共查询到20条相似文献,搜索用时 15 毫秒
1.
Tiba 《Applied Mathematics and Optimization》2008,47(1):45-58
Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W
0
l,p
(Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary
value problems of arbitrary order, in arbitrary dimension and with general cost functionals. 相似文献
2.
Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ
Ω
and λ
Π
denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type
are considered where M
n
(z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that
if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh.
Furthermore, we show that
holds for arbitrary simply connected domains whereas the inequality 2
n-1
≤ C
n
(Ω,Π) is proved only under some technical restrictions upon Ω and Π . 相似文献
3.
Gao Jia Xiao-ping Yang 《应用数学学报(英文版)》2006,22(4):589-598
Let Ωbelong to R^m (m≥ 2) be a bounded domain with piecewise smooth and Lipschitz boundary δΩ Let t and r be two nonnegative integers with t ≥ r + 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and (-Δ)^T on the right-hand side. Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized. 相似文献
4.
On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains
Tomoyuki Suzuki 《manuscripta mathematica》2008,125(4):471-493
Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R
3. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg
in R
3 and Sohr-von Wahl in exterior domains to general domains. 相似文献
5.
Let Ω⊂R
n
be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W
1,p
(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W
0
1,p
(Ω). 相似文献
6.
Guyan Robertson 《K-Theory》2004,33(4):347-369
Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H
0(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H
0(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K
0 group of the boundary crossed product C
*-algebra C(Ω)Γ. If the Tits system has type ?
2, exact computations are given, both for the crossed product algebra and for the reduced group C
*-algebra. 相似文献
7.
We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R
N
, N=2,3, surrounded by a thin layer Σ
ε
, along a part Γ2 of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ
ε
with Reynolds’ number of order 1/ε in Σ
ε
. Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier
law involving a matrix of Borel measures having the same support contained in the interface Γ2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider
an optimal control problem within this context. 相似文献
8.
The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density.
The Neumann problem is examined in a bounded n-dimensional domain Ω+ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω− = R
n
\Ω+. Let Γ be the common boundary of the domains Ω±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R
n
, and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω− with a cusp of an inward peak may be represented as Vρ−, where ρ− ∈ Tr(Γ)* is uniquely determined for all Ψ− ∈ Tr(Γ)*. If Ω+ is a domain with an inward peak and if Ψ+ ∈ Tr(Γ)*, Ψ+ ⊥ 1, then the solution of the Neumann problem for Ω+ has the representation u
+ = Vρ+ for some ρ+ ∈ Tr(Γ)* which is unique up to an additive constant ρ0, ρ0 = V
−1(1). These results do not hold for domains with outward peak. 相似文献
9.
In this paper, we study the asymptotic behavior of the solutionsu
ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ωε=Ω−∪Ω
ε
+
∪γ one part of which (Ω
ε
+
) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω+) is a homogeneous medium; γ=∂Ω
ε
+
∩∂Ω+. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ωε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission
condition on γ. The estimates for the difference betweenu
ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997. 相似文献
10.
11.
M. N. Demchenko 《Journal of Mathematical Sciences》2010,166(1):11-22
The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω
T
:= {x ∈ Ω| dist(x, ∂Ω) < T}, Ω ⊂⊂
\mathbbR \mathbb{R}
3, to the space of transversal fields. The latter space consists of vector fields in Ω
T
tangential to the equidistant surfaces of the boundary ∂Ω. In papers devoted to the dynamical inverse problem for the Maxwell
system, in the framework of the BC-method, the operator M
T
was defined for T < T
ω, where T
ω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M
T
is partially isometric, and its intertwining properties are established. Bibliography: 6 titles. 相似文献
12.
For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;Ω=w
2Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds
2=w(z)-2
dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;Ω
2, with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs
go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter
formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a
result is established concerning the convergence of the series ∑
ω∈G
(1-|ω0|2)
s
for G the covering group of the uniformization map of Ω and 0<s<1.
Received: August 21, 2000?Published online: October 30, 2002
RID="*"
ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005. 相似文献
13.
R. Abreu-Blaya J. Bory-Reyes F. Brackx H. De Schepper F. Sommen 《Complex Analysis and Operator Theory》2012,6(2):359-372
We consider H?lder continuous circulant (2 × 2) matrix functions G12{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in
\mathbbR2n{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis.
This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued
differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from
which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where
the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define
a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian
monogenicity of a circulant matrix function G12{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g12=G12|G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular. 相似文献
14.
We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in RN (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution uλ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of uλ with the exact second term and the ‘optimal’ estimate of the third term. 相似文献
15.
On positivity of solutions of degenerate boundary value problems for second-order elliptic equations
In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B
u
=gon Ω∂Г where ω is a domain in ℝ
n
,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary.
The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability
of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue,
the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this
problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet
boundary value problem, where Γ=∂Ω, were examined intensively by many authors. 相似文献
16.
Abstract In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain Ω with a Lipschitz boundary. We prove that, if the boundary datum a is square summable, then the problem admits a solution which tends to a in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as r–k if and only if a satisfies k compatibility conditions, which we are able to explicit when Ω is the exterior of an ellipse.
Keywords: Dirichlet problem, Asymptotic behavior, Potential theory
Mathematics Subject Classification (2000): 31A05, 31A10 相似文献
17.
18.
We prove that the Schr?dinger equation defined on a bounded open domain of
and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L2(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L2(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in
a given L2(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically
rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L2(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local
machinery [L-T-Z.2, Section 10], to shift down the more natural H1(Ω)-level energy estimate to the L2(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the
general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior
and) boundary dissipation. 相似文献
19.
We prove global Lipschitz regularity for solutionsu : Ω → ℝ
N
of some relaxed variational problems in classes of functions with prescribed Dirichlet boundary data. The variational integrals
under consideration are of the form ∫Ω
W(▽
u
)dx withW of quadratic growth. 相似文献
20.
Dian K. Palagachev 《Journal of Global Optimization》2008,40(1-3):305-318
We derive W
2,p
(Ω)-a priori estimates with arbitrary
p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular
coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent
to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.
相似文献