Isoperimetric variational techniques and applications

Project Abstract

<p> Epigraph 2<br>0 Introduction and Motivations 8<br>1 Preliminaries<br>Notations, Elementary notions and Important facts. 1<br>1.1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br>1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br>1.3 Dierential Calculus in Banach spaces . . . . . . . . . . . . . . 6<br>1.4 Sobolev spaces and Embedding Theorems . . . . . . . . . . . 9<br>1.5 Basic notions of Convex analysis . . . . . . . . . . . . . . . . . 13<br>2 Minimization and Variational methods 18<br>3 Existence Results of Periodic Solutions of some Dynamical Systems.<br>28<br>Bibliography 49<br>7 <br></p>

Project Overview

<p> </p><p>Preliminaries:<br>Notations, Elementary notions and Important facts.<br>1.1 Banach Spaces<br>Denition 1.1.1 Let X be a real linear space, and k:kX a norm on X and dX<br>the corresponding metric dened by dX(x; y) = kx ô€€€ ykX 8x; y 2 X:<br>The normed linear space (X; k:kX) is a real Banach space if the metric space<br>(X; dX) is complete, i.e., if any Cauchy sequence of elements of space (X; k:kX)<br>converges in (X; k:kX). That is, every sequence satisfying the following Cauchy<br>criterion:<br>8″ &gt; 0; 9n0 2 N : p; q n0 ) dX(xp; xq) ”<br>converges in X:<br>Denition 1.1.2 Given any vector space V over a eld F ( where F = R or C),<br>the topological dual space (or simply) dual space of V is the linear space of<br>all bounded linear functionals. We shall denote it by V :<br>V := f’ : ‘ : V ô€€€! F; ‘ linear and bounded g<br>Remark 1.1.1<br>1)The topological dual space of V is sometimes denoted V 0:<br>1<br>2 Banach Spaces<br>2)The dual space V has a canonical norm dened by<br>kfkV = sup<br>x2V;kxk6=0<br>jf(x)j<br>kxk<br>; 8f 2 V :<br>3)The dual of every real normed linear space, endowed with its canonical norm<br>is a Banach space.<br>In order to dene other useful topologies on dual spaces, we recall the following<br>Denition 1.1.3 (Initial topology)<br>Let X be a nonempty set, fYigi2I a family of topological spaces (where I is an<br>arbitrary index set) and i : X ô€€€! Y ; i 2 I; a family of maps.<br>The smallest toplogy on X such that the maps i; i 2 I are continous is called<br>the initial topology.<br>Next, we dene the weak topology of a normed vector space X and the weak<br>star topology of its dual space X which are special initial topologies.<br>Denition 1.1.4 (weak topology)<br>Let X be a real normed linear space, and let us associate to each f 2 X the<br>map<br>f : X ô€€€! R<br>given by<br>f (x) = f(x) 8x 2 X:<br>The weak topology on X is the smallest topology on X for which all the f are<br>continous.<br>We write ! ô€€€ topology for the weak topology.<br>Denition 1.1.5 (weak star topology)<br>Let X be a real normed linear space and X its dual. Let us associate to each<br>x 2 X the map<br>x : X ô€€€! R<br>given by<br>x(f) = f(x) 8f 2 X:<br>The weak star topology on X is the smallest topology on X for which all the<br>x are continous.<br>We write ! ô€€€ topology for the weak star topology.<br>2<br>3 Banach Spaces<br>Proposition 1.1.6 Let X be a real normed linear space and X its dual space.<br>Then, there exists on X three standard topologies, the strong topology given by<br>the canonical norm k:kX on X; the weak topology (! ô€€€topology) and the weak<br>star topology ! ô€€€ topology such that :<br>(X; !) ,! (X; !) ,! (X; k:kX ) :<br>The following part of this section is devoted to reexive spaces.<br>For any normed real linear space X; the space X of all bounded linear functionals<br>on X is a real Banach space and as a linear space, it has its own corresponding<br>dual space which we denote by (X) or simply by X and often refer to as the<br>the second conjugate of X or double dual or the bidual of X:<br>There exists a natural mapping J : X ô€€€! X dened , for each x 2 X by<br>J(x) = x<br>where<br>x : X ô€€€! R<br>is given by<br>x(f) = f(x)<br>for each f 2 X:<br>Thus<br>hJ(x); fi f(x) for each f 2 X:<br>J is linear and kJxk = kxk for all x 2 X; (i.e.) J is an isometry embedding .<br>In general, the map J needs not to be onto. Since an isometry is injective,we<br>always identify X to a subspace of X:<br>The mapping J is called canonical embedding. This leads to the following<br>denition.<br>Denition 1.1.7 Let X be a real Banach space and let J be the canonical embedding<br>of X into X: If J is onto, then X is said to be reexive. Thus, a<br>reexive real Banach space is one for which the canonical embedding is onto.<br>We now state the following important theorem.<br>Theorem 1.1.8 (Eberlein-Smul’yan theorem)<br>A real Banach space X is reexive if and only if every ( norm ) bounded sequence<br>in X has a subsequence which converges weakly to an element of X:<br>3<br>4 Hilbert spaces<br>1.2 Hilbert Spaces<br>Denition 1.2.1<br>A map : E E ô€€€! C is sesquilinear if:<br>1) (x + y; z + w) = (x; z) + (x;w) + (y; z) + (y;w)<br>2) (ax; by) = ab(x; y) where the bar indicates the complex conjugation<br>for all x; y; z;w 2 E and all a; b 2 C:<br>A Hermitian form is a sesquilinear form : E E ô€€€! C such that<br>3) (x; y) = (y; x) ;<br>A positive Hermitian form is a Hermitian form such that<br>4) (x; x) 0 for all x 2 E ;<br>A denite Hermitian form is a Hermitian form such that<br>5) (x; x) = 0 =) x = 0 :<br>An inner product on E is a positive denite Hermitian form and will be<br>denoted h: ; :i := (: ; :). The pair (E; h: ; :i) is called an inner product space.<br>We shall simply write E for the inner product space (E; h: ; : i) when the inner<br>product h: ; : i is known.<br>In the case where we are using more than one inner product spaces, specication<br>will be made by writting h: ; :iE when talking about the inner product space<br>(E; h: ; :i):<br>Denition 1.2.2 Two vectors x and y in an inner product space E are said to<br>be orthogonal and we write x ? y if hx; yi = 0: For a subset F of E; then we<br>write x ? F if x ? y for every y 2 F:<br>Proposition 1.2.3 Let E be an inner product space and x; y 2 E:<br>Then<br>jhx; yij2 hx; xi:hy; yi :<br>4<br>5 Hilbert spaces<br>For an inner product space (E; h: ; :i); the function k:kE : E ô€€€! R dened<br>by<br>kxkE =<br>p<br>hx; xiE<br>is a norm on E.<br>Thus, (E; k:kE) is a normed vector space, hence a metric space endowed with<br>the distance dE : E E ô€€€! R dened by dE(x; y) = kx ô€€€ ykE :<br>Denition 1.2.4 (Hilbert Space)<br>An inner product space (E; h: ; :i) is called a Hilbert space if the metric space<br>(E; dE) is complete.