Isoperimetric Variational Techniques and Application

Isoperimetric Variational Techniques and Application.

ABSTRACT

The exploitation of nature’s propensity oers us ample opportunities to achieve or deal with an optimal objective concerning constrained shape, volume, time, velocity, energy or gain. This vivi es the need to study Optimization Theory and related topics.

In order to make the concepts clear, let us recall some keywords. Given a nonempty set X and a function f : X ! R which is bounded below, computing the number
inf Xf := infff(x) : x 2 Xg (F1) represents a minimization problem posed in X: namely that of nding a minimizing sequence, i.e. (xk)k X such that

lim k!1 f(xk) = inf Xf : The number inf X f is often called the inmal value of f or more simply the inmum of f over X. The function f is usually called the objective function or also inmand. By analogy we have the concepts of supremal value (supremum) and supremand.

TABLE OF CONTENT

Epigraph 2
0 Introduction and Motivations 8
1 Preliminaries:
Notations, Elementary notions and Important facts. 1
1.1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Dierential Calculus in Banach spaces . . . . . . . . . . . . . . 6
1.4 Sobolev spaces and Embedding Theorems . . . . . . . . . . . 9
1.5 Basic notions of Convex analysis . . . . . . . . . . . . . . . . . 13
2 Minimization and Variational methods 18
3 Existence Results of Periodic Solutions of some Dynamical Systems……28
Bibliography 49

INTRODUCTION

Definition 1.1.1 Let X be a real linear space, and k:kX a norm on X and dXthe corresponding metric dened by dX(x; y) = kx ? ykX 8x; y 2 X: The normed linear space (X; k:kX) is a real Banach space if the metric space (X; dX) is complete, i.e., if any Cauchy sequence of elements of space (X; k:kX) converges in (X; k:kX). That is, every sequence satisfying the following Cauchy criterion: 8″ > 0; 9n0 2 N : p; q n0 ) dX(xp; xq) ” converges in X:

Definition 1.1.2 Given any vector space V over a eld F ( where F = R or C), the topological dual space (or simply) dual space of V is the linear space of all bounded linear functionals. We shall denote it by V : V := f’ : ‘ : V ?! F; ‘ linear and bounded.

BIBLIOGRAPHY

[1] Melvyn S. Berger; Periodic Solutions of a Second order Dynamical Systems and Isoperimetric Variational Problems, American Journal of Mathematics, Vol. 93, No 1 (Janv 1971), pp. 1-10.

[2] C.E. Chidume; Applicable functional analysis, International Centre for Theoretical Physics Trieste, Italy, July 2006.
[3] N. Djitte; Lecture Notes on Sobolev spaces and Linear Elliptic Partial Differential Equations, African University of Science and Technology, 2010.

[4] G. Degla Lecture notes on Dierential Analysis,African University of Science and Technology, 2010.

[5] L. Thibault; Lecture Notes on Convex Analysis, African University of Scienceand Technology, 2010.

[6] N. Djitte; Lecture notes 2005-2006 on Optimisation en dimension Calcul des Variations- Contrôle Optimal: Principe de Pontryagin, Université Gaston Berger,Senegal.

[7] R. Tyrell Rockafellar and Rogeo J-B Wets; Variational Analysis , Volume 317, Springer-Verlag, Berlin 1998.

[8] H. Brézis; Analyse Fonctionnelle- Théorie et Applications

[9] L. Todjihounde; Calcul Dierentiel-cours et exercices corrigés. Editions Cépaduès, 2004.

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