , X
p
may depend on x.
Assume that . It is proved that for almost all irrational α, π( x) = number of primes up to x.
Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA
T46993. 相似文献
5.
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f ( x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
6.
In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation under which its solution satisfying the initial condition stabilizes to zero, i.e., there exists the limit uniform in x from every compact set K in ℝ N for any function u
0( x) belonging to a certain uniqueness class of the problem considered and growing not rapidly than
with a > 0 and b < 0 at infinity.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 79–97, 2006. 相似文献
7.
Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ
2
, and let
be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:
, where
with
is the scalar product in ℝ
3
and
is the external loading per unit surface. We take into account the fact that an infinite amount of energy is required to compress
a finite surface of the plate into zero surface, i.e.,
Mathematics Subject Classification (2000) 49J45 相似文献
8.
Two Beurling generalized number systems, both with and k > 0, are constructed. The associated zeta function of the first satisfies the RH and its prime counting function satisfies
π( x) = li ( x) + O( x
1/2). The associated zeta function of the second has infinitely many zeros on the curve σ = 1−1/log t and no zeros to the right of the curve and the Chebyshev function ψ( x) of its primes satisfies
and
A sharpened form of the Diamond–Montgomery–Vorhauer random approximation and elements of analytic number theory are used
in the construction. 相似文献
9.
The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special
class of functionals: with respect to the convergence u n → u in measure, v n ⇀ v in L p(Ω;ℝ d)
in W −1,p(Ω), and χ n ⇀ χ in L p(Ω), where χ n ∈ Z:= {χ ∈ L ∞(Ω): 0 ≤ χ(x) ≤ 1 for a.e. x}. Here
is a constant-rank partial differential operator. The main result is that if the characteristic cone of
has the full dimension, then the l.s.c. is equivalent to the fact that the F ± are both
-quasiconvex and for a.e. x ∈ Ω and for all u ∈ ℝ d. As a corollary, we obtain several results for the functional with respect to the same convergence. We show that this functional is l.s.c. iff Bibliography: 14 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 100–119. 相似文献
10.
We study the limit as n goes to +∞ of the renormalized solutions u
n
to the nonlinear elliptic problems
where Ω is a bounded open set of ℝ
N
, N≥ 2, and μ is a Radon measure with bounded variation in Ω. Under the assumption of G-convergence of the operators , defined for , to the operator , we shall prove that the sequence ( u
n
) admits a subsequence converging almost everywhere in Ω to a function u which is a renormalized solution to the problem
相似文献
11.
Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅)
H
. Let us consider three linear bounded operators, We define the functions where a
i
∈ H and α
i
∈ℝ. In this paper, we discuss the closure and the convexity of the sets Φ
H
⊂ℝ 2 and F
H
⊂ℝ 3 defined by Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case. 相似文献
12.
We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality where c > 0, α ≥ 0 and p > 0 are constants, and is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study
the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation
where V and f are Borel measurable functions conditioned by the generalized Kato class. 相似文献
13.
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈ R+ / Q, h( t) ∈ L∞ [0, 2π ] is 2π-periodic, x±=max {± x, 0 }.
Received: 23 September 2004 相似文献
14.
In an unbounded domain Ω in ℝ
n
( n ≥ 2) with a compact boundary or Ω = ℝ
n
, we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as
where Δ is the Laplacian and c > 0 and p > 0 are constants. The result is applicable to positive solutions of semilinear elliptic equations of Matukuma type.
This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 19740062), Japan Society for the Promotion
of Science. 相似文献
15.
If denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ 1( x) = ∫
x
0 Δ( u) du. The latter bound is, up to ‘ɛ’, best possible.
Received: 8 February 2007 相似文献
16.
For the singular Cauchy problem, the authors find some sufficient conditions for the existence of continuously differentiable
solutions x: (0, ρ] → ℝ ( ρ > 0 is sufficiently small) of the form
where m ≥ 2 and c
1,…, c
m
are definite constants.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal
Conference–2006, Part 3, 2008. 相似文献
17.
Let f( x, y) be a periodic function defined on the region D
with period 2π for each variable. If f( x, y) ∈ C
p ( D), i.e., f( x, y) has continuous partial derivatives of order p on D, then we denote by ω
α,β( ρ) the modulus of continuity of the function and write For p = 0, we write simply C( D) and ω( ρ) instead of C
0( D) and ω
0( ρ).
Let T( x,y) be a trigonometrical polynomial written in the complex form We consider R = max( m
2 + n
2) 1/2 as the degree of T( x, y), and write T
R( x, y) for the trigonometrical polynomial of degree ⩾ R.
Our main purpose is to find the trigonometrical polynomial T
R( x, y) for a given f( x, y) of a certain class of functions such that attains the same order of accuracy as the best approximation of f( x, y).
Let the Fourier series of f( x, y) ∈ C( D) be and let Our results are as follows
Theorem 1 Let f( x, y) ∈ C
p( D ( p = 0, 1) and
Then
holds uniformly on D.
If we consider the circular mean of the Riesz sum S
R
δ
( x, y) ≡ S
R
δ
( x, y; f): then we have the following
Theorem 2 If f( x, y) ∈ C
p ( D) and ω
p( ρ) = O( ρ
α (0 < α ⩾ 1; p = 0, 1), then
holds uniformly on D, where λ
0
is a positive root of the Bessel function J
0( x)
It should be noted that either or implies that f( x, y) ≡ const.
Now we consider the following trigonometrical polynomial Then we have
Theorem 3 If f( x, y) ∈ C
p( D), then uniformly on D,
Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem
of Zygmund, which can be extended to the multiple case as follows
Theorem 3′ Let f( x
1, ..., x
n) ≡ f( P) ∈ C
p
and let
where
and
being the Fourier coefficients of f(P). Then
holds uniformly.
__________
Translated from Acta Scientiarum Naturalium Universitatis Pekinensis, 1956, (4): 411–428 by PENG Lizhong. 相似文献
18.
Abstract This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF
(the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF
consists of formulas of the form where φ is an -formula, φ
<x
is the -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include:
Theorem
The following five conditions are equivalent for a complete first order theory T in a countable language
with a distinguished linear order:
| (1) |
Some model of T has an elementary end extension with a first new element.
|
| (2) |
T ⊢ REF
.
|
| (3) |
T has an ω
1-like model that continuously embeds ω
1.
|
| (4) |
For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ.
|
| (5) |
For some regular uncountable cardinal κ, T has a κ-like model
that has an elementary extension in which the supremum of M exists.
|
Moreover, if κ is a regular cardinal satisfying κ = κ
<κ
, then each of the above conditions is equivalent to:
| (6) |
T has a κ
+ -like model that continuously embeds a stationary subset of κ.
|
相似文献
19.
This paper deals with conditions for the existence of solutions of the equations considered in the whole space ℝ n, n ≥ 2. The functions A
i
( x, u, ξ), i = 1,…, n, A
0( x, u), and f( x) can arbitrarily grow as | x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝ n. We prove the existence theorem for a solution u ∈ W
loc
1,p
(ℝ n) under the condition p > n.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 133–147, 2006. 相似文献
20.
In this paper we study the Dirichlet problem in Q
T
= Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:
The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α( u), and the term grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework
of the distributional solutions under the hypotheses that both f and the initial datum u
0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem
in the case that the principal part of the operator is of fast-diffusion type, i.e. α( u) = u
m
, with −1 < m < 0.
相似文献
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