<br>Remark 1.2.1<br>1)Hilbert spaces are thus a special class of Banach spaces.<br>2)Every nite dimension inner product space is complete and simply called<br>Euclidian Space.<br>Proposition 1.2.5<br>Let H be a Hilbert space. Then, for all u 2 H; Tu(v) := hu; vi denes a<br>bounded linear functional, i.e. Tu 2 H. Furthermore kukH = kTukH :<br>Theorem 1.2.6 (Riesz Representation theorem)<br>Let H be a Hilbert space and let f be a bounded linear functional on H: Then,<br>(i) There exists a unique vector y0 2 H such that<br>f(x) = hx; y0i for each x 2 H;<br>(ii) Moreover, kfk = ky0k:<br>Remark 1.2.2 The map T : H ô€€€! H dened by T(u) = Tu is linear,(antilinear<br>in the complex case) and isometric. Therefore the canonical embedding is<br>an isometry showing that any Hilbert space is reexive .<br>At the end of this part, we state this important proposition which is just a<br>corollary of Eberlein-Smul’yan theorem.<br>Proposition 1.2.7 Let H be a Hilbert space, then any bounded sequence in H<br>has a subsequence which converges weakly to an element of H:<br>5<br>6<br>Dierential Calculus<br>in Banach spaces<br>1.3 Dierential Calculus in Banach spaces<br>In this section, we dene the derivative of a map dened between real Banach<br>spaces.<br>Denition 1.3.1 ( Directional Dierentiability)<br>Let f be a function dened from a real linear space X into a real normed linear<br>space Y and let x0 2 X and v 2 Xnf0g:<br>The function f is said to be dierentiable at x0 in the direction v if the function<br>t 7ô€€€! f(x0 + tv) is dierentiable at t = 0: i.e.<br>t 7ô€€€!<br>f(x0 + tv) ô€€€ f(x)<br>t<br>; t 6= 0;<br>has a limit in Y when t tends to 0: This limit, when it exists is denoted f0(x0; v)<br>or @f<br>@v (x0):<br>Denition 1.3.2 ( Gâteaux Dierentiability)<br>A function f dened from a real linear space X into a real normed linear space<br>Y is Gâteaux Dierentiable at a point x0 2 X if :<br>1) f is dierentiable at x0 in every direction v 2 Xnf0g and<br>2) there exists a bounded linear map A : X ô€€€! Y such that f0(x0; v) = A(v); in<br>other words, the map<br>v 7ô€€€! f0(x0; v)<br>is a bounded linear map from X into Y:<br>In this case the map f0(x0; <img alt="" src="https://s.w.org/images/core/emoji/11/svg/1f642.svg">&nbsp;is called the Gâteaux dierential of f at x0 and is<br>denoted by DGf(x0; <img alt="" src="https://s.w.org/images/core/emoji/11/svg/1f642.svg">&nbsp;or f0G<br>(x0):<br>Denition 1.3.3 (Fréchet Dierentiability)<br>A map f : U X ô€€€! Y whose domain U is an open set of a real Banach<br>space X and whose range is a real Banach space Y is ( Fréchet ) dierentiable<br>at x 2 U if there is a bounded linear map A : X ô€€€! Y such that<br>lim<br>kukô€€€!0<br>kf(x + u) ô€€€ f(x) ô€€€ Auk<br>kuk<br>= 0;<br>or equivalently<br>f(x + u) ô€€€ f(x) ô€€€ Au = o(kuk) :<br>Proposition 1.3.4 If f : U X ô€€€! Y is Fréchet Dierentiable, then f is<br>Gâteaux Dierentiable.<br>6<br>7<br>Dierential Calculus<br>in Banach spaces<br>Proof. Indeed by taking u = tv; in the denition of Fréchet Dierentiability we<br>have<br>f(x + tv) ô€€€ f(x)<br>t<br>=</p><p>A(v) +<br>o(ktvk)<br>ktvk</p><p>by the Fréchet Dierentiability. And since as t ô€€€! 0; u ô€€€! 0; so<br>lim<br>tô€€€!0<br>f(x + tv) ô€€€ f(x)<br>t<br>= A(v)<br>and we are done.<br>Proposition 1.3.5 Let X be a real Banach space and Y be a real normed linear<br>space.Then<br>1) The set of Gâteaux dierentiable mappings from X into Y is a linear subspace<br>of the linear space of all the mappings dened from X into Y space is contained<br>in B(X; Y );<br>2) The set of Fréchet Dierentiable mappings from X into Y is also a subspace<br>of B(X; Y ):<br>Theorem 1.3.6 (Mean Value Theorem in Banach Spaces) Let X and Y be Banach<br>spaces, U X be open and let f : U ! Y be Gâteaux dierentiable. Then<br>for all x1 ; x2 2 X, we have<br>kf(x1) ô€€€ f(x2)k sup<br>t2[0;1]<br>kDGf(x1 + t(x2 ô€€€ x1)k kx1 ô€€€ x2k<br>provided that sup<br>t2[0;1]<br>kDGf(x1 + t(x2 ô€€€ x1)k is nite.<br>Proof. Suppose that the assumptions of Theorem 1.3.6 hold. Let g 2 Y (the<br>dual of Y ) such that jjgjj 1. Then the real-valued function ‘ : [0; 1] ô€€€! R<br>dened by<br>‘(t) = g f(x1 + th) where h = x2 ô€€€ x1<br>is dierentiable on [0; 1] in the usual sense. Moreover we see that<br>‘0(t) = gô€€€<br>DGf(x1 + th)(h)</p><p>; 8 t 2 (0; 1) :<br>It follows from the classical mean valued theorem that<br>j'(1) ô€€€ ‘(0)j sup<br>0j’0(t)j ;<br>7<br>8<br>Dierential Calculus<br>in Banach spaces<br>that is<br>kg f(x1) ô€€€ g f(x2)k sup<br>0j’0(t)j :<br>Moreover for all t 2 (0; 1), we have<br>j’0(t)j =</p><p>gô€€€<br>DGf(x1 + th)(h)</p><p>jjgjj kDGf(x1 + th)k khk<br>kDGf(x1 + th)k khk<br>And so<br>kgô€€€<br>f(x1)ô€€€f(x2)</p><p>k = kgof(x1)ô€€€gf(x2)k</p><p>sup<br>0kDGf(x1 + th)k</p><p>khk :<br>But it is well known as a consequence of the Hahn-Banach theorem that<br>kyk = supfu(y) ; u 2 Y ; kuk 1 g:<br>Therefore we nally have<br>kf(x1) ô€€€ f(x2)k sup<br>t2[0;1]<br>kDGf(x1 + t(x2 ô€€€ x1)k kx1 ô€€€ x2k :<br>Remark 1.3.1 : The intereted reader is refered to [8] for another approach of<br>the proof.<br>Sucient conditions for the Fréchet Dierentiability is given by the following<br>Theorem 1.3.7 Suppose that f : U X ô€€€! Y is a Gâteaux Dierentiable<br>function dened from an open subset of a real Banach space X into a real Banach<br>space Y: If the Gâteaux derivative f0G<br>: U X ô€€€! B(X; Y ) is continous at<br>x 2 U; then f is Fréchet Dierentiable at x and f0(x) = f0G<br>(x):<br>Proof. Let x 2 U: Since U is open, there exixts &gt; 0 such that B(x; ) U:<br>Now for h 2 B(x; ); we dene<br>r(h) = f(x + h) ô€€€ f(x) ô€€€ f0<br>G(x)h: (1.3.1)<br>8<br>9<br>Sobolev spaces and<br>Embeddings Theorems<br>The Gâteaux Dierentiability of f at x implies that r is also Gâteaux Dierentiable,<br>and<br>r0<br>G(h) = f0<br>G(x + h) ô€€€ f0<br>G(x):<br>Applying theorem 1.3.6 on the segment line connecting 0 and h; we have that<br>kr(h)k M(h)khk;<br>where<br>M(h) = sup<br>0t1<br>kr0<br>G(th)k:<br>The continuity of the Gâteaux Dierential of f at x implies that M(h) ! 0 as<br>h ! 0; so r(h) = o(h): Relation 1.3.1 assures that f is Fréchet Dierentiable at<br>x; and so f0(x) = f0G<br>(x):<br>1.4 Sobolev spaces and Embedding Theorems<br>We recall the following notations and basic results from Distridutions Theory.<br>Let<br>RN be an open subset of RN:<br>A multi-index is a vector (1 ; ; N ) 2 NN: The length of is<br>jj = 1 + + N :<br>Let u 2 L1<br>loc(<br>); where L1<br>loc(<br>) is the set of functions which are integrable on<br>every compact subset of<br>: If is a multi-index, we set<br>D :=<br>Djj<br>@x1<br>1 @x<br>N<br>N<br>We also recall that we denote by D(<br>) the set of C1 ô€€€functions dened on</p><p>with compact support in<br>:<br>Denition 1.4.1<br>We say that the function v is the -th weak partial derivative of u if :<br>1) v 2 L1<br>loc(<br>);<br>2) v = Du in the sens of distribution , i.e.<br>Z</p><p>u(x)D (x)dx = (ô€€€1)jj<br>Z</p><p>v(x) (x)dx; 8 2 D(<br>):<br>9<br>10<br>Sobolev spaces and<br>Embeddings Theorems<br>Denition 1.4.2 Let f; g 2 L1<br>loc(RN): We dene the convolution product f g<br>of f and g by<br>(f g)(x) =<br>Z<br>RN<br>f(x ô€€€ y)g(y)dy<br>Theorem 1.4.3 Let (n)n be a sequence of functions such that :<br>n 2 D(RN); supp n = B(0;<br>1<br>n<br>);<br>Z<br>RN<br>n(x)dx = 1; n 0 on RN:<br>(Such a sequence of smooth functions is called Friedrich mollier ).<br>If f 2 L1<br>loc(RN) then the convolution product<br>f n(x) =<br>Z<br>RN<br>f(x ô€€€ y)n(y)dy<br>exists for each x 2 RN:<br>Moreover<br>1. f n 2 C1(RN);<br>2. If K is a compact set of points of continuity of f; then f n ô€€€! f uniformly<br>on K as n ô€€€! 1:<br>Proof. Since supp n = B(0; 1<br>n); ( which is compact ), and using f 2 L1l<br>oc(RN)<br>we get<br>jfn(x)j = j(fn)(x)j =</p><p>Z<br>B(0; 1<br>n )<br>f(x ô€€€ y)n(y)dy</p><p>=<br>Z<br>B(0; 1<br>n )<br>jf(xô€€€y)jn(y)dy &lt; 1:<br>Further, since<br>supp</p><p>@n<br>@xi</p><p>B(0;<br>1<br>n<br>) and<br>@<br>@xi<br>[f(y)n(x ô€€€ y)] =<br>@n(x ô€€€ y)<br>@xi<br>f(y);<br>we get<br>@n(x ô€€€ y)<br>@xi<br>f(y)</p><p>Mnjf(y)j<br>B(0; 1n<br>)<br>and using a corollary of Lebesgue dominated convergence theorem, we have :<br>@<br>@xi<br>Z<br>RN<br>f(y)n(xô€€€y)dy =<br>Z<br>RN<br>@n(x ô€€€ y)<br>@xi<br>f(y)dy =<br>Z<br>RN<br>@n(y)<br>@xi<br>f(xô€€€y)dy = f<br>@n<br>@xi<br>:<br>10<br>11<br>Sobolev spaces and<br>Embeddings Theorems<br>Let us prove now that fn ! f as n ! 1; uniformly on compact subsets of<br>RN:<br>Let K be a compact set of points of continuity of fn: So, for any &gt; 0; there<br>exists &gt; 0; such that for x; z 2 K<br>kx ô€€€ zk &lt; =) kf(x) ô€€€ f(z)k &lt; :<br>Now,<br>fn(x) ô€€€ f(x) =<br>Z<br>B(0; 1<br>n )<br>(f(x ô€€€ y) ô€€€ f(x))n(y)dy;<br>because<br>f(x) = f(x):1 = f(x)<br>Z<br>RN<br>n(y)dy =<br>Z<br>RN<br>f(x)n(y)dy and<br>Z<br>RN<br>n(y)dy = 1;<br>Hence, for n n0 with n0 =<br>1</p><p>+ 1,<br>jfn(x) ô€€€ f(x)j<br>Z<br>B(0; 1<br>n )<br>jf(x ô€€€ y) ô€€€ f(x)jn(y)dy</p><p>Z<br>RN<br>n(y)dy = for each x 2 K:<br>Indeed<br>n n0 =) n<br>1</p><p>=)<br>1<br>n</p><p>so that<br>k(x ô€€€ y) ô€€€ xk = kyk<br>1<br>n</p><p>and the result follows from the uniform continuity of fn:<br>We then conclude that f n ô€€€! f uniformly on each compact.<br>Denition 1.4.4 Let 1 q +1; m 2 N: The Sobolev space Wm;p(<br>) is<br>dened by<br>Wm;p(<br>) = fu 2 Lp(<br>); jDu 2 Lp(<br>) for all jj mg:<br>Clearly, Wm;p(<br>) is a real vector space .<br>The case p = 2 will play a special role. The Sobolev spaces Wm;2(<br>) are denoted<br>by Hm(<br>); i.e.<br>Hm(<br>) := Wm;2(<br>):<br>11<br>12<br>Sobolev spaces and<br>Embeddings Theorems<br>The spaces Hm(<br>) have a natural inner-product dened by<br>hu; viHm =<br>jjm<br>Z</p><p>Du(x)Dv(x)dx; 8u; v 2 Hm(<br>)<br>and are Hilbert spaces with the inner-product dened above. We will be more<br>interested in our work by H1(<br>):<br>Concerning Sobolev spaces, we will give here two important results, Rellich-<br>Kondrachov compact embedding theorem (which is crucial in regularity<br>analysis) and the Poincaré Inequality .<br>Theorem 1.4.5 (Rellich-Kondrachov)<br>Let<br>be a C1ô€€€bounded open subset of RN; 1 p &lt; 1 and p := Np<br>Nô€€€p :<br>The followings embeddings are compact:<br>a. If 1 p &lt; N then W1;p(<br>) Lq(<br>); 8q 2 [1; p[;<br>b. If p = N then W1;p(<br>) Lq(<br>); 8q 2 [1;1[;<br>c. If p &gt; N then W1;p(<br>) C(<br>):<br>We have D(<br>) Wm;p(<br>) 8m 2 N; 8p 1; and we dene Wm;p<br>0 (<br>) := D(<br>).<br>Proposition 1.4.6 (Poincaré Inequality)<br>Let 1 p &lt; 1 and<br>a bounded open subset of RN: Then there exists a constant<br>C = C(p;<br>) such that<br>kuk<br>L<br>p<br>(<br>)<br>CkOuk<br>L<br>p<br>(<br>)<br>; 8u 2 W1;p<br>0 (<br>)<br>If<br>is connected and satises a C1 boundary condition, then there exists a constant<br>C = C(p;<br>) such that<br>ku ô€€€ uk<br>L<br>p<br>(<br>)<br>CkOuk<br>L<br>p<br>(<br>)<br>; 8u 2 W1;p(<br>)<br>where<br>u =<br>1<br>j<br>j<br>Z</p><p>u(x)dx:<br>12<br>13<br>Basic notions of<br>Convex analysis<br>1.5 Basic notions of Convex analysis<br>Denition 1.5.1 Let X be a real normed vector space, x0 2 X and<br>f : X ô€€€! R = R [ fô€€€1;+1g an extended real-valued function. One says<br>that f is lower semicontinuous (lsc) at x0 when for any real number r such<br>that r &lt; f(x0); there exists some neighborhood V of x0 such that for all x 2 V;<br>r &lt; f(x):<br>We next connect the lower semicontinuity to some geometric concept. For an<br>extended real-valued function f : X ô€€€! R, we dene its epigraph epi f by<br>epi f := f(x; r) 2 X R : f(x) rg:<br>We also introduce the concept of lower level set for r 2 R by ff(:) rg<br>where for r 2 R,<br>ff(:) rg := fx 2 X : f(x) rg:<br>We therefore give the following characterisation ;<br>Theorem 1.5.2 Let X be a real normed vector space and f : X ô€€€! R an<br>extended real-valued function. The following assertions are equivalent<br>a) f is lower semicontinous (lsc) ;<br>b) The epigraph epi f of f is closed in X R ;<br>c)For any r 2 R; the lower level set ff(:) rg is closed in X:<br>Denition 1.5.3 Let C be a nonempty subset of a real normed vector space X:<br>One says that the set C is convex provided that for x; y 2 C; and 2 [0; 1]; one<br>has x + (1 ô€€€ )y 2 C:<br>Through the epigraph of an extended real-valued function over a real vector<br>space, one can dene the concept of convex function as follow:<br>Denition 1.5.4 Let f : X ô€€€! R an extended real valued function. Ones says<br>that the function f is convex provided that its epigraph is a convex set in XR:<br>13<br>14<br>Basic notions of<br>Convex analysis<br>We also give the following important results.