/pdf/asymptotic-analysis-of-periodic-structures-14wjonu4ba.pdf

Asymptotic Analysis of Periodic Structures

A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L
 loc
 1
 (0, T; X) and if {∂fn/∂t} is bounded in L
 loc
 1
 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p<q.

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Compact sets in the spaceL p (O,T; B)

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Convexity conditions and existence theorems in nonlinear elasticity

Ce traite a marque une date dans le progres des mathematiques et de la physique en levant l’ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens et l’inacceptabilite de leurs formules au regard de la rigueur mathematiques Le merite revient a Laurent Schwartz d’avoir englobe dans une theorie qui est a la fois une synthese et une simplifications, des procedes heterogenes et souvent incorrects utilises dans des domaines tres divers
Une definition correcte et une etude systematique de ces etres nouveaux, les distributions, leur ont donne droit de cite dans l’usage courant Leur utilisation extensive dans de nombreuses branches des mathematiques pures et appliquees, de la physique et des sciences de l’ingenieur fait de ce livre un classique des mathematiques modernes

Laurent Schwartz

Papers

Théorie des distributions

Radon measures on arbitrary topological spaces and cylindrical measures

Méthodes mathématiques pour les sciences physiques

Mathematics for the physical sciences.

Cours d'analyse