Isoperimetric Variational Techniques and Applications
TABLE OF CONTENTS
0 Introduction and Motivations 8
Notations, Elementary notions and Important facts. 1
1.1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Differential 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.
Notations, Elementary notions and Important facts.
1.1 Banach Spaces
Definition 1.1.1 Let X be a real linear space, and k:kX a norm on X and dX the 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 g
1)The topological dual space of V is sometimes denoted V 0:
2 Banach Spaces
2)The dual space V has a canonical norm dened by
kfkV = sup
; 8f 2 V :
3)The dual of every real normed linear space, endowed with its canonical norm is a Banach space.
In order to dene other useful topologies on dual spaces, we recall the following
Definition 1.1.3 (Initial topology)
Let X be a nonempty set, fYigi2I a family of topological spaces (where I is an arbitrary index set) and i : X ?! Y ; i 2 I; a family of maps.
The smallest topology on X such that the maps i; i 2 I are continuous is called the initial topology.
Next, we dene the weak topology of a normed vector space X and the weak star topology of its dual space X which are special initial topologies.
Definition 1.1.4 (weak topology)
Let X be a real normed linear space, and let us associate to each f 2 X the map f : X ?! R given by f (x) = f(x) 8x 2 X:
The weak topology on X is the smallest topology on X for which all the f are continuous.
We write ! ? topology for the weak topology.
Definition 1.1.5 (weak star topology)
Let X be a real normed linear space and X its dual. Let us associate to each
x 2 X the map
x : X ?! R
x(f) = f(x) 8f 2 X:
The weak star topology on X is the smallest topology on X for which all the
x are continuous.
We write ! ? topology for the weak star topology.
3 Banach Spaces
Proposition 1.1.6 Let X be a real normed linear space and X its dual space.
Then, there exists on X three standard topologies, the strong topology given by the canonical norm k:kX on X; the weak topology (! ?topology) and the weak star topology ! ? topology such that :
(X; !) ,! (X; !) ,! (X; k:kX ) :
The following part of this section is devoted to revive spaces.
For any normed real linear space X; the space X of all bounded linear functionals on X is a real Banach space and as a linear space, it has its own corresponding dual space which we denote by (X) or simply by X and often refer to as the the second conjugate of X or double dual or the bi-dual of X:
There exists a natural mapping J : X ?! X dened , for each x 2 X by
J(x) = x
x : X ?! R
is given by
x(f) = f(x)
for each f 2 X:
hJ(x); fi f(x) for each f 2 X:
J is linear and kJxk = kxk for all x 2 X; (i.e.) J is an isometry embedding .
In general, the map J needs not to be onto. Since an isometry is injective, we always identify X to a subspace of X:
The mapping J is called canonical embedding. This leads to the following definition.
Definition 1.1.7 Let X be a real Banach space and let J be the canonical embedding of X into X: If J is onto, then X is said to be reexive. Thus, a reexive real Banach space is one for which the canonical embedding is onto.
We now state the following important theorem.
Theorem 1.1.8 (Eberlein-Smul’yan theorem)
A real Banach space X is reexive if and only if every ( norm ) bounded sequence
in X has a subsequence which converges weakly to an element of X:
4 Hilbert spaces
1.2 Hilbert Spaces
A map : E E ?! C is sesqui linear if:
1) (x + y; z + w) = (x; z) + (x;w) + (y; z) + (y;w)
2) (ax; by) = ab(x; y) where the bar indicates the complex conjugation
for all x; y; z;w 2 E and all a; b 2 C:
A Hermitian form is a sesqui linear form : E E ?! C such that
3) (x; y) = (y; x) ;
A positive Hermitian form is a Hermitian form such that
4) (x; x) 0 for all x 2 E ;
A definite Hermitian form is a Hermitian form such that
5) (x; x) = 0 =) x = 0 :
An inner product on E is a positive definite Hermitian form and will be denoted h: ; :i := (: ; :). The pair (E; h: ; :i) is called an inner product space.
We shall simply write E for the inner product space (E; h: ; : i) when the inner product h: ; : i is known.
In the case where we are using more than one inner product spaces, specification
will be made by writing h: ; :iE when talking about the inner product space
(E; h: ; :i):
Definition 1.2.2 Two vectors x and y in an inner product space E are said to
be orthogonal and we write x ? y if hx; yi = 0: For a subset F of E; then we
write x ? F if x ? y for every y 2 F:
Proposition 1.2.3 Let E be an inner product space and x; y 2 E:
jhx; yij2 hx; xi:hy; yi :
5 Hilbert spaces
For an inner product space (E; h: ; :i); the function k:kE : E ?! R dened by
is a norm on E.
Thus, (E; k:kE) is a normed vector space, hence a metric space endowed with the distance dE : E E ?! R dened by dE(x; y) = kx ? ykE :
Definition 1.2.4 (Hilbert Space)
An inner product space (E; h: ; :i) is called a Hilbert space if the metric space (E; dE) is complete.
1)Hilbert spaces are thus a special class of Banach spaces.
2)Every nite dimension inner product space is complete and simply called
Let H be a Hilbert space. Then, for all u 2 H; Tu(v) := hu; vi denes a bounded linear functional, i.e. Tu 2 H. Furthermore kukH = kTukH:
Theorem 1.2.6 (Riesz Representation theorem)
Let H be a Hilbert space and let f be a bounded linear functional on H: Then,
(i) There exists a unique vector y0 2 H such that f(x) = hx; y0i for each x 2 H;
(ii) Moreover, kfk = ky0k:
Remark 1.2.2 The map T : H ?! H dened by T(u) = Tu is linear,(antilinear in the complex case) and isometric. Therefore the canonical embedding is an isometry showing that any Hilbert space is reexive .
At the end of this part, we state this important proposition which is just a corollary of Eberlein-Smul’yan theorem.
Proposition 1.2.7 Let H be a Hilbert space, then any bounded sequence in H has a subsequence which converges weakly to an element of H: