Le Théorème de Lions & Stampacchia fête ses 40 ans

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1 Le Théorème de Lios & Stampacchia fête ses 40 as Michel Théra Uiversité de Limoges ad XLIM (UMR 6172) Reportig a joit work with E. Erst Fuded by the ANR project NT05 1/43040 Sixièmes Jourées Fraco-Chiliees La Lade des Maures 20 mai, 2008 M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

2 Cotets 1 Positio of the problem 2 The Lios & Stampacchia Theorem, Goal of the talk ad mai result 4 A techical propositio ad a sketch of the proof of the mai theorem M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

3 Cotets 1 Positio of the problem 2 The Lios & Stampacchia Theorem, Goal of the talk ad mai result 4 A techical propositio ad a sketch of the proof of the mai theorem M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

4 Cotets 1 Positio of the problem 2 The Lios & Stampacchia Theorem, Goal of the talk ad mai result 4 A techical propositio ad a sketch of the proof of the mai theorem M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

5 Cotets 1 Positio of the problem 2 The Lios & Stampacchia Theorem, Goal of the talk ad mai result 4 A techical propositio ad a sketch of the proof of the mai theorem M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

6 «Ce que j aime das les mathématiques appliquées, c est qu elles ot pour ambitio de doer du mode des systèmes ue représetatio qui permette de compredre et d agir. Et, de toutes les représetatios, la représetatio mathématique, lorsqu elle est possible, est celle qui est la plus souple et la meilleure. Du coup, ce qui m itéresse, c est de savoir jusqu où o peut aller das ce domaie de la modélisatio des systèmes, c est d atteidre les limites.» M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

7 Positio of the problem Let us cosider a real Hilbert space X with scalar product, ad associated orm. We assume give a liear ad cotiuous operator A : X X, (i short, A L(X )), a closed ad covex subset K of X ad a fixed elemet f X. By a variatioal iequality we mea the problem V(A, K, f ) of fidig u K such that Au f, v u 0 for each v K. Variatioal iequalities were itroduced by Fichera i his aalysis of the Sigorii problem o the elastic equilibrium of a body uder uilateral costraits (see for istace the survey by S. Mazzoe). G. Fichera, Problemi elastostatici co vicoli uilaterali : il problema die Sigorii co ambigue codizioi al cotoro, Mem. Accad. Naz. Licei 8 (1964), p S. Mazzoe, Variatioal aalysis ad applicatios. Proceedigs of the 38th Coferece of the School of Mathematics G. Stampacchia" i memory of Stampacchia ad J.-L. Lios held i Erice, Jue 20 July 1, Edited by Fraco Giaessi ad Atoio Maugeri. Nocovex Optimizatio ad its Applicatios, 79, Spriger-Verlag, New York, M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

8 Positio of the problem I their celebrated 1967 paper, Lios ad Stampacchia exteded Fichera s aalysis to abstract variatioal iequalities associated to biliear forms which are coercive or simply o egative i real Hilbert spaces as a tool for the study of partial differetial elliptic ad parabolic equatios. They had i view applicatios to problems with uilateral costraits i mechaics (for these problems we refer to the book by Duvaut & Lios for details). The theory has sice bee expaded to iclude various applicatios i differet areas such as ecoomics, fiace, optimizatio ad game theory. J.-L. Lios, G. Stampacchia, Variatioal iequalities, Comm. pure ad appl. Math. 20 (1967), p G. Duvaut, J. L. Lios, Les iéquatios e mécaique et e physique, Duod, Paris, F. Facchiei & Jog-Shi Pag, Fiite-dimesioal variatioal iequalities ad complemetarity problems, Vol. I & II. Spriger Series i Operatios Research. Spriger-Verlag, New York, 2003 ad the refereces therei. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

9 The Lios & Stampacchia Theorem Theorem ( ) Lios & Stampacchia For every bouded closed ad covex set K ad f X, the liear variatioal iequality V(A, K, f ) admits at least oe solutio provided A is coercive, that is Au, u a u 2 for every u X ad some a > 0. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

