Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Distance functional and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A - d(x, A) is continuous for each x X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) - {d(ta, b): a A, b B} is continuous on CL(X) × CL(X).Visiting the University of Minnesota.Visiting California State University, Los Angeles.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

An algebraic approach to the vector ε-algorithm

The vector -algorithm is obtained from the scalar -algorithm by taking the pseudo-inverse of a vector instead of the inverse of a scalar. Thus the vector -algorithm is known only through its rules contrarily to the scalar -algorithm and some other extrapolation algorithms.The aim of this paper is to provide an algebraic approach to the vector -algorithm.     

Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [–^–, ^–], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0⁺, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as the center of the instanton is far from the endpoints of the interval. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit 0⁺ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

On the Incompleteness of (k, n)-arcs in Desarguesian Planes of Order q Where n Divides q

We investigate the completeness of an ( nq – q + n – , n)-arc in the Desarguesian plane of order q where n divides q. It is shown that such arcs are incomplete for 0< n/2 if q/n3. For q = 2n they are incomplete for 0 < < 0.381n and for q = 3n they are incomplete for 0 < < 0.476n. For q odd it is known that such arcs do not exist for = 0 and, hence, we improve the upper bound on the maximum size of such a ( k, n)-arc.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.     

Singular perturbations as a selection criterion for periodic minimizing sequences

Summary Minimizers of functionals like subject to periodic (or Dirichlet) boundary conditions are investigated. While for =0 the infimum is not attained it is shown that for sufficiently small > 0, all minimizers are periodic with period ^1/3. Connections with solid-solid phase transformations are indicated.     

Approximation by infinitely divisible distributions in the multidimensional case

Let be the collection of parallelepipeds in R with edges parallel with the coordinate axes and let be the collection of closed sets in R. Let (G, H)=inf {G{A}H{A}+, H{A}G{A}+ for any; L(G, H)= inf {G{A}H{A}+, H{A}G{A}+ for any, where G, H are distributions in . In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR,253, No. 2, 277–279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances (·, ·), L(·, ·) and. It is proved that, where 0p_i1, ; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 89–103, 1983.     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Lokalkonvexe Unterräume in Topologischen Vektorräumen und Das ε-Produkt

It is well-known that the algebraic tensor product E Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called -product EY (a space of continuous linear mappings from Y into E). So, whenever EY is complete, even the completed tensor product is (isomorphic to) a subspace of EY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding jUE and a continuous linear map ûFU such that u=j·û. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders C_j,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the C_j almost fills in the sense that any two contiguous cylinders C_j are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-_o has thickness of the order of (_o is obtained from by formally setting =0). The cylinder from which the C_j are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction.For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders C_j. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder _o.It is shown that the limiting problem consists of an interior diffusion in _o and a boundary diffusion on the lateral boundary S of _o. The interior diffusion is governed by the 2-dimensional heat equation in _o, for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u_S. Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u_S. As a consequence of the geometry of , local arguments do not suffice to prove convergence in _o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ^N, by means of the Kirzbraun–Pucci extension technique.The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed. Mathematics Subject Classification (2000) 35B27, 35K50, 92C37     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).