Nonlinear Functional Analysis - A First Course

94 Pages·2004·44.869 MB·English

by  S.Kesavan

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Advlsory Edltor
C. S. Scshadri, Chennai Mathematical lnst., Chennai.
Managlng Editor Nonlinear Functional AnalYsis
Itajendra Bhatia, Indian Statistical Inst., New Delhi.
A First Course
Edltors
V. S. Borkar, Tata lnst. of Fundamental Research, Mumbai.
Itrohal Chaudhuri, lndian Statistical Inst., Kolkata.
R. L. Karandikar, Indian Statistical Inst., New Delhi.
M. Ram Murty, Queen's University, Kingston.
C. Musili, Vignan School of Sciences, Hyderabad.
V. S. Sunder, Inst. of Mathematical Sciences, Chennai.
M. Vanninathan, TIFR Centre, Bangalore.
T. N. Venkataramana, Tata Inst. of Fundamental Research, Mumbai.
Already Published Volumes
Il. B. Bapat: Linear Algebra and Linear Models (Second Edition) S. Kesavan
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H. Helson: Linear Algebra (Second Edition)
I). Sarason: Notes on Complex Function Theory
M. G. Nadkarni: Basic Ergodic Theory (Second Edition)
H. Helson: Harmonic Analysis (Second Edition)
K. Chandrasekharan: A Course on Integration Theory
K. Chandrasekharan: A Course on Topological Groups
Il. Illratia (ed.): Analysis, Geometry and Probability
K. lì,. l)aviclson: C* - Algebras by Example
M. Ilhirttacharjee et al.; Notes on Infinite Permutation Groups
V. S. Srrneler: Functional Analysis - Spectral Theory
V. S. Varirdarajan: Algebra in Ancient and Modern Times
M. (;. Naclkarni: Spectral Theory of Dynamical Systems
A. llorcl: Senrisinple Groups and Riemannian Symmetric Spaces
M. Mrrrcolli: Seiberg-Witten Gauge Theory
A. lìortcher and S. M. Grudsky: Toeplitz Matrices, Asymptotic Linear
r\lgt'hra and Functional Analysis
A, l{. l{;ro lnrl [). lJhirlasankaram: Linear Algebra (Second Edition)
('. Ì\{rrrili: Algt'hraic (ìcometry for Beginners
A. ll. tlljrr';rrle : (lotrvcx Polyhedra with Regularity Conditions and
I lillrt'rt's'l'lrird Problem
S. tirrrrr,rrt.srrrr: A ('ourst' irt [)ifferential Geometry and Lie Groups
Srcl 'l iis: lrrtrotlttt'liott ttt ()tttrtc Thcory
ll. Srrr1':'l lrt' ( ongrtl('tìt.(' Stthgrttttp I)roblcn't HINDUSTAN
l{, lìlr.rri.r (t'rl.); ('ottttt't'tt'tl irt lrrl'irrity
BOOKAGENCY
l(. lf lrrl.lr,.r'j,.;r: l)illì'rcrrtill ('lle trltts itt Norrttt'.1 Litrcltr Spitce s
S,rt\'.r l)r'o: l\lgt'l'ririr.'l'.t1t,tl,tg1': A Itritttt'r
Preface
I'rrl'lr',lrr,l I'V Nonlinear Functiollal Analysis studies the properties of (con-
lI ' llrlrrt, lr(r r'ttr, rrr rt t lllrr,',r't[l. yl'\rgl1c'1t1t1r t1o' t(tlrrtliir) ,lirrrrlrsu lttots )s omlvaep ptìiolìgnsli rìbeeatrw eeeqnu antoiormnse din lvinoelvairn gsp ascuecsh a mnda epvpoillvlgess ' nTlewtloì-
N, rr ll, llrr lltì tll(r ilr;r.ior approaclìes to the solution of nolllinear equatiotls could be
lrr,lr,r
,lcs<;ribed as topologi,cal and. uariati,onal Topological rnethods are
, trr,rtl lrlt.rr,t t'sttl.tpttt rkrrived frorn fixecl point theorerns and are usually based 1;1ì the
lrll;' rr rr rr'.lrttr.llt,xr[.tr)ll n( )t iorì of the topological clegree. Variational methods describe the
:iolutions as critical points of a suitable functional and strtdy ways
,,1' locating them.
