共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
Y. V. Kurylev 《Journal of Mathematical Sciences》1996,79(4):1231-1246
In this paper a uniqueness theorem is proved for the wave equation in the domain Q2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given
on a part Γ0 × (0, 2T) of the space-time boundary of the cylinder Q2T, Γ0. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the
wave equation. Bibliography:24 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 113–136.
Translated by T. N. Surkova. 相似文献
3.
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
4.
M. A. Berezhnoi 《Ukrainian Mathematical Journal》2009,61(3):361-382
We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid
filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood
of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ. 相似文献
5.
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
6.
M. S. Agranovich 《Functional Analysis and Its Applications》2009,43(3):165-183
We consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω+ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained
in the standard torus $
\mathbb{T}^n
$
\mathbb{T}^n
. In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces H
p
σ
and B
p
σ
without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface
potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω+ and the complementing domain Ω− are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular
operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission
problems. 相似文献
7.
Jiří Neustupa 《Annali dell'Universita di Ferrara》2009,55(2):353-365
We prove the existence of a weak solution to the steady Navier–Stokes problem in a 2D domain Ω, whose boundary ∂Ω consists of two unbounded components Γ
− and Γ
+. We impose an inhomogeneous Dirichlet—type boundary condition on ∂Ω. The condition implies no restriction on fluxes of the solution through the components Γ
− and Γ
+. 相似文献
8.
S. A. Nazarov 《Journal of Applied and Industrial Mathematics》2009,3(3):377-390
Taking various viewpoints, we study the selfadjoint extensions $
\mathcal{A}
$
\mathcal{A}
of the operator A of the Dirichlet problem in a 3-dimensional region Ω with an edge Γ. We identify the infinite dimensional nullspace def A with the Sobolev space H
−ϰ(Γ) on Γ with variable smoothness exponent −ϰ ∈ (−1, 0); while the selfadjoint extensions, with selfadjoint operators $
\mathcal{T}
$
\mathcal{T}
on the subspaces of H
−ϰ(Γ). To the boundary value problem in the region with a “smoothed” edge we associate a concrete extension, which yields a
more precise approximate solution to the singularly perturbed problem. 相似文献
9.
Yves Benoist 《Inventiones Mathematicae》2006,164(2):249-278
Divisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient
M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:
The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.
Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these
triangles is dense in the boundary of Ω (see Figs. 1 to 4).
Moreover, we construct examples of such divisible convex open sets Ω.
相似文献
10.
Marc Briane Juan Casado-Díaz 《Calculus of Variations and Partial Differential Equations》2007,29(4):455-479
In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F
n
, , defined in L
2(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F
n
is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains
the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by
some minimizers of the equicoercive sequence F
n
, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density
is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question
of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional
conductivity. 相似文献
11.
E. M. E. Zayed 《数学学报(英文版)》2000,16(4):627-636
Abstract
Small-time asymptotics of the trace of the heat semigroup
where {μ
ν
} are the eigenvalues of the negative Laplacian
in the (x
1, x
2)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and
Robin boundary conditions, on the piecewise smooth parts Γ
i
(i = 1, ..., n) of ∂Ω such that
, are considered. Some geometrical properties associated with Ω are determined. 相似文献
12.
The authors study the inverse problem of recovering damping coefficients for two
coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional
measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion
Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp
conditions on Γ0 = ΓnΓ1, T > 0, the uniqueness and stability of the damping coefficients
are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in
2000, together with a convenient tactical route “post-Carleman estimates” suggested by
Isakov in 2006. 相似文献
13.
S. V. Poborchii 《Vestnik St. Petersburg University: Mathematics》2010,43(3):175-182
The problem of representing the solution of the Dirichlet problem for the Laplace equation as a single-layer potential V
ϱ with unknown density ϱ is known to lead to the equation V
ϱ = f for density ϱ, where f is the Dirichlet boundary data. Let Γ be the boundary of a bounded planar domain with an outward or inward peak and T(Γ) be the space of the traces on Γ of functions with finite Dirichlet integral over R
2. It is shown that the operator $
L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right)
$
L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right)
is continuous, and the operator $
\varrho \to V\varrho - \overline {V\varrho }
$
\varrho \to V\varrho - \overline {V\varrho }
(where $
\bar u
$
\bar u
denotes u averaged over Γ) can be uniquely extended to the isomorphism
相似文献
14.
Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations
of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of
the constructed representations is in fact a representation of Γ ⋉λ
C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉λ
C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum
of exactly two irreducible, inequivalent Γ-representations.
This research was supported by the Italian CNR. 相似文献
15.
We consider a steady-state heat conduction problem P
α
with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ
1
of the boundary of a given bounded domain in R
n
. We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op
α
and its corresponding system u_ g_ op
α
α and adjoint p_ g_ op
α
α states for each α are strongly convergent to g
op
, u_ g
op
and p _ g
op
, respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control,
and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation
with a different boundary condition on the portion Γ
1
. We use the fixed point and elliptic variational inequality theories. 相似文献
16.
We consider a steady-state heat conduction problem P
α
with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ
1
of the boundary of a given bounded domain in R
n
. We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op
α
and its corresponding system u_ g_ op
α
α and adjoint p_ g_ op
α
α states for each α are strongly convergent to g
op
, u_ g
op
and p _ g
op
, respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control,
and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation
with a different boundary condition on the portion Γ
1
. We use the fixed point and elliptic variational inequality theories. 相似文献
17.
Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W
2
2
(Ω) of the equation Δ
x
2
u = f with the boundary conditions u = Δxu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained
by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported
polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If
the variable opening angle α ∈ C∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the
iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case
α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel
and co-kernel of the operators and determining the solution u ∈ W
2
2
(Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a
nonnegative function f ∈ L2(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to
introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical
situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property
remain open. Bibliography: 46 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198. 相似文献
18.
This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H1(Ω) × L2(?Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H1(Ω) into L2(Ω), or L2(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy. 相似文献
19.
The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ
of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an Lp-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity
fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed.
Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution
of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure. 相似文献
20.
G. Avalos 《Applied Mathematics and Optimization》1997,35(2):203-219
In this paper the regularity properties of second-order hyperbolic equations defined over a rectangular domain Θ with boundary
Γ under the action of a Neumann boundary forcing term inL
2 (0,T;H
1/4 (Γ)) are investigated. With this given boundary input, we prove by a cosine operator/functional analytical approach that
not only is the solution of the wave equation and its derivatives continuous in time, with their pointwise values in a basic
energy space (in the interior of Ω), but also that a trace regularity thereof can be assigned for the solution’s time derivative
in an appropriate (negative) Sobolev space. This new-found information on the solution and its traces is crucial in handling
a mathematical model derived for a particular fluid/structure interaction system. 相似文献