<br>Proposition 1.5.5 Let X be a real normed vector space. If f : X ô€€€! R is lsc<br>at x 2 X and fxng is a sequence in X which converges (strongly) to x then ,<br>lim inf<br>n!1<br>f(xn) f(x):<br>Proposition 1.5.6<br>Let f : X ô€€€! R be any map.<br>Then , f is convex and lsc () f is convex and weakly lsc.<br>And we obtain the following corollary<br>Corollary 1.5.7 Let f : X ô€€€! R be a convex and weakly lsc mapping. Suppose<br>fxng is a sequence in X which converges weakly to x: Then,<br>lim inf<br>n!1<br>f(xn) f(x):<br>Denition 1.5.8 Let X be a real normed vector space and C a nonempty convex<br>subset of X: A function f : C ô€€€! R [ f+1g is said to be convex relative to C,<br>provided for all 2]0; 1[; x; y 2 C<br>f(x + (1 ô€€€ )y) f(x) + (1 ô€€€ )f(y);<br>and f is said to be strictly convex relative to C if for x; y 2 C with x 6= y and<br>f(x); f(y) nite, we have<br>f(x + (1 ô€€€ )y) &lt; f(x) + (1 ô€€€ )f(y):<br>Lemma 1.5.9 (Slope inequality for convex functions)<br>Let I be an unterval of R and h : I ô€€€! R [ f+1g be a proper convex function.<br>Let r1; r2; r3 2 I such that r1 &lt; r2 &lt; r3 and h(r1) and h(r2) are nite. Then<br>h(r2) ô€€€ h(r1)<br>r2 ô€€€ r1</p><p>h(r3) ô€€€ h(r1)<br>r3 ô€€€ r1</p><p>h(r3) ô€€€ h(r2)<br>r3 ô€€€ r2<br>:<br>Furthermore, these inequalities for all such r1; r2; r3 2 I characterizes the convexity<br>of f relative to I:<br>If we have<br>h(r2) ô€€€ h(r1)<br>r2 ô€€€ r1<br>&lt;<br>h(r3) ô€€€ h(r1)<br>r3 ô€€€ r1<br>&lt;<br>h(r3) ô€€€ h(r2)<br>r3 ô€€€ r2<br>for all r1; r2; r3 2 I such that r1 &lt; r2 &lt; r3 and h(r1); h(r2) and h(r3) are nite,<br>we obtain a characterisation of the strict convexity of f relative to C:<br>14<br>15<br>Basic notions of<br>Convex analysis<br>Through the above lemma, we can characterize the convexity of dierentiable<br>functions of one real variable as follows.<br>Proposition 1.5.10 Let I be an open interval of R and h : I ô€€€! R be a realvalued<br>dierentiable function on I: The following assertions are equivalent :<br>(a) h is convex on I;<br>(b) the derivative function h0 is nondecreasing on I;<br>(c) h0(r)(s ô€€€ r) h(s) ô€€€ h(r) for all r; s 2 I:<br>Similarly, the following are equivalent<br>(a’) his strictly convex on I;<br>(b’) the derivative function h0 is increasing on I;<br>(c’) h0(r)(s ô€€€ r) &lt; h(s) ô€€€ h(r) for all r; s 2 I with r 6= s:<br>Proof. (a) ) (b) Let r &lt; t in I: According to the above lemma, we have<br>h0(r) = lim<br>s#r<br>h(s) ô€€€ h(r)<br>s ô€€€ r</p><p>h(t) ô€€€ h(r)<br>t ô€€€ r<br>lim<br>s”t<br>h(t) ô€€€ h(s)<br>t ô€€€ s<br>= lim<br>s”t<br>h(s) ô€€€ h(t)<br>s ô€€€ t<br>= h0(t);<br>which ensures the nondecreasing property of the derivative h0 on I:<br>(b) ) (c) Fix r 2 I and set ‘(s) := h(s) ô€€€ h(r) ô€€€ h0(r)(s ô€€€ r) for all s 2 I:<br>The function ‘ is dierentiable on I and ‘0(s) = h0(s)ô€€€h0(r): By the assumption<br>(b), taking s 2 I; we have that ‘0(s) 0 if s r and ‘0(s) 0 if s r: We<br>then deduce that ‘(s) ‘(r) = 0 for all s 2 I and we are done.