( ,,; '1 r rlilrt ,r ' .'1)1ì.f lry' l lirrdtrstan Book Agcncy ( Incliu)
Nrr lr,rrr ,rl 1l1q'nr;rtt'riirl l)r()tccted by this copyriglìt n()tice nìiìy be reproduced The airn of this book is to present the basic theory of these
,,t ttltlt.'r'rl ttt,tttl'lìrrnt or by any means, electronic or mechanical, including ln(ìtlìod.s. It is nìeant to be a primer of nonlinear analysis and is
Ir'rlrt,lrlrl'.rrilrrll rr\v'utirlllc:,t rrp('(('(nììrì(ilsirsìgi( )n()frr obmy tahney icnofoprymrigahtitoonw nsetor,rwagheo haansd raeltsroiethveaslo sleyrsitgehmt, ,lcsigned to be use(l as a text or referelxìe book by students at
Io [:r.rrtl litt'rttt's lì)r triurslrtion irrto other languages and publication thereof. llI(: masters or doctoral levr:l in Indian urliversities' The prereq-
;\ll , r;','rt liglrts lor tlris t'tlitiorr vt'st t'xcltrsively witl'r Hindustan Book Agency uisite for following this book is knowledge of functional analysis
(lrr,lr,r). I ltr;rttllt,rril:r'.1 .'xlt,trt is ;t viol;rtiorr ol' ('opyright I-au,irncl is srrhjcct to ;ìn(l topology, tlsually part of the curriculunì at the rnasters level
l, l:,rl ,rt I t, tt r. irr rtrost universities in India.
I't,', lrr, r'rl lt()nt (iilìì(.t'it rt':r.ly c,r;ty srrPPlir.rl lry thc Atrtlr,rr,
The first chapter covers the preliminaries needed from the dif-
l'.ltN Hl t{5() ll-4().1 lr.r'rlrtial calculus in normed linear spaces. It introduces th'e no-
Iiorr of the Fréchet derivative, which generalizes the notion of the
,k.r'ivntive of a real valued function of a single real variable' Sorne
ll;rssical theorerns which are repeatedly used in the sequel' like the
irrrlrlir:it funr:tion theorem ancl Sard's theorenl' are proved here.
'l'hc seconcl chapter develops the theorv of the topological de-
ilt filite dimensions. The Brouwer fixed point theorem and
l,,r'r,r,
lirrsrrk's tltet)rent are proved a1d Solne of their applications are
I r r ( 's( 't tt,c< l.
't'ltc tt(ìxt, (:hapter extends the notion of the topological de-
r,,rr,r. l,o irrf irrit,g rligretìsional spaces for a special class of mappings
Irrown ;r,s ('()nll)a(:t perturbations of the identity. Again, fixed
vll
lr rnrl I lrl'tr'tn:; ( itr 1r;rt'l,icttl;rt'. Sr:lr;rrrrkrr''s tlxrrlrem) are proved an<l ;r (:ertain ,lack of cornpactness' has been built. Another instauce
;r;r;,lrr ;rl rr )nri ;lt r, 1',ivr.tr. is tlre tlreory of I - conuergence. This theory studies the conver-
of the rnininra and ntinirnizers of a family of fttnctionals.
fl(ìrr(:e
'l'lr,' lirtn'l lr clr;r.;rlct' rkr;rls wit,lr lrifrrrt:ation theory. This stu<lies Aga,irr. while the theory can be developed in the very general con-
llrr rurlurr,ol'l,lrc sr.l, of'solrrt,iorrs l,o c<1rt:r,t,ions dependent on a pa- Ir.xt, of a topological space, alot of technical results in Sobolev
runlllr', itr l,lrc rrciglrlrorrrlroo<l of ir ttriviir,l solrrtiont. Science anrl sl)ir(:(ìs are tteeded in order to present reasonably interesting re-
t'n,t,irtlct inlì ;r.r'(' lirll of itrsl,a,tr<:cs of' srr<:lr llrolrlcrns. A variety of srrll,s. The applications of this theory are rnvriacl., ranging from
ttrll lrorls lìrt' l,lrc i<lcrrl,ifit:;rl,iorr of' lrifìrr'<:;r,l,ir)n Ixrirrts - topological rr,rrrlirr<ra,r clasticity to fuolrogelization theory' Such topics, in my
;rtr,l r';r.t'i;rl,iotr;r,l - ;Lt'(' pt'csctrl,rrrl. opirrioil, would be ideal for a secluel to this volurne, Ireant for an
;r,ll,;r,rrr:c<l <:ottrstl <ltr tronlillcar atralysis, specifically aimed at stu-
'l'lrt' r'ottt'lrrrlirrg clrallt,ct' <lc;lls wil,lr l,lrc cxisl,crrr:c :urrl rrlrlti- ,lcrrl,s workirrg irr aplllir:atiorrs Of tnatheulatics.
grlilil,y ,f'r't'il,ir';rl lroittt,s of'lìrrr<:l,iotr;lls rlcfirur<l orr IJ:r,rrilr:h sllnt:cs.
Wlrik' tnitritrriz;rliott is otxt trurl,lul<1, otlx:r <:riti<:nl ltoints,'like sad- 'l'lrc r1ir,l,t'r'i;ll ;lrcscrrl,c<l ltct'c is r:limsi<:ill a,tr<l no claill is rnade
,llr';roittls;rt'r'lìrtttt<l lr.y rtsitrg rcsults like the urountairì pass theo- l,wi.1.rls ,r.igirr;rlil,.y ol' lrrr.scrrt,;r,l,iott (cx<:c1lt, filt' stlttrtl tlf l]ry own
r'r,ttr. r)r'. ilr()t'(. l](ìlrct'a,ll.y, what are known as min - max theorems. rr',,t li itt.lttrl.rl irr ( llt;t;rlr'l' l)' M.y l't'c;t'l'lttt'lrl' of' t'lrtr srtlr'ject has
lr.r,' illlrtr,rrcr',1 1,1' llr. w,t'l<s,l'(l;r,t'1,;rtt ['1], Dtliruling [7],
1,,r'r.;rll.y
Norrlirrr,;r,r' A rr;r,l.ysis, t,orlily, has a bewildering array of tools. In lr,rr,irrrr ll ll. Nir(,rrl,r,r';q llt)l :rrrrl lì;rlritt(}witz [2{)].
slh,r'l,irr11 llrr';rlrovc l,opics, ir, <:orrs<:iorts r:hoice has been nÌade with
l lrc lolk rwirrll olr.iccl ivcs irr rnirr<l: l'lrir;1,,,,,|i f,,t't'w otll, ol'l,lrc llol,t's 1lt'rlllilrtltl ftlr t:OtrfSeS that I
. I,o;rt'ovirlr';r, l,r'xl, lrook wlrir:lt r:;rtr lrc ttsc<l fìrr':ur irrl,rorlrr<:tory 1,.;r\r.,1 \,;l1s.ttS,,t't';t,Si'lts I,, rltlt'l,tlt';r,l Sl,tt<ltrtll,S ift thC TIFR' Cen-
(,n(, sr,nr('sl,(,,' ('r)ut's(, r:ovct'itrg r:lir,ssir:ill tttitl,ct'iill; Irr. ll,rtr;,,,r1,,t,' lrr,li;r (rvlrrt'r' I w,,l'l<r'tl r';rt'lict'), l'lttr Dipartillento
,lr l\l;rl'rrr,rltr,r ( I (';t:,1,'ltttl,,t',,. llrrivcrsil,i rlt'gli Strr<li di ROrna
. lrr lrc of ittl,r't't'st, lr t it (l('It,('trr,/ sl,tt<lcrrl, of'lriglrcr tnitt,lxruratics. I ,r i,,llltr tti ir ll,,nrr,, ll;rlt' ;rtr,l llr,' l,;rlrol';t.l,t,irc MMAS, Urliver-
,,1t.,l, \lr lr \1, t,, l,'r,rrr,,. I rvorrl,l lilil l,o l;rlic llris ollllttrturlity
'l'lrc cx;r,trrlrkrs ;r,tr<l cxct'r:iscs tlult arc f<ntncl throughout the text lrr Ilr,rrrl, Iltr ,,r ttt,,lllllllill1,r lill Ilr|tt l;rt tltlir':l :ttrtl lroslril;rlity'
Ir;rvr. lrcr,n r:lroscrr t,o lrr: irr trrnc with these objectives (though, from
linrc lo l,itn('. nr.y owtr lrias towards clifferential equations does show
I r.,,.1,1 ltl,, t, llrlrrI llr, lrr',lrlr1l',l Nl;rlltcrtt;lli<:;tl St:ietrcels,
rr1r.
) I l, r,rt l',lt,r frrr 1l r r r, r lllrrl l.r, rlrlrr,:i ;rrrrl t't'str;lt't:h eDvirol-
ll is lìrr l,lris t'crì,s()rr t,hat sorne of tlre tools rklveloperl nrrlre re- r'r tr'fl * lrtr lr 1,. Irrrrllr,l .' lrr lrl ttty,, ,tll Ilris 1,99k. I a,lSO tfiaffk
lctrll.\' lr;rvc lrccn (r'cgrct,t,alrl.y) oruittc<1. Two cxrllrlrles sllring to tlrr 1,rrlrlt,,l, I llrrr,lrr',l,rrr ll,',,1, \t,,r'trr,\' ;rll(l tlrc Ma,rragiffg EflitOf
()nr'li ntitt,l. 'l'lrc lirsl is tlrc rrlr:flt,od o.f rrtrt,rrrt,tttt,ti,ort, r:ornpactness ,,1 llrr rr llll\l 1,,'t1', l't,,1 ll'r;,'tt,lt;t lÌlr;rli;r. lìrr tlrtlir (:oopera-
(wlrillr n',rrr llrr. l,'ickls Mc<lill firr P. L. Liorrs). It <leals with the ll,tr irtrrl ,rulrlrrt I I ,rl:,,, llr,rrrL llrr' l\\'() ;l,ll()ll.Ylllolts l'cf'erees who
lotr\'r'rl',r,n('r, ol's(,(lu('n{'cs itr Soll<llcv slltt<:t:s. Its rnain application r, ,r,l lls1,11,,lr llrr, r.rrlltr. nr;ulnli('r ilrt rrtt,l ltt;t<lt'sttv(lral helpful sUg-
tr in llrc sltrrlì' ol' rrrirrirrrizirrg sc(lllorìcos for functionals associated 1,,r':,1 tr)1:, \\'lrr, lr l,',1 1,, llrr rlllltlt'\'('lll('lrl ol'l,lris trlxt. At a I)erSOIlal
1,, rt,nr rir'trrilirrr';r,r'r'llilrl,ir: piurt,ia,l <lifferential equations into which 1,,r,,1. I rr,,rrrlrllrlir, l1llr;r1k rrry l'r'icttrl;r,rrrltrrstwhilecolleague, Prof.
\ l,l'',i' lllhrrxrhr ril, wlr, r'r,,,1(,(l rrrc orr to give the lectures at Bangalore
Irr ;rlrr.r,,'r,r(r kr,1rr, irrsisti'g'that I pubrish the n'tes. I
nln, 11',rhlr r, rlr;rilk ,,nr, of' nr.y sunìrner stu4ents, Mr. sltiva,rtand
llw'lr',',1r, wlr,,. wrrir. rc;r,r'rrirrg tlrc nraterial from the ma'rrsr:ript,
rrln, rlr,l
'lrlrr;r.lrlr' Prrrf'rrr;lrrirrg. Finall.y, for numerous perso'al
rrvurrrH, I r,lr;rrrk r,lrc rrrcrrrlrcrs ol'rrr.y f:rrnil.y and fondly àedicate Contents
Ilrtr lrlulr l,o l,lrlrrr.
('ll,trturi,
S. Kesavan
( )r'l nlrr,r'. ?l)llil.
Differential Calculus on Normed Linear Spaces 1
1.1 The Fbéchet Derivative . 1
I.2
Higher Order Derivatives 16
1.3 Some Important Theorerns 22
I.4
Extrenra of Real Vahred Functions 29
The Brouwer Degree 36
2.1 Definition of the Degree 36
2.2 Properties of the Degree 45
2.3 Brouwerts Theorern and Applications . 50
2.4 Borsuk's Theorern 57
2.5 The Genus 63
The Leray - Schauder Degree 69
3.1
Preliminaries 69
3.2 Definition of the Degree 72
3.3
Properties of the Degree 74
3.4 Fixed Point Theorerns 78
3.5 The Index 82
3.6 An Application to Differential Equations . 87
Bifurcation Theory 92
4.L Introduction . 92
4.2 The Lyapunov - Schmidt Method . 97
4.3 Lemrna
Morse's 99
4.4 A Perturbation Method I07
I l, It t ;rirr,,sr,lsk'ii's'I'llcol'r'nt 111
I t; I I;r I rr rrorvi l,'t,'' I'lrcol'r'nr 113
I'i ,\ \';rri;rl i,,rr;rl Mcl,lrorl rt7
iril, icrr l I )oirrt,s of Functionals 125
I
l\lirrirrriz;r,l,iorr ol' l,'rrrrr:l,iorr;rls 725
Chapter
S;r,,l,llr, l'oirrls 132 1
'l'lrl
l';rl;ris - Srrr;r,lr' ( lorrrliliorr r37
'l'lrr.
| )r,lirrrrr;rl iolr l,r,nlnt;t, r44 Differential Calculus on Normed Linear
'l'lrc
l\lornrl,;r,irr l';ws'l'lrcr)r'(,nr
150
l\l rrll,ilrlicil,.y ol' ( lrit,ir.;r,l )oirrl,s Spaces
I 1l-r5
( lril ir';rl l 'oirrls wil ll ( lorrsl,r.tuiltls ll'r9
.
llilrliogrrrplr.y L7t
1.1 The Fréchet Derivative
I rrrlr.x 175
In this chapter we will review sonre of the irnportant results of the
differential calculus cln nortned linear spaces.
