Anisotropic singular perturbations—the vectorial case

1994 ◽  
Vol 124 (3) ◽  
pp. 527-571 ◽  
Author(s):  
Ana Cristina Barroso ◽  
Irene Fonseca
Keyword(s):  
Singular Perturbations ◽  
Blow Up ◽  
The Family ◽  
Image Position ◽  
Vector Valued Functions ◽  
Nonconvex Functional ◽  
Vector Valued

We obtain the Γ(L1(Ώ))-limit of the sequencewhere Eε is the family of anisotropic perturbationsof the nonconvex functional of vector-valued functionsThe proof relies on the blow-up argument introduced by Fonseca and Müller.

The gradient theory of phase transitions for systems with two potential wells

1989 ◽  
Vol 111 (1-2) ◽  
pp. 89-102 ◽  
Author(s):  
Irene Fonseca ◽  
Luc Tartar
Keyword(s):  
Phase Transitions ◽  
Interfacial Area ◽  
Gradient Theory ◽  
Compactness Result ◽  
Potential Wells ◽  
The Family ◽  
Two Phases ◽  
Image Position ◽  
Nonconvex Functional ◽  
Vector Valued

SynopsisIn this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbationsof the nonconvex functionalwhere W:RN→R supports two phases and N ≧1. We obtain the Γ(L1(Ω))-limit of the sequenceMoreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of Eε(·) converging in L1(Ω) to a minimiser of E0(·) with minimal interfacial area.

scholarly journals On weak approximation and convexification in weighted spaces of vector-valued continuous functions

1989 ◽  
Vol 31 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Marek Nawrocki
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Weak Approximation ◽  
Completely Regular ◽  
Hausdorff Topological Vector Space ◽  
The Family ◽  
Image Position ◽  
Positive Functions ◽  
Vector Valued

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

Differential Operators with Abstract Boundary Conditions

1978 ◽  
Vol 30 (02) ◽  
pp. 262-288 ◽  
Author(s):  
R. C. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Vector Space ◽  
Differential Operators ◽  
Dimensional Vector ◽  
Absolutely Continuous ◽  
Abstract Boundary ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

Suppose F is a topological vector space. Let ACm ≡ ACm[a, b] be the absolutely continuous m-dimensional vector valued functions y on the compact interval [a, b] with essentially bounded components. Consider the boundary value problem where A0, A are respectively... operator with range in F.

Local minimisers and singular perturbations

1989 ◽  
Vol 111 (1-2) ◽  
pp. 69-84 ◽  
Author(s):  
Robert V. Kohn ◽  
Peter Sternberg
Keyword(s):  
Surface Area ◽  
Singular Perturbations ◽  
Transition Layer ◽  
Model Problem ◽  
Variational Problems ◽  
Γ Convergence ◽  
Image Position ◽  
Nonconvex Domains ◽  
Vector Valued

SynopsisWe construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problemIt is shown that in certain nonconvex domains Ω ⊂ ℝn and for ε small, there exist nonconstant local minimisers uε satisfying uε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.

Best L∞-approximation of measurable, vector-valued functions

1984 ◽  
Vol 96 (3) ◽  
pp. 477-481 ◽  
Author(s):  
Abdallah M. Al-Rashed ◽  
Richard B. Darst
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Convex Banach Space ◽  
Equivalence Classes ◽  
Uniformly Convex ◽  
Integrable Functions ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Measurable Vector

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

scholarly journals The theory of functional least squares

1983 ◽  
Vol 34 (3) ◽  
pp. 336-355 ◽  
Author(s):  
Sándor Csörgő
Keyword(s):  
Least Squares ◽  
Sufficient Condition ◽  
Slope Parameter ◽  
Necessary And Sufficient Condition ◽  
Continuous Vector ◽  
The Family ◽  
Strong Uniform Consistency ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractThe functional least squares procedure of Chambers and Heathcote for estimating the slope parameter in a linear regression model is analysed. Strong uniform consistency for the family of these estimators is proved together with a necessary and sufficient condition for weak convergence in the space of continuous vector valued functions. These results are then used to develop the asymptotic normality of an adaptive version of the functional least squares estimator with minimum limiting variance.

scholarly journals The Fourier transform of vector-valued functions

1985 ◽  
Vol 26 (2) ◽  
pp. 181-186
Author(s):  
Susumu Okada
Keyword(s):  
Fourier Transform ◽  
Integrable Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bounded Continuous Function ◽  
The Fourier Transform ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Convergence In Mean

For each natural number n, let un(x)=(1—cos nx)/πnx2(xɛℝ). It is well–known that a bounded continuous function f on the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined byform a Cauchy sequence in the space L1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the space L1(ℝ) is complete with respect to convergence in mean.

scholarly journals Mean Value Theorems for Vector Valued Functions

1965 ◽  
Vol 14 (3) ◽  
pp. 197-209 ◽  
Author(s):  
Robert M. McLeod
Keyword(s):  
Differential Calculus ◽  
Closed Interval ◽  
Open Interval ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we havefor some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to bewhere cka, b, 0 k, and .

scholarly journals Abelian Ergodic Theorems for Vector-Valued Functions

1976 ◽  
Vol 17 (2) ◽  
pp. 154-154
Author(s):  
P. E. Kopp
Keyword(s):  
Ergodic Theorems ◽  
Definition Of ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

The definition of Ωf,a on p. 57 should read a

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