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1.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
2.
Elliott H Lieb 《Journal of Functional Analysis》1983,51(2):159-165
Let ψ1, …,ψN be orthonormal functions in d and let , or , and let . Lp bounds are proved for p, an example being , with p = d(d ? 2)?1. The unusual feature of these bounds is that the orthogonality of the ψi, yields a factor instead of N, as would be the case without orthogonality. These bounds prove some conjectures of Battle and Federbush (a Phase Cell Cluster Expansion for Euclidean Field Theories, I, 1982, preprint) and of Conlon (Comm. Math. Phys., in press). 相似文献
3.
Tomas Schonbek 《Journal of Differential Equations》1985,56(2):290-296
New and more elementary proofs are given of two results due to W. Littman: (1) Let . The estimate cannot hold for all u?C0∞(Q), Q a cube in , some constant C. (2) Let n ? 2, p ≠ 2. The estimate cannot hold for all C∞ solutions of the wave equation □u = 0 in ; all t ?; some function C: → . 相似文献
4.
David Gurarie 《Journal of Mathematical Analysis and Applications》1985,108(1):223-229
For elliptic operators on Rn and certain of their singular perturbations relative compactness of B with respect to A is established. This result applies to the study of Lp-spectra of elliptic operators for different p. 相似文献
5.
Walter Rudin 《Journal of Functional Analysis》1983,50(1):100-126
Let B be the open unit ball of n, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let be the norm closure, in L∞ = L∞(S), of X. Some results: set I is dense in the unit ball of H∞(B) in the compact-open topology. On is weak1-dense in does not contain . (When .) Every unimodular is a pointwise limit a.e. of products . The zeros of every in the ball algebra (but not of every H∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if . However, cannot be bounded in B if u ? I is non-constant. 相似文献
6.
Steven Zelditch 《Journal of Functional Analysis》1983,50(1):67-80
We prove a Szegö-type theorem for some Schrödinger operators of the form with V smooth, positive and growing like . Namely, let πλ be the orthogonal projection of L2 onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights and with total symbol a(x, ξ); and let f∈C(). Then (assuming one limit exists). 相似文献
7.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,43(2):419-440
Let ψ be convex with respect to ?, B a convex body in Rn and f a positive concave function on B. A well-known result by Berwald states that (1) if ξ is chosen such that .The main purpose in this paper is to characterize those functions f : B → R+ such that (1) holds. 相似文献
8.
Philip W. Smith 《Journal of Approximation Theory》1974,10(4):337-357
In [3] Golomb describes, for 1 < p < ∞, the Hr,p(R)-extremal extension of a function (i.e., the Hr,p-spline with knots in E) and studies the cone of all such splines. We study the problem of determining when is in Wr,p ≡ Hr,p ∩ Lp. If , then is called a Wr,p-spline, and we denote by the cone of all such splines. If E is quasiuniform, then if and only if . The cone with E quasiuniform is shown to be homeomorphic to lp. Similarly, is homeomorphic to hr,p. Approximation properties of the Wr,p-splines are studied and error bounds in terms of the mesh size are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of are derived. 相似文献
9.
Elliptic operators , α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on or a quotient space are considered. It is shown that the Lp-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L1 radial convolution kernel. Some consequences are: A can be closed in all Lp (1 ? p ? ∞), and is essentially self-adjoint in L2 if it is symmetric; A generates an analytic semigroup e?tA in the right half plane, strongly Lp and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability , with φ analytic, is proved for , with convergence in Lp and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients. 相似文献
10.
Jeng-Eng Lin 《Journal of Functional Analysis》1979,31(3):321-332
Consider a smooth solution of and is C1, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, , and for simplicity, qr ? 0. Then the local energy decays faster than exponentially. 相似文献
11.
Herbert Kamowitz 《Journal of Functional Analysis》1975,18(2):132-150
For Hp, 1 ? p < ∞, composition operators C?, defined by for , ? analytic on are considered, and their spectra determined in the case where ? is analytic on an open region containing D?. 相似文献
12.
Pierre A. Vuillermot 《Journal of Mathematical Analysis and Applications》1982,89(1):327-349
Necessary and sufficient conditions are proved for a (2)-Young function G (with independent variable t) to be convex (resp. concave) in t2 in terms of inequalities between the second derivative of G and the first derivative of its Legendre transform G? (with independent variable s). It is then proven that a Young function G is convex (resp. concave) in t2 if and only if G? is concave (resp. convex) in s2. These results, along with another set of inequalities for functions G convex (resp. concave) in t2, allow the proof of the uniform convexity and thereby of the reflexivity with respect to Luxemburg's norm of the Orlicz space over an open domain Ω ?N with Lebesgue measure dξ. When applied to and with p?1 + (p′)?1 = 1, the preceding results lead to the shortest proof to date of two Clarkson's inequalities and of the reflexivity of Lp-spaces for 1 < p < +∞. Finally, some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear Hill's equations; these variational problems are formulated on suitable Orlicz-Sobolev spaces and thereby allow for nonlinear terms which may grow faster than any power of the variable. 相似文献
13.
14.
An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λθ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ2 < θ1. Let (0.4) , and denote by the solutions of (0.1), (0.3), respectively, when K is replaced by . The following results are well known for the loading problem (0.1):(0.5) ; (0.6) the plastic set is connected to the boundary. In this paper we show that, in general, (0.7) ; (0.8) the plastic set is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ1, θ2. 相似文献
15.
Zeev Schuss 《Journal of Mathematical Analysis and Applications》1977,59(2):227-241
Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?2(m ? n)A + B + λ2nI)u? = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given. 相似文献
16.
Teruo Ikebe 《Journal of Functional Analysis》1975,20(2):158-177
A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R3, is obtained, where V(x) is a long-range potential: , grad , being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator from PL2(R3) onto being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ), . 相似文献
17.
Hubert Kalf 《Journal of Functional Analysis》1976,21(4):389-396
For a class of potentials including the Coulomb potential q = μr?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem is shown to hold, u being an eigenfunction of the operator , (+3 := ?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation. 相似文献
18.
Robert S Strichartz 《Journal of Functional Analysis》1973,12(4):341-383
The regular representation of O(n, N) acting on is decomposed into a direct integral of irreducible representations. The homogeneous space is realized as the Hyperboloid . The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □h on H, which is the tangential part of the operator □ = Δx ? Δt on Rn + N. The spectrum of □h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region , and a continuous part ?. As a representation of O(n, N), ? ⊕ ? is unitarily equivalent to the regular representation on L2 of the cone , and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R). 相似文献
19.
Douglas Hensley 《Journal of Number Theory》1984,18(2):206-212
For a > 0 let , the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about () that , where pa(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of pa(u) is worked out. 相似文献
20.
Philippe Delanoe 《Journal of Functional Analysis》1982,45(3):403-430
Let (Vn, g) be a C∞ compact Riemannian manifold without boundary. Given the following changes of metric: , where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)±, (2)±. We first solve Eq. (2)?, under some simple assumptions on F?C∞. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)?, we extend to Eq.(1)? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)+ is not locally invertible, and does not even admit a solution for all , a similar change of functions leads to partial results about Eq. (1)+, via C2 and C3 estimates for Eq. (2)+. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353). 相似文献