Given a function f : R. + IR we know what is rneant by its
derivative (if it exists) at a point a € IR. It is a number denoted
by f '(o) (or D f(a) or #@) such that
f(a+h)-f(")
lim - f'(a) (1.1.1)
h-+0
or, equivalently,
lf @+tr)- f(")- f'(o)hl -o(h) (1.1.2)
where, by the syrnbol o(h) we unclerstand that the right-hancl side
is equal to a function e(h) such that
-+0. (1. 1.3)
#-+oaslhl
If we wish to generalize this notion of the derivative to a function
defined in an open set of IM' or, rLore generally, to a fitnction
defined in an open set of a normed linear space .E and taking
values in another normed linear space F, it will be convenient
(:H.1 DIFFERENTIAL CALCULUS 1.1 The Fréchet Deri,uat'i,ue
1,, r'r'l',;r'(l .l'tlt,)lt.;Ls r,lx) *lsrrlt of a linear operatio' on à. Thus, that point. I
.['(,,.1 is rrow corrsirkrrc<l as a bounded linearoperator on ]R. which
s;rlislics ( 1.1.2). Wtr ttow define the notion of differeltiability for Example 1.1.L Let E - IR2 and F - IR. Define
lìrrrcl,iorrs rlcf irrrlrl ()rì a norrned linear space.
l,cl, I') ;r,rrrl F bc normed linear spaces (over R).we denotc by : (*t
[.(l';, /") l,lrc sl)iì,(:o of bounded linear transforrnations of E into tr'. / ('" Y) t ry*'rl'rilí\![3, 3ì
Dlx'crtf,i rtfri,iutti:ort,n fu 1,r.t1c.t1io nL.e Tt htle cfu nEct ,bioen a fn io,sp esar,li ds, etto a bned d tiefft ef re: nllt ia-+b lFe jTolirrerrinr gif h(: rt, oA )t h-+e o(0r,ig0i)n a),lo wnge gaenyt tdhiraetc ttihoen lhir'n (ii't. ein. a(l1o.n1g. 6t)h eex liisntes
Ltt,(l, It,t), . € It)l, {n u'i,:,fl tt,h t,lrt:,raet eri,sts a bounded li,near transforntat,ion f ,(o) e and is equal to zero. Thus, if / were differentiable, f '(0,0): : 0.
However, if we pass to the same limit along the parabola A tr2,
ll,f (" + h) - f (") - f,(o)hll : ,fllhll). (1.1.4) the limit turns out to be unity, which contradicts (1.1.5). Thus, /
is not differentiable at the origin even though it possesses a Gàteau
I
I,) qr r,'i,rtu,l,t,rt,l,l,'y,'tDe can write derivative at that point along every directiorr.
f(a + h) - f(") - f'(o)h - e(h) (1.1.5) DefinitionL.L.2 Let f :U c E -+ F beag'iuenfuncti,on. If
f '(") eri,sts for each a € U, we say that f i's d'ifferenti,able 'in U. If
rulttx: ffi - 0 as llhll -+ 0.f the mappi,ng a ,-+ f '(o) i,s conti,nuous frorn U i,nto L(8, F), we
say that f, i,s of class Ct.a
Remark 1-.1.1 The following facts are simple conseque'ces of
l,lrrr a,lrovr: rl<:finition: (i)if / is differentiable at a e U, then / is We now give exarnples to illustrate the Fréchet derivative.
r:rrrrl,irurorrs at that point; (ii)if / is differentiable at a € u,, then
l,lrtr <lrl'ivative f'(") € L(E,r') is uniquely defined. It is for the Example L.L.z Let E,f'be normed linear spaces and let C e
tttti<lttcttcss tlf the derivative that it is convenient to assume that L(E,l7). For b € F, define
l,lrc rlorrrain of dcfinition is an open set. f
'l'ltc <lcrivittive defined above is called the Fréchet derivative f (") -Cr*b.
of' ./' ;r,1, l,lur point a. we can also define the Gàteau derivative of
Then / is differentiable in .E and
.l' it"l, u, ,lrnt,q rt qi,uen uector h e E by means of the limit
f("+th) - f(") f'(r)h - Clt,. for every r,h, e E.
lim
(1.1.6)
t-+0 :
Tlrus f' C.a
Rer.:rrk L.1.2 If'./ is Fréchet differentiable at a point a, then,
fìrr.v.r'.y /r ( Il, it, is Gàteau rlifferentiable at that point along Example 1-.L.3 Let E be a Hilbert spar:e and a : E x.E -+ R a
/r, iunrl llrr'(l:ul,oiur <krrivative is given by f,(")h.The converse is s.yrrrrnetric and continuous bilinear form ot E. Let b e E. Define
rrol, l,r'rrc. A lìrrrr:l,ir)u nray possess a Gàteau derivative at a point
;llottg cvcl'.Y <lit'cr:l,iotr lrttt r:a,n fail to be Fréchet differentiable at f (*) -Iro(*,r) - (b,*), for r € E.
I
I (,,11. I)II,'T,'ERENTIAL CALCULUS 1.1 The Fréchet Deriuat'iue
wlrr,rr.(., ) r.;l;rrr,ls lor. l,lrc irrrrrrr-llrodrrct in E. Then result, continuous fronr L'2 @) into itself (since, O being bounded,
,-(o) is contained in L'@)). Then, N is also differentiable and
(., t lt)
I .l' (t:) - u(t:, h,) + **rO, h) - (b, h,) . if h e L2 (f'L), thern the function l{/ (tt)h, e L'z(O) is given bv
Sitrr',, r/(.. . ) is r.orrt,irrrulrs, .nú,(,u) (à) (r) - l! A, u(r))h,(r).
lo,(h,,fr,) I < tulllh,l|2 . To scc this, notice that by the classical mean value theorem for
funt;tiorrsofseveralvariables,tlrereexists0(,)suclrtlrat0<
ll.rr.r' il, lirlkrws r,hilt / is clifferentiabkr i, E,,'<l th.t
0(*) < 1 ancl
.l't(.t:)lt, - u,(r,h,) - (b,tt,), f<rr ovor.y :r:.Ìt, e E.a f (r,rt(r) + h(:r)) - f (r,u,(r)) - ff A,,,(r) + o (r)h(r))h(*).
Hence, denoting the rì.orrn in ,2 (CI ) by ll.ll, we get
rIl'r)rxrrr;rrr,irrrr prlrc,rr <rl .lcr.t 4l' (:Toh ex lRr[ e-n+r yrRts bkeii ao pgeivreant ofru) lcLteiot lC sI rcrc ;1p t,h^ra tb eth ea lllr("+h) -lr(") - lf/(tr,)hll llg, ,, toh)-ffr,,,,11
rr;rr)lrirrg:r: ,+ .f (r,t) is rneasurable for all fixecl ú € R ancl the llhll
ilr;rf )lrirr11 I r ) .f (:r,,t) is cclntirìuols for alnost all r € o. such By the contiuuity of the Nernytskii operator associatecl to #, ,,
rr l'rrrrcl iorr is r:allcd a Carath,éodory ,fttnct,i,on Let W l-te a vector follows that the term on the riglrt tends to zero as h, -+ O in f2(fl)
sf );r,('(' ol' r'r';rl - valued functions on l). The l{enugt.sk,i,i, oper.ator ancl this proves ortr claint. I
;r,ssrrci;r,l,r,tl 1,,./'is a nonlinear rnapping clefined on w by
N(u)(r)_ Example 1.1.5 Let Q C RN be a bounded dotnain and let /:
f(r,u(r)).
fì x R -+ IR. be function as in the preceding exanrple. Let ai (O) Ue
,\ r'r'rrr;rrli;rlrkr t,lxrorenr, due to Krasnoselsk,ii [1a] (see also Joshi the usual Sobolev space (cf. Kesavan [13]) of functions in f2(O; att
;rrrrl llrsr'[10] firr a proof), is that if (tlfl +Olqi:1, where 1 ( ef wfiose first derivatives are also in that space and which vanish,
t),(t \. ;r,rrrl if'.4/ nraps Lp@) into trq(e), then this rnapping is in the sense of trace on the bortndary AO. Then, the following
t'rrrtl,ittttotls ;l,ll(l lrortttded, i.e. it rnaps boundecl sets i'to boundecl problem has a unique (weak) solution (cf. Kesavan [13]):
s.ls' A l.vlrir';tl t'orr<lition ou f woulcl be a growth colditiol of the -Au(r) _ f(r,u(r)) re O
l1'1rc u(r) : r€0Q.
