共查询到20条相似文献,搜索用时 750 毫秒
1.
Hans Engler 《Journal of Functional Analysis》1985,64(3):412-435
Consider the heat equation ?ru ? Δxu = 0 in a cylinder Ω × [0,T] ? n+1 smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P, 6▽xu(·, t)6Lp will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p, it is shown how to construct an equivalent Lp-norm, such that ▽x(·, t) is non-increasing in this norm. 相似文献
2.
John Michael S Rassias 《Journal of Mathematical Analysis and Applications》1982,85(1):106-113
In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in 3 and n + 1(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x1, x2, t}-space and {x1, x2,…, xn, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2]. 相似文献
3.
M.Z Nashed 《Journal of Mathematical Analysis and Applications》1976,53(2):359-366
Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let be the linear integral operator induced by the kernel K(s, t) on the space 2[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6Kx?y6 in the 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝01K(si, t) x(t) dt = y(si), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating (where 2 is the generalized inverse of ), without recourse to the normal equation , that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind. 相似文献
4.
Stephen Bancroft 《Journal of Mathematical Analysis and Applications》1975,50(2):384-414
In this paper we discuss the problem of determining a T-periodic solution of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space Rk. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?0 > 0 such that the differential equation has a T-periodic solution for all ? satisfying 0 < ? < ?0. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ0 where λ0 is a fixed vector in Rk. This means that attention is confined in the perturbation procedure to examining the dependence of on λ as λ varies along a line segment terminating at the origin in the parameter-space Rk. The results established here generalize this previous work by allowing one to study the dependence of on λ as λ varies through a “conical-horn” whose vertex rests at the origin in Rk. In the process an implicit-function formula is developed which is of some interest in its own right. 相似文献
5.
Abraham Boyarsky 《Journal of Mathematical Analysis and Applications》1978,63(2):490-501
Let xtu(w) be the solution process of the n-dimensional stochastic differential equation dxtu = [A(t)xtu + B(t) u(t)] dt + C(t) dWt, where A(t), B(t), C(t) are matrix functions, Wt is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rn is (?, δ) attainable if there exists an admissable control u such that P{xtu?S?(x)} ? δ, where S?(x) is the closed ?-ball in Rn centered at x. The set of all (?, δ) attainable points is denoted by (t). In this paper, we derive various properties of (t) in terms of K(t), the attainable set of the deterministic control system . As well a stochastic bang-bang principle is established and three examples presented. 相似文献
6.
Allen Devinatz 《Journal of Functional Analysis》1979,32(3):312-335
Let Ω be a domain in Rn and T = ∑j,k = 1n(?j ? ibj(x)) ajk(x)(?k ? ibk(x)), where the ajk and the bj are real valued functions in , and the matrix (ajk(x)) is symmetric and positive definite for every . If T0 is the same as T but with bj = 0, j = 1,…, n, and if u and Tu are in , then T. Kato has established the distributional inequality ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rn. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rn. 相似文献
7.
Hendrik J Kuiper 《Journal of Mathematical Analysis and Applications》1977,60(1):166-181
In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),…, vm(x)) from into Rm which satisfy ψi(x, t) ? vi(x) ? θi(x, t) for all , where ψiand θi are extended real-valued functions on . We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),…, um(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem. 相似文献
8.
In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C1, and g(x, t)?L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (1) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method. 相似文献
9.
Palle E.T. Jørgensen 《Advances in Mathematics》1982,44(2):105-120
Let Ω be an arbitrary open subset of n of finite positive measure, and assume the existence of a subset Λ ? n such that the exponential functions eλ = exp i(λ1x1 + … + λnxn), λ = (λ1,…, λn) ∈ Λ, form an orthonormal basis for with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of (n, +) by K = Λ0 = {γ ∈ n:γ·λ ∈ 2π}, A = {a ∈ n:Ua U1a = }, where Ut is the unitary representation of n on given by Ute = eitλeλ, t ∈ n, λ ∈ Λ, and where is the multiplication algebra of on L2. Assume that A is discrete. Then there is a discrete subgroup D ? A of dimension n, a fundamental domain for D, and finite sets of representers RΛ, RΓ, , each containing 0, RΛ for in K0, and for in A such that Ω is disjoint union of translates of : Ω = ∪a∈RΩ (a + ), neglecting null sets, and Λ = RΛ ⊕ D0. If RΓ is a set of representers for in D, then Γ = RΓ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = n, direct sum, (neglecting null sets). The case A = n corresponds to Ω = , Λ = D0 and Γ = K. This last case corresponds in turn to a function theoretic assumption of Forelli. 相似文献
10.