10 Pseudomootoicity i the sese of Brezis (1968) A importat otio i the study of variatioal iequalities was provided by Brezis, who proved that the Lios & Stampacchia Theorem actually holds withi the settig of reflexive Baach spaces, ad for a very large class of (o-liear) operators, called pseudo-mootoe operator. Precisely, let X be a reflexive Baach space with cotiuous dual X. Let us deote by, the duality product betwee X ad X ad by the symbol the weak covergece o X. We say that A is pseudo-mootoe, if it is bouded ad if it verifies Au, u v lim if Au, u v v X (1) wheever {u } N is a sequece i X such that u u ad lim sup Au, u u 0. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

11 Pseudomootoicity i the sese of Brezis (1968) Mootoe operators which are hemicotiuous are pseudomootoe T (x 0 + t y) T (x 0 ) as t 0 for all y A : X X is F-hemicotiuous if for all v X, the fuctio u Au, u v is weakly lower semicotiuous. F-hemicotiuity = pseudomootoicity. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

12 Lemme ( ) Let X be a real Hilbert space ad A L(X ). The A is pseudo-mootoe if ad oly if [ ] [u 0] = lim if Au, u 0. (2) M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

13 Proof of the Lemma Let A be a L(X )-pseudo-mootoe operator, ad u 0. We oly eed to prove relatio (2) whe lim if Au, u 0. I this case we may also suppose, by takig, if ecessary, a subsequece, that lim sup Au, u 0. Takig v = 0 i the defiitio of pseudomootoicity we derive : 0 = A0, 0 0 lim if Au, u, that is 0 = lim if Au, u. Relatio (2) holds accordigly for every L(X )-pseudo-mootoe operator. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

14 Proof of the Lemma Coversely take A L(X ) such that relatio (2) is verified. Pick u 0 such that O oe had, as lim sup Au, u u 0. lim Au, u u = 0, (3) the previous relatio implies that lim sup A(u u), (u u) 0. (4) O the other had, applyig relatio (2) to the sequece {u u} yields lim if A(u u), (u u) 0. (5) Combiig relatios (3), (4) ad (5) we deduce that wheever u u ad lim sup Au, u u 0. lim Au, u u = 0, (6) M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

15 Proof of the Lemma Ay L(X )-operator is also cotiuous with respect to the weak topology o X ; As u u we deduce that lim Au, w = Au, w. Whe applied for w = u v, the previous relatio shows that Au, u v = lim Au, u v = lim if Au, u v, (7) for every sequece u u. Summig up relatios (6) ad (7), we deduce that relatio (1) holds wheever u u ad lim sup Au, u u 0 ; i other words, the operator A is pseudo-mootoe. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

16 Remark It is well kow that, as log as o-liear operators are cocered, problem V(A, K, f ) may admit solutios for every bouded closed ad covex set K, eve if the operator A is ot pseudo-mootoe. I a real Hilbert settig, there exists a cotiuous ad positively homogeeous operator which is ot pseudo-mootoe but for which the variatioal iequality V(A, K, 0) has solutios provided K is a closed ad covex bouded set. Example Let X be a separable Hilbert space with basis {b i : i N }. As customary, for every real umber a, let us set a + = max(a, 0) for the positive part of a. For every i N, let us defie A i : X X, A i (x) = (3 x, b i 2 x ) + b i, ad set A(x) = Σ i=1a i (x). The A is a cotiuous ad positively homogeeous mappig which fails to be pseudo-mootoe, while the variatioal iequality V(A, K, 0) admits solutios for every bouded closed ad covex set K. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

17 Remark We eed ow to prove that the variatioal iequality V(A, K, 0) has solutios for every bouded closed ad covex set K. Let us cosider first the case whe the domai K of the variatioal iequality is ot etirely cotaied withi oe of the coes K i. As every covex set is also a coected set, ad sice {K i : i N } form a family of disjoit ope sets, it follows that K cotais some poit x which does ot belog to ay of the coes K i. Accordigly, A( x) = 0, fact which meas that x is a solutio of the problem V(A, K, 0). Cosider ow the case of a bouded closed ad covex set K cotaied i some coe K p, p N. The A ad A p coicide o the coe K p, ad thus o K. Therefore, A is pseudo-mootoe. The existece of a solutio to problem V(A, K, 0) is guaratied by Brezis s theorem. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