0
(:r, t)l
l.f
cwoltrtsclr,;.t trol. is;r, n()rr-rrcg;r,t,ivc function in Lq(q ancl b is a positive S/,irrr(cOe )É v4i a(O th) ec rLe'l2a(t0io),l w?e( 2ca)n : t'htpus. dLeefti,n eh t€ heI ,m2(aCpIp).i nIge t? z : € -LH2(jf(lO) -)+
f ,r'l f I lrr';r lrorrrr<lc<l <lornain and !"t p: Q:2. Assume that, rr. tr*r ''iq.e so:i:;" IT;,;,
irr ;r,l,lit,i,rrr. ./' is irr f ) x ll( ;'r<l that ff @,u,(r)) is in r*(CI) if u e th(,):r € r)
/,''(f l). 't'lrrrs, t,lrc rrr;uplring ,u, ts+ #(",u (.)) is, by Kras'oselsk,ii,s z(t:) 0 :re 0ll.
-
T
t;
(, I I. I DIT,'T,'ENENTIAL
CALCULT]S
1.1 Th,e Fréchet Deri,uatiue
Wr, r.l;r,irrr l,lr;rl,/,
't, 'l'(rr, I lr,), l,lrc1i1s, rCli f:li rrurr r-r tiua)l r_le z a vnadn !ihs.ha*t, TotQì ró)óh :* rr2a. Isnactliesfeieds, if il differentiable at the origin. I
-AC - f(.,u + h) _ f(,u) _ The Fré<:lurt clerivative follows the usual rules of the calc;ulus.
fff ,,rt)tt For instanr:c, if ./ and g are two fuuctious whicfi are differentiable
it ]i;,,",L:ll'ìi1x[]j'Tii;:' we k'ow rhar ,,*urntis bo''ded at a point c, anrl if we rlefine f + g and À/ (for À € R) by
*ti:,::Hil, jilil,Hf ;d;lit.::;,:ll:i::"if: U + g)(*) - f(r) + g(r), (À/)(r) - À/(r),
i:,T.!ff
then
fffi:Ii, (f + s)'@,) - .f'(o) + g'(o), (À/)'(.) - Àf'(o)
i;l;i"t:l * [rT",l" a bounded cron,ain, wh,:re z r 3 as can be easily seen. Another unportant mle rerlates to the cleriva-
r:ontinrrous incl T ;ì ,;
r *re,,,,,.ì,.lij?')-'ii *f; j; tJ:À !! afr: í i| ; tive of the cotrtposition of two differentiable fittrt;tiotts.
::
'r,r or u € Hó (tù frv Proposition L.1.1 Let E,F an,d G be nornt,ed l,ine.ar spaces, U
J(r) - ; Itvu!2dr - * Iu6dr - Irud,r Gan soupcehn,, tshea.tt i,fno, rE a a gniudeVn apno ino[t) eun, € seUt i,n w Fe .h aLueet .,ff (:oU) --+ bF € ,gV. .V O-+n,
is <liffer.entiable and that the opert, setl,{t - Í-t()/), whic:h, conta'ins u, rlefirte
< J,(u)tu ) - ," It,: g o .f :I/, _+ G.
/ vurlr _
_
Iu.udr lrf ud,r
whelr: <\ .). /\ If f is rlifferenti,o,bk: at o, an,d g at b, then h i,s di.ffe.r'en,tiable at a
bracker and
rlrra,l (denoted ffi;::,:ljl.dllrity berwe en H](e)
ancr irs h! (o) - g' (f(o)) o f' (o). ( 1.1.7)
frfiiffii;f MJ?, Proof: We have
l(i,rilr)r ., S4rr2l.rs(onrrw,a lr Rtthoap).to lGeosgL'"y (:L rr;e, itpR Gt)) Li*s(u rartR' r,rR) do)e pba€Ireo rtesh er"th ;ien; ;s ;Mpi,a,(c ne ,oRt'ru)a" 1r(pr irbrolrxvenid rnenrdaa rrrwriicciteehss where e(r - fga(,f)r u:) ) o:-(ll" sf- ((b"")l) l +)+ sa.f' n'((dba )(q)u((r y- - - b r)br )*) + :q @e (; r- - b -)rr, ,)l.,ll). Now,
"(lly
iii)' i -9 r ryj j"L,':,11,:';,;,^,:^ii#, R) -+ a,t (n, R) dennecr by h'(r) - tt(a) _ ttu @)) - gff kù)
- g'(.f(a))(/(r,) -f(o))+rt(.f(") - f(,,)).
