共查询到20条相似文献,搜索用时 15 毫秒
1.
Let be a domain. Suppose that f ∈ W1,1loc(Ω,R2) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of Jf. We show that f-1 ∈ W1,1loc(f(Ω),R2) and that Df−1(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f−1 ∈ W1,q(f(Ω),R2) for some 1<q≤2 are also given. 相似文献
2.
Stanislaus Maier-Paape Thomas Wanner 《Archive for Rational Mechanics and Analysis》2000,151(3):187-219
This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation
where Ω⊂ℝ
n
, n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u
3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear
semiflow in certain affine subspaces of H
2(Ω). In a neighborhood U
ε with size proportional to ε
n
around the constant solution , where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold containing there exists a finite-dimensional invariant manifold
which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1. The dimension of is proportional to ε−
n
and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to ε.
(Accepted May 25, 1999) 相似文献
3.
Jiří Neustupa 《Archive for Rational Mechanics and Analysis》2010,198(1):331-348
We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary
∂Ω consists of M unbounded components Γ1, . . . , Γ
M
and N − M bounded components Γ
M+1, . . . , Γ
N
. We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L
3-extension to Ω with the gradient in L
2(Ω)3×3. We assume that the fluxes of α through the bounded components Γ
M+1, . . . , Γ
N
of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ1, . . . , Γ
M
. 相似文献
4.
We show that each quasi-light mapping f in the Sobolev space W
1n
(, R
n
) satisfying ¦Df(x)¦
n
K(x, f)J(x, f) for almost every x and for some KL
r
(), r>n-1, is open and discrete. The assumption that f be quasilight can be dropped if, in addition, it is required that f W
1p
(, R
n
) for some p > = n + 1/ (n-2). More generally, we consider mappings in the John Ball classes Axxx
p,q
(), and give conditions that guarantee their discreteness and openness. 相似文献
5.
Antonin Chambolle 《Archive for Rational Mechanics and Analysis》2003,167(3):211-233
We show that in a two-dimensional bounded open set whose complement has a finite number of connected components, the vector
fields uH
1
(Ωℝ2) are dense in the space of fields whose symmetrized gradient e(u) is in L
2
(Ωℝ4). This allows us to show the continuity of some linearized elasticity problems with respect to variations of the set, with
applications to shape optimization or the study of crack evolution.
(Accepted September 18, 2002)
Published online February 4, 2003
Commmunicated by V. Šverák 相似文献
6.
Theodore Yaotsu Wu 《Acta Mechanica Sinica》2011,27(2):135-151
This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture. 相似文献
7.
R.J. Weinacht 《Journal of Elasticity》1999,57(2):165-170
The linear elastostatic displacement boundary-value problem is considered for a bounded simply connected region Ω in R
n
in the case of a homogeneous isotropic medium. For n = 2 it is shown that if all solenoidal forcing terms result in solenoidal displacements, then Ω is a disk. It is likely that
the result is true for n ≥ 3 but that problem is not resolved.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
8.
Let Ω be a bounded Lipschitz domain in ℝ
n
with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x
n) with x′=(x1, ..., x
n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special
eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with
Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in
3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
9.
Hyunseok Kim 《Archive for Rational Mechanics and Analysis》2009,193(1):117-152
We consider the stationary Navier–Stokes equations in a bounded domain Ω in R
n
with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in L
n
(Ω) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete L
q
-regularity results on very weak solutions in L
n
(Ω). If n = 2, then similar results are also proved for very weak solutions in with any q
0 > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results. 相似文献
10.
We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space
We show, among other things, that there are two positive constants
and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to Lq(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with
and u ∈ L∞(Ω) if a ∈ L∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W1-1/q,q(∂Ω), with
then ∇u, p ∈ Lq(Ω) and if a ∈ C0,μ(∂Ω), with μ ∈ [0, α), then
also, natural estimates holds. 相似文献
11.
Nicolae Tarfulea 《应用数学和力学(英文版)》2000,21(3):283-290
IntroductionInthispaper,weconsidertheellipticsystem(1λ) -Δu=f(λ,x,u)-v (inΩ),-Δv=δu-γv(inΩ),u=v=0(onΩ),whereΩisasmoothboundeddomaininRN(N≥2)andλisarealparameter.Thesolutions(u,v)ofthissystemrepresentsteadystatesolutionsofreactiondiffusionsystemsderivedfromseveralap… 相似文献
12.
Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in
RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0¥(W){C_0^\infty(\Omega)} with respect to the norm:
||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x. 相似文献
13.
In this paper, we consider v(t) = u(t) − e
tΔ
u
0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data
u0 ? L2(\mathbb Rn)?Ln(\mathbb Rn){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L
2 norm of D
β
v(t) decays like
t-\frac |b|-1 2-\frac n4{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u
1(t) such that the L
2 norm of D
β
(v(t) − u
1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions. 相似文献
14.
In this paper, we extend a classical result by J. Serrin [15] to exterior domains , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of satisfying the overdetermined boundary conditions that and u are constant on , and such that , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in [13]. The main result here covers new cases like with . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet
condition that u is constant on . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical
position.
(Accepted May 30, 1997) 相似文献
15.
I. M. Grod 《Nonlinear Oscillations》2005,8(2):163-171
We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ
m
, to have solutions in the space of bilateral number sequences.
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005. 相似文献
16.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2004,6(4):389-404
A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t)
α, y = x/(t* ~ t)β,α,β> 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H3(Ω,R3 X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that
solutions of the Euler equations blow up at a timet = t*, t* < +∞. 相似文献
17.
Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed
in this paper. In particular, it is shown that if {un} is bounded in and if u ∈ BV(Ω;ℝN) are such that un→u in L1(Ω;ℝN) and
in the sense of measures, then for
This result is sharp, and counterexamples are provided in the cases where the regularity of {un} or the type of weak convergence is weakened. 相似文献
18.
We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}
19.
Zdeněk Skalák 《Journal of Mathematical Fluid Mechanics》2010,12(4):503-535
In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show,
for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω
⊆ R3 and β ∈[1/2, 1) , then there exist C0 > 1 and δ0 ∈ (0, 1) such that
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