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1.
We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L2(R,X), is Fredholm if and only if the not well-posed equation u(t)=D(t)u(t), tR, has exponential dichotomies on R+ and R and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.  相似文献   

2.
We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L1-setting.  相似文献   

3.
The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d2/dx2+Bx2+Ax−2+λxα (B>0, A?0) in L2(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L2(0,∞), namely C0(0,∞) and D(T2,F)∩D(Mλ,α), where the latter is a subspace of the Sobolev space W2,2(0,∞). Adjoints of these differential operators on C0(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C0(0,∞) in D(T2,F)∩D(Mλ,α) in L2(0,∞) lead to the other adjoints. Further density properties C0(0,∞) in D(T2,F)∩D(Mλ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T2,F)∩D(Mλ,α).  相似文献   

4.
Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤rd, we define obstruction groups Er(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) Er(A,AEs(A,A)→Er+s(A,A). For a projective A-module Q of rank nd, with an orientation , we define a Chern class like homomorphism
w(Q,χ):Edn(A,L)→Ed(A,LL),  相似文献   

5.
The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces Lp(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ Rd > 1) on the variable Lebesgue spaces Lp(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Msγδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator Msγδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).  相似文献   

6.
In this paper we consider nonautonomous elliptic operators A with nontrivial potential term defined in I×Rd, where I is a right-halfline (possibly I=R). We prove that we can associate an evolution operator (G(t,s)) with A in the space of all bounded and continuous functions on Rd. We also study the compactness properties of the operator G(t,s). Finally, we provide sufficient conditions guaranteeing that each operator G(t,s) preserves the usual Lp-spaces and C0(Rd).  相似文献   

7.
该文研究周期椭圆算子sun from(j,l=1) to d D_(jw)(x)a_(jl)D_l+V(x)在R~d(d≥3)中的谱性质,其中A=(a_(jl))是d×d阶的实常值正定矩阵,V(x)和w(x)是关于相同格点的周期标量函数,并且w(x)是正的.利用文中第一作者建立的d-环面上的一致Sobolev不等式,证明了该算子的谱是纯绝对连续的,如果V∈L_(loc)~(2pd/(d+2p))(R~d)且w∈A_(1+α)~(p,∞)(T~d)∩L~∞(T~d)(α0,p≥d),或者V∈L_(loc)~(2d/3)/(R~d),ω∈C~1(T~d),或者V∈L_(loc)~(d/2)(R~d),w∈L_(2,loc)~(d/2)(T~d).  相似文献   

8.
In this paper we first introduce Ls(μ)-averaging domains which are generalizations of Ls-averaging domains introduced by S.G. Staples. We characterize Ls(μ)-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A-harmonic tensors in Ls(μ)-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.  相似文献   

9.
Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each BL(D((ωA)1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator BL(D(γ(ωA)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator BL(D(A),X) of rank-1, then A generates a holomorphic C0-semigroup. If A+B generates a C0-semigroup for each operator BL(D(γ(ωA)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T(t)‖=O(tα) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.  相似文献   

10.
This paper deals with some models of mathematical physics, where random fluctuations are modeled by white noise or other singular Gaussian generalized processes. White noise, as the distributional derivative od Brownian motion, which is the most important case of a Lévy process, is defined in the framework of Hida distribution spaces. The Fourier transformation in the framework of singular generalized stochastic processes is introduced and its applications to solving stochastic differential equations involving Wick products and singularities such as the Dirac delta distribution are presented. Explicit solutions are obtained in form of a chaos expansion in the Kondratiev white noise space, while the coefficients of the expansion are tempered distributions. Stochastic differential equations of the form P(ωD) ◊ u(xω) = A(xω) are considered, where A is a singular generalized stochastic process and P(ωD) is a partial differential operator with random coefficients. We introduce the Wick-convolution operator which enables us to express the solution as u = sA ◊ I◊(−1), where s denotes the fundamental solution and I is the unit random variable. In particular, the stochastic Helmholtz equation is solved, which in physical interpretation describes waves propagating with a random speed from randomly appearing point sources.  相似文献   

11.
We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A2(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.  相似文献   

12.
In this paper we obtain an estimate of the norm of the Bergman projection from L p (D, dλ) onto the Besov space B p , 1 < p < + . The result is asymptotically sharp when p → + . Further for the case P : L 1(D, dλ) → B 1, we consider some weak type inequalities with the corresponding spaces.  相似文献   

13.
This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of d and let (D), (D) and (D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1].  相似文献   

14.
Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by αl,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that αl,d−1(B(d,n))=dn,αl,d−1(K(d,n))=αl,d(K(d,n))=dn+dn−1 if d≥3 and nd−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound αl,d−1(B(d,n))≥d2 if n≥3 and d≥5.  相似文献   

15.
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L space into another weighted L space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.  相似文献   

16.
The Banach-Lie algebra L(A) of multiplication operators on the JB-triple A is introduced and it is shown that the hermitian part Lh(A) of L(A) is a unital GM-space the base of the dual cone in the dual GL-space (Lh(A)) of which is affine isomorphic and weak-homeomorphic to the state space of L(A). In the case in which A is a JBW-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space Lh(A) satisfy
0?D(u,u)+D(v,v)?idA,  相似文献   

17.
Let be a family of elliptic differential operators with unbounded coefficients defined in RN+1. In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G:=ADs generates a semigroup of positive contractions (Tp(t)) in Lp(RN+1,ν) for every 1?p<+∞, where ν is an infinitesimally invariant measure of (Tp(t)). Here, under some additional conditions on the growth of the coefficients of A, which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator Gp of (Tp(t)) in Lp(RN+1,ν) for p∈[1,+∞), and we also give a partial characterization of D(Gp). Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T>0.  相似文献   

18.
Stepanov  S. E. 《Mathematical Notes》2004,75(3-4):420-425
We construct a strong Laplacian D * D by using the third operator in the basis {d,d *,D} of the space of natural first-order operators acting on the differential forms of a Riemannian manifold (M,g). We study the properties of the Laplacian D * D and obtain Weitzenbock's formula relating the three strong Laplacians dd *, d * d, and D * D to the curvature of the manifold (M,g).  相似文献   

19.
A space Apq^s (R^n) with A : B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp(Г), where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D(R^n/Г) is dense in it. Related dichotomy numbers are introduced and calculated.  相似文献   

20.
Summary Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all AA(X) and some B L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.  相似文献   

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