Eng-Bin Lim 《Journal of Differential Equations》1978,30(1):49-53
The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x0, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution of degree k (cf. [1]) for a given positive integer k. Furthermore, , where ¦ · ¦ is any norm in Rn. 相似文献
11.
The authors consider irreducible representations of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝Nφ(n)πndn, regarding the π as modeled in L2(Rk) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels Kφ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rn, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rn → Rn with the following properties. The Fourier transforms F1φ = Kφ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ? f is Schwartz class on Rn. If polynomial operators T?P(N) are transferred to operators on Rn such that is transformed isomorphically to P(Rn). 相似文献
12.
Donald C. Solmon 《Journal of Mathematical Analysis and Applications》1979,71(2):351-358
Let Π be a k-dimensional subspace of Rn, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by . In this article certain a priori inequalities are established which show in particular that if , , then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces Lp(Rn), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rn are also obtained. 相似文献
13.
The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) Lnx(t) + δq(t) f(x[g1(t)],…, x[gm(t)]) = 0, where δ = ± 1 and L0x(t) = x(t), Lkx(t) = ak(t)(Lk ? 1x(t))., , is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y.(t) + c1(t) f(y[g1(t)],…, y[gm(t)]) = 0 and (an ? 1(t)z.(t)). ? c2(t) f(z[g1(t)],…, z[gm(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos. 相似文献
14.
John Palmer 《Journal of Functional Analysis》1978,27(3):308-336
In this paper a Cohen factorization theorem x = at · xt (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? at is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L1(+) into B such that ∥ θ ∥ = 1 and θ(L1(+)). B = B = B · θ(L1(+)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm. 相似文献
15.
Susanne M. Kuen Krzysztof P. Rybakowski 《Journal of Mathematical Analysis and Applications》1985,112(2):378-390
Let b: [?1, 0] → be a nondecreasing, strictly convex C2-function with b(? 1) = 0, and let g: n → n be a locally Lipschitzian mapping, which is the gradient of a function G: n → . Consider the following vector-valued integro-differential equation of the Levin-Nohel type . (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components. 相似文献
16.
17.
Béla Uhrin 《Journal of Number Theory》1981,13(2):192-209
Given a lattice and a bounded function g(x), x ∈ Rn, vanishing outside of a bounded set, the functions ?(x)maxu∈Λg(u +x), ?(x)?Σu∈Λ g(u +x), and ?+(x)?Σu∈Λ maxv∈Λ min {g(v + x); g(u + v + x)} are defined and periodic mod Λ on Rn. In the paper we prove that ?(x) + ?+(x) ? 2?(x) ≥ ?(x) + h?+(x) ? 2?(x) holds for all x ∈ Rn, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved. 相似文献
18.
A weighted translation semigroup {St} on L2(+) is defined by for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on +. It is shown that {St} is subnormal if and only if φ2 is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed. 相似文献
19.
Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L2(k) are analyzed in terms of the elementary generator, , for n even. Integral operators are defined using the fundamental solutions pn(x, t) to ut = Au and using real polynomials ql,…, qk on m by the formula, for q = (ql,…, qk), m. It is determined when, strongly on L2(k), . If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form. 相似文献
20.
R.A MacLeod 《Journal of Number Theory》1982,14(2):185-227
Elementary methods are used to study sums of the form for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form for integers p and t, t > 0, where Pt(x) = Bt({x}) and Bt(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σn≤xntσa(n) and Σn≤xEt(n) for integers t and a, a ≥ 0, where σa(n) is the sum of the ath powers of the divisors of n and Et(x) is the error term in the sum Σn≤xntσa(n). 相似文献