f '(A)H - -A-r HA-r.a
Exercise Thus
1.1.8_Let E
bea norrned Ii::u.
nriìl) ,f : E _+ R defined av-^f @) : llrll fostr) aaclle r. S€ hoEw tish anre vrheer h(") - h(") - s' ( f (o,)) f ' (") (* - o) + 9' ff ( a) ) e (.r, - u) +r 1 ( J l, ) - (.1f .(1 ". r)J )).
T
CII. 1 DI IItr'ENEÌVTIAL CALCULUS
I I 'l'ln'
Irr'ór:het Deriuatiue
llrrl,
I'r'oof: If / is differentiable, so is fi, since it is the composition
Iki (.1'k,))e (:r; - ")ll s ìlg,ff (o))ll.llr(" _ a)ll _ ,( Il* _ oll). (1.1.e) , 'l ./' :r,rr<l a continuous linear map (which is always differentiable;
I,'rrrt,lrcr', if M > llf ,@)f l, then, for llr _ all small enough, , l'. l,)x:unple I.I.2).Thus, by Proposition 1.1.1,
|f -f(")ll<MIl:r-all, fi@):piof,("). (1.1.13)
@)
irrrd so llf (r) - f @)ll _+ 0 as l:r - rtll -+ 0. Tlrus ll. r'onversely, fi is differentiable for each we get, from (1.1.11),
f
llr;rl,
llryU@ll:r) - - afl l(a))!l .r 'r*vrll ,wrU @) - .t'kù)-l+l o,r ;Ls ll'' -'ll -+ o f :iuto ft
i:I
whic;h provr:s that
;rrrrl again, as ui is a linear ilap, it is differentiable and (1.1.12)
llnU@) - f("))ll _ ofl l, _ olD. (1.1.10) lirllows.This cornpletes the proof. I
[,ct us r].ow consider the case where E is the product of norrned
The. _relations (1.1.g)_(1.1.10) prove (1.1.2). I lirrrr;rr spaces. Let E - E1 x ... x En andU c E an open set and
spacwees .l oLoekt artr ss oansrseu rsnpee ctihara st,itF'a t_io nFs 1wxh..e.r ex i Frunr,r athf,e aprreo cplruocdtu octf l\', :: Ell i--++ EFb ay given map. Given a - (at,...,an) e E, we define
trrrrrrred linear spaces. For 1 < i < m, d,efine the projectiorr
:
\t@r) (at, ..ai-r, ri, ai+r, ..an).
h:FlFt,
arrrl lr:t 1r; : F; -+ F be the injection defined by tl.' r.'lo' op )o,,id, ,iitsi odniff eLre.1nt.i3,a bIfle f ait, sa d;.if fFeurertnhteiarb,le at a e Ll, then for each
u,;(r;) : (0, ..,0, ri, 0, .., 0),
(with 0 eve'ywhere except i' the z,-th place). The' f '(o)(hr, ...h,): É ff o À,),(o,)h, (1.1.14)
i:l
_
t Pio'tt't Jp; I'tn' url,'!l h - (ht,...,hn) e Er x ... x En: E.
I D',?r II; o t)i _ Ip (1.1.11)
(where 16' de'otr:s trre icltrntit.y map in a nornred ri'ear space .E). Ifr ';rr,'vrcrof: If u; is the injection of E,; into E as defined previously, we
Proposition r.L.z Lt:t r'r c E be an open, set a,d, .f , Il -+ F Àírt):olut(*t-at)'
.Il lln't;. i .:sr t 'r Irttfr i.ios''tr.:'.ln' ' :n l'm{ 'i+l. "IF r;tr :'rist, r.tri.'/ Jiser .rerrifitf,eia,ertnr,,et i,aatb rr.re, , faotr ae,a er: hr,l, i,,iL,f , {a ni,, d{ orrtr,rt,,.y Iir,tf,, (<:f. Exarnple \[email protected])) ,: ut, for all ri e Ei.
.f'(n):É u; oJ!fu,). (1.1.12) if'./ is differentiable, so is / o ,\; and
i:l
(/ " Àr)'(ot) : f' (") o rr,i.