共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider the general Cauchy problem with initial data in a Hilbert space and with a formal dissipative linear generator. A complete parametrization is known of the (abstract) boundary conditions which make this problem well set. We exhibit a distinguished subset E of the set of boundary conditions and demonstrate explicitly that the evolution associated with each B in can be represented as a (time independent) average over the evolutions associated with B′ in E. Applications are discussed to Schrödinger equations in bounded regions or with singular potentials. 相似文献
2.
Various initial-boundary value problems and Cauchy problems can be written in the form , where ?:R→ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e?tA in an L1 space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if acting in L1() we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′2 ? ν??′' ? M?′2, where ν ? {1,?1}, then (roughly speaking), the norm of may be estimated in terms of the initial data u0 in L1. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest. 相似文献
3.
Pierre Martin 《Journal of Combinatorial Theory, Series B》1978,24(3):318-324
Let Γ be a finite multigraph; we denote by χ(Γ, x, y) the dichromatic polynominal of Γ, as defined by W. T. Tutte in 1953. We prove that, for any planar multigraph Γ with m edges, χ(Γ, ?1, ?1) = (?1)m · (?2)k, where . Furthermore, if Γ is connected, s = k ? 1 turns out to be a pertinent invariant of the medial of Γ. 相似文献
4.
J.H Michael 《Journal of Mathematical Analysis and Applications》1981,79(1):203-217
We consider the mixed boundary value problem , where Ω is a bounded open subset of n whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on . The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset of the reals such that, if then for is a Fredholm operator if and only if s ∈ . Moreover, = ?xewx, where the sets x are determined algebraically by the coefficients of the operators at x. If n = 2, x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, x is either an open interval of length 1 or is empty; and finally, if n ? 4, x is an open interval of length 1. 相似文献
5.
6.
For a closed densely defined operator T on a complex Hilbert space and a spectral measure E for of countable multiplicity q defined on a σ-algebra over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space of functions which are L2 with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7). 相似文献
7.
K Mahler 《Journal of Number Theory》1980,12(1):20-26
Over 50 years ago, when I was his student at the University of Frankfurt a.M., C. L. Siegel explained to me how to apply Mellin's integral , where the integration is over a line parallel to the imaginary axis and to the right of s = 0, to the study of the function f(z) = Σn=0∞z2n in the neighborhood of roots of unity on the complex unit circle |z| = 1. I later could obtain similar results by means of Poisson's or Euler's summation formula. In the present note I return to this old problem and obtain estimates by means of a very elementary method. It has the further advantage that it allows the study of f(z) in the neighborhood of points on the unit circle which are not roots of unity. 相似文献
8.
L.Thomas Ramsey 《Journal of Functional Analysis》1977,25(3):306-316
Let G be a compact abelian group whose dual group Γ has a finite torsion subgroup. Let μ?M(G) such that ¦μ ¦assigns no mass to any coset of any closed subgroup of G whose index is infinite. Then there is d > 0, dependent only on ∥ μ ∥, such that if for each , then the set is finite. An upper bound on the cardinality of this set is obtained in terms of ∥ μ ∥and the cardinality of the torsion subgroup of Γ. 相似文献
9.
Allan M Krall 《Journal of Differential Equations》1977,24(2):253-267
This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in np[0,1] × np[0,1], 1 ? p < ∞. The adjoint relation in nq[0,1] × nq[0,1], , is derived. Green's formula is also found. Self-adjoint relations are found in n2[0,1] × n2[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of np[0,1] × n2[0,1] is established. 相似文献
10.
E.B Davies 《Journal of Differential Equations》1985,60(1):103-130
We obtain a positive lower bound to the spectrum of certain second-order elliptic operators H on N has a Lipschitz boundary and the coefficients of H become singular as one approaches the boundary. We also find a general formula for the order of the asymptotic eigenvalue distribution of H in some situations where the classical limit formula of Weyl, Courant, Titchmarsh, and others is not applicable. 相似文献
11.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
12.
H.H. Hung 《Topology and its Applications》1982,14(2):163-165
We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family of closed sets, interior-preserving over each member C of which is a countable family {n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈v(Γ))?U. 相似文献
13.