18 Ideed, remark that ay two sets from the family of ope covex coes K i = {x X : 3 x, b i > 2 x }, i N are disjoits. This fact proves that the defiitio of the operator A is meaigful, as at ay poit x, at most oe amog the values A i (x), i N, may be o-ull. A is cotiuous ad positively homogeeous, as is each of the operators A i. A is ot pseudo-mootoe. A(b i ) = b i, so 0 = A0, 0 0 > lim if Ab i, b i 0 = 1; i this iequality proves that relatio (1) does ot hold for b i istead of u i, ad 0 istead of u ad v. Fially let us remark that b i 0 ad lim sup Ab i, b i 0 = 1 0, i to ifer that the operator A is ot pseudo-mootoe. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

19 V(A, K, 0) has solutios for every bouded closed ad covex set K. Let us cosider first the case whe the domai K of the variatioal iequality is ot etirely cotaied withi oe of the coes K i. As every covex set is also a coected set, ad sice {K i : i N } form a family of disjoit ope sets, it follows that K cotais some poit x which does ot belog to ay of the coes K i. Accordigly, A(x) = 0, fact which meas that x is a solutio of the problem V(A, K, 0). Cosider ow the case of a bouded closed ad covex set K cotaied i the coe K p for some p N. Remarkig that the operators A ad A p coicide o the coe K p, ad thus o K, we deduce that A is pseudo-mootoe. Accordigly, the existece of a solutio to problem V(A, K, 0) is garateed i this case by Brezis s theorem. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

20 Goal of the talk ad mai result The aim of this ote is to establish that, i the origial liear settig of the Lios-Stampacchia Theorem, the pseudo-mootoicity of the operator A, which, i geeral, is oly a sufficiet coditio for the existece of solutios for every bouded covex set K, becomes also a ecessary oe. Theorem ( ) - Mai result - Let X be a ifiite dimesioal real Hilbert space ad A a liear ad cotiuous operator. The followig statemets are equivalet. (i) A is pseudo-mootoe ; (ii) The variatioal iequality V(A, K, f ) admits at least a solutio for every bouded closed ad covex set K ad f X. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

21 A techical propositio The followig techical result proves that a L(X )-operator is pseudo-mootoe oly if its restrictio to some closed hyperplae of X amouts the opposite of a mootoe symmetric L(X )-operator. Propositio ( ) Let X be a ifiite dimesioal real Hilbert space ad suppose that the operator A L(X ) is ot pseudo-mootoe. The we ca costruct a ifiite-dimesioal ad separable closed subspace H of X such that the restrictio of A to H is both symmetric, Au, v = Av, u u, v H, (8) ad egatively defied, for some α > 0. Au, u α u 2, (9) M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

22 Sketch of the proof (i) (ii) Brezis (ii) (i) Let A L(X ) be a operator which is ot pseudo-mootoe. Accordig to the Propositio the biliear ad cotiuous form Θ : H H R, Θ(x, y) = Ax, y is symmetric ad positively defied for some ifiite-dimesioal separable closed subspace H of X. Edow the vector space H with the ier product [x, y] = Θ(x, y), ad cosider B = {b i : i N } a Hilbert basis of (H, [, ]). As usually, if x H, let x i deote the i-th coordiate of x with respect to B, x i = [x, b i ] for every x H, i N. We prove that the set K { K = x H : x i 1 2 i ad ( ) } i xi 2 2, is a bouded closed ad covex subset of X such that the variatioal iequality V(A, K, 0) does ot have solutios. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21 i=1

23 The case of ubouded sets K Theorem (ø) [Maugeri & Raciti, 2009] Let X be a Hilbert space ad A : X X a liear ad cotiuous operator. Let K be a closed ad covex subset of E ad assume that V(A, K, f ) admits a solutio. The A is such that if u u, u K, the lim if Au, u 0. Propositio Let K be a arbitrary closed covex set of a Hilbert space X ad A a cotiuous liear operator o X. TFAE : 1 A is F-hemicotiuous o K ; 2 If u 0, u K, the lim if Au, u 0. M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21

24 M. Théra (Uiv. Limoges ad XLIM) O the Lios & Stampacchia Theorem may 20, / 21