<br>(c) ) (a) For s xed in (c), we<br>h(s) sup<br>r2I<br>[h0(r)(s ô€€€ r) + h(r)] h(s)<br>that is<br>h(s) = sup<br>r2I<br>[h0(r)(s ô€€€ r) + h(r)]<br>Further, setting H(s) = [h0(r)(s ô€€€ r) + h(r)] ; for s1; s2 2 I and 2 [0; 1]; we<br>have that<br>H(s1 + (1 ô€€€ )s2) = h0(r)(s1 + (1 ô€€€ )s2 ô€€€ r) + h(r)<br>= h0(r)(s1 + (1 ô€€€ )s2 ô€€€ r + (1 ô€€€ )r) + h(r) + (1 ô€€€ )h(r)<br>= [h0(r)(s1 ô€€€ r) + h(r)] + (1 ô€€€ ) [h0(r)(s2 ô€€€ r) + h(r)]<br>= H(s1) + (1 ô€€€ )H(s2)<br>15<br>16<br>Basic notions of<br>Convex analysis<br>that is, H is convex and hence, h is convex on I as the pointwise supremum of<br>a family of convex functions on I:<br>The case of the strict convexity of h follows the same arguments.<br>Proposition 1.5.11 Let I be an open interval of R and h : I ô€€€! R be a realvalued<br>dierentiable function on I:<br>If the function h is twice dierentiable on I; then h is convex on I if and only if<br>h00(r) 0 for all r 2 I:<br>Similarly if h is twice dierentiable on I and h00(r) &gt; 0 for all r 2 I; then h is<br>strictly convex on I: The converse does not hold, that is, the strict convexity of<br>a twice dierentiable function h on I does not entail the positivity of h00 on I:<br>Proof. Since h is twice derivable, we have<br>h00(r) 0 8r 2 I () h0 is nondecreasing () h is convex<br>and we are done.<br>The case of the strict convexity of h follows the same arguments.<br>We will consider now the more genaral case of dierentiable functions on an<br>open convex set of a normed vector space .<br>Theorem 1.5.12 Let U be an open set of a real normed space (X; k:k) and<br>f : U ô€€€! R be a function which is (Fréchet) dierentiable on U: Then the following<br>assertions are equivalent:<br>(a) f is convex torelative U;<br>(b) hf0(y) ô€€€ f0(x); y ô€€€ xi 0 for all x; y 2 U;<br>(c) hf0(x); y ô€€€ xi f(y) ô€€€ f(x) for all x; y 2 U:<br>Similarly, the following are equivalent :<br>(a’) f is strictly convex relative to U;<br>(b’) hf0(y) ô€€€ f0(x); y ô€€€ xi &gt; 0 for all x; y 2 U with x 6= y;<br>(c’) hf0(x); y ô€€€ xi &lt; f(y) ô€€€ f(x) for all x; y 2 U with x 6= y.<br>Proof. For xed x; y 2 U with x 6= y; consider the open interval<br>I := fs 2 R : x + s(y ô€€€ x) 2 Ug<br>and set h(s) := f(x + s(y ô€€€ x) for all s 2 I: Observing that 0 2 I and 1 2 I<br>with h(0) = f(x) and h(1) = f(y): we have<br>f is convex relative to U if and only if the function h is convex relative to I:<br>16<br>17<br>Basic notions of<br>Convex analysis<br>Indeed, since 0 2 I and 1 2 I and I is an interval, then [0; 1] U;<br>so for all 2 [0; 1] U<br>f(y + (1 ô€€€ )x) = f(x + (y ô€€€ x))<br>= h()<br>= h(:1 + (1 ô€€€ ):0)<br>h(1) + (1 ô€€€ )h(0)<br>= f(y) + (1 ô€€€ )f(x)<br>We then apply proposition 1.5.10.<br>Theorem 1.5.13 Let U be an open set of a real normed space (X; k:k) and<br>f : U ô€€€! R be a function which is (Fréchet) dierentiable on U:<br>If f is twice dierentiable on U; f is convex relative to U if and only if for each<br>x 2 U the bilinear form associated with f00(x) is positive semidenite, i.e.,<br>hf00(x):v; vi 0 for all v 2 X:<br>Similarly assuming the twice dierentiabilty of f on U; a sucient (but not necessary)<br>condition for the strict convexity of f on U is for each x 2 U the positive<br>deniteness of f00(x); i.e., hf00(x):v; vi &gt; 0 for all v 2 X with v 6= 0X<br>Proof. It follows the same arguments as in the proof of the above theorem, but<br>in this case, we apply proposition 1.5.11<br>17</p> <br><p></p>