Pascal Cherrier 《Journal of Functional Analysis》1984,57(2):154-206
Nonlinear Neumann problems on riemannian manifolds. Let () be a C∞ compact riemannian manifold of dimension n ? 2 whose boundary B is an (n ? 1)-dimensional submanifold and let be the interior of . Study of Neumann problems of the form: , where, for every are bounded by . Application to the determination of a conformal metric for which the scalar curvature of M and the mean curvature of B take prescribed values. 相似文献
14.
We show that for a C1-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to if and only if every nonzero closed ideal in G × αA has a nonzero intersection with A. Denote by GJ the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups GJ are discrete, then GXαA is simple if and only if A is G-simple and . This result is applicable not only when G is discrete but also when G? or G? provided that A is not primitive. Specializing to single automorphisms (i.e., G=) we show that if (the transposed of) α acts freely on a dense set of points in , then Λ(α)=. The converse is only proved when A is of type I. 相似文献
15.
Thomas I Seidman 《Journal of Differential Equations》1985,60(2):151-173
We obtain a strict coercivity estimate, (generalizing that of T. I. Seidman [J. Differential Equations 19 (1975), 242–257] in considering spatial variation) for second order elliptic operators A: u ? ?▽ · γ(·, ▽u) with γ “radial in the gradient” ?γ(·, ξ) = a(·, |ξ|)ξ for ξ ? m. The estimate is then applied to obtain existence of solutions of boundary value problems: with Dirichlet conditions. 相似文献
16.
In this paper we establish the L2 convergence of a polynomial collocation method for the solution of a class of Cauchy singular linear integral equations, which we term the generalized airfoil equation. Previous numerical results have shown that if the right hand side is smooth then convergence is rapid, with 6 decimal accuracy achievable using 8–10 basis elements. Practical problems in aerodynamics dictate that this equation be solved for discontinuous data. The convergence rate is numerically demonstrated to be , where N is the number of basis elements used. Simple extrapolation is shown to be effective in accelerating the convergence, 4–5 decimal accuracy being achieved using 16 basis elements. 相似文献
17.
Alain A. Lewis 《Mathematical Social Sciences》1985,9(3):197-247
Let 1M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal 1finite subset of 1N such that , we define a class of 1finite cooperative games having the form , where A(F) is the internal algebra of the internal subsets of F, and is a set-function with , , and . If is the space of S-imputations of a game ΓF(1ν) such that , for some , then we prove that contains two nonempty subsets: and , termed the quasi-kernel and S-bargaining set, respectively. Both and are external solution concepts for games of the form ΓF (1ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L(Θ) is the Loeb space generated by the 1finitely additive measure space 〈F, A(F), UF〉, and if a game ΓF(1ν) has a nonatomic representation on L(Θ) with respect to S-bounded transformations, then the standard part of any element in is Loeb-measurable and belongs to the quasi-kernel of defined in standard terms. 相似文献
18.
Let be a smooth second order differential operator on n written in Hörmander form, and be a bounded open set with smooth noncharacteristic boundary. Under a global condition that ensures that the Dirichlet problem is well posed for L on and a nondegeneracy condition at the boundary (precisely: the Lie algebra generated by the vector fields V0, V1,…, Vd is of full rank on the boundary) then the harmonic measure for L starting at any point in has a smooth density with respect to the natural boundary measure. Estimates on the derivatives of this density (the Poisson kernel) similar to the classical estimates for the Poisson kernel for the Laplacian on a half space are given. 相似文献
19.
In this paper we study linear differential systems (1) matrix-valued function defined on the k-torus Tk and (θ, t) → θ + ωt is a given irrational twist flow on Tk. First, we show that if A ? CN(Tk), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class CN on Tk. Next we assume that à is sufficiently smooth on Tk and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix in (1) is the Jacobian matrix of a nonlinear vector field evaluated along a quasi-periodic solution x = φ(t) of (2) . We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of , the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system. 相似文献
20.
Let xm ? a be irreducible over F with char and let α be a root of xm ? a. The purpose of this paper is to study the lattice of subfields of and to this end is defined to be the number of subfields of F(α) of degree k over is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then . The irreducible binomials xs ? b, xs ? c are said to be equivalent if there exist roots βs = b, γs = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x2e ? a which are normal over F (char F ≠ 2) together with their Galois groups are characterized. 相似文献