Introduzione al Calcolo delle Variazioni
Roberto Monti
Matematica – Anno Accademico 2016-17 Appunti del Corso – 30 Maggio 2017 E-mail address: monti@math.unipd.it
Impaginazione e file pdf a cura di Marco De Zotti
Indice
(1) Metodo diretto del Calcolo delle variazioni p.1 (2) Funzionali classici
(a) Equazioni di Eulero-Lagrange p.6
(b) Equazione di Du Bois-Reymond p.11
(c) Metodo di convessit`a (metodi indiretti) p.13 (d) Principio di Fermat per l’ottica geometrica p.15
(e) Problema della brachistocrona p.20
(f) Funzionali del solo gradiente. Condizione di pendenza limitata p.26 (3) Funzionali sugli spazi di Sobolev
(a) Elementi essenziali sugli spazi di Sobolev p.39 (b) Convessit`a e semicontinuit`a inferiore in W1,p p.48
(c) Esistenza dei minimi in W1,p p.53
(d) Esempi p.55
(4) Funzioni a variazione limitata
(a) Definizione e Teorema di Riesz p.66
(b) Decomposizione della misura gradiente distribuzionale p.75 (c) Semicontinuit`a inferiore e approssimazione p.78 (d) Teorema di compattezza e disuguaglianza di Poincar´e p.82
(e) Tracce ed estensioni p.87
(f) Propriet`a fini e funzioni SBV p.89
(g) Funzionale di Mumford-Shah p.93
(5) Insiemi di perimetro finito
(a) Definizione ed esempi p.98
(b) Una soluzione del problema di Plateau p.103 (c) Frontiera ridotta e stime di densit`a p.107
(d) Blow-up della frontiera ridotta p.115
(e) Struttura della frontiera ridotta p.119
(6) Formule di integrazione geometrica
(a) Formula dell’area p.120
(b) Formula di coarea p.127
(7) Γ-convergenza
(a) Rilassamento p.130
(b) Γ-limiti p.132
(c) Convergenza dei minimi p.134
(d) Funzionale di Modica-Mortola p.138
(8) Teorema isoperimetrico e applicazioni
(a) Riarrangiamento di Steiner p.146
(b) Propriet`a isoperimetrica della sfera p.155 (c) Problema della frequenza fondamentale minima p.159
(d) Riarrangiamento di Schwarz p.162
(9) Cenni di teoria del trasporto ottimo
(a) Problema di Monge p.170
(b) Formulazione di Kantorovic p.173
(c) Problema duale p.177
(d) Teorema di Brenier p.183
(e) Applicazione alla disuguaglianza isoperimetrica p.183 (10) Cenni sulla teoria delle correnti
(a) Richiami sulle algebre esterne p.186
(b) Correnti, massa e bordo p.188
(c) Correnti rettificabili. Problema di Plateau p.194
(d) Teorema di deformazione p.201
(e) Cenni sulla regolarit`a p.209
(f) Coni di Simon. Subcalibrazioni p.210
(g) Le variet`a olomorfe sono minime p.217
(11) Superfici minime
(a) Superfici minime p.221
(b) Formula di rappresentazione di Weierstrass p.223 (12) Esercizi
(13) Bibliografia
M ti
ODO
1)I R.~TTo ùt l CALCOLO Ùtllf \/ AR.\f.4?:l_?_~ I
.).... '
! •
' (
.)'t'"lvY'
""b
Ot\/\l .V or avvno
rì~l;
'O)ce
t,; x _.,
Ifjro~'d
I(-oc oo
.)l
_,ol; LvVì~
I
I X
E X~.
[) ~ s-~ s t01)t:a . G~ ~
:i~Q k}6 f·-r ol-; m
or)h-à" €
1~11 ~-e-vì t:a
o' e\
lf\'\ \ \I\ 1' \tvvOet ~I ~
VY1 {~ ol-o cl~
te&rio
o(f...(e~ t.~ ~t-llv
Ce r cJ.tt ~
1'vvvo~
'jte t° ~~
occ
i'1..v X e.,O'V\., ~ t'e.
'
' L 1,..
~ ~r'Y' >7C.d ~
; j F ; 'X - ; (-Oc)
0oJ e~ ~'Yì ~
c..ovt.h
ìA.L.t. d H~~~~
M.e--w+-(,OV\J ~y-.!)
V~G!R. ~td
i ~g r n ~ v.tl\..~
'àr~ .
I , I
!\~ID \ \...\./'('\ ~ \
U, ol-u..R.. ~r~·~ .. Jal. ~ ~Vii ~rh"h"~ ~~
t ~ ,·.;: ~ b, !,.,.:{t vt.t. ( X ~· ì rr: ~ ~ ~
1 \;\".,~ ~ IAO\,~
~ tì~ \-° ~ ~ r ~ ( ~ ; ok hot..,'') , -:
\!\# t>\vJV\~\o
( ~ \J/'fi"J \;
f.vr \ e: oh l~· \.;\. ~ t,·K F ~I
Qs. e
I~
•I
s ; a (X ;r) ~r ò-H
oe
1)1.s
X n"~ t--ibtt:> (:}
t:"' •/~ n F
~ X.
1) \
n-i. s,; d
F=' : X - ') (-
Ccl I 00J
8SSVVVY\e
W\ :::
~ n t t F
rx)f (-O\\,~ : X'~ X~ ,
s h·-a.~o lu. r r';) ~~ F ~ oo ~ °'i~ \..-\,_ç~' W\ e [-oo) O<JJ .
~ a ( 'N(l ) o hru, t.X,-..t,,
w '"!'vtl'N).,fv
oj. I< '); {, l:m ?v - - m_
i -')
QO ~Bf ,
I VV'T\ ' {/\/vi.. \,
~ - - l
X€)('
I . I 'Fr)(1 > ~~ ~
1)-ovvo
o ~
e,A~+I .) A~ ,
f e.- a s ~
l.(.-rcl
orr:
ifcx; ~ """ V><
€:X
~
X = u A~ .
4. ='
re~ U>v'n 'r '\ tt
e,~t:a ~-Il t
't(V E IN f ru ~
N
X = L) A" :: AN .
"'
:: \~ ,'wo~ ~ F
(X)> ~"' ) m lrl )(
E; )(~tro {~
ol0~~vi : t;: ~-c. o\~
VYl -o
-2 -
f" ~è\
os-: 1'
V\-o ( X
Icl )
INVW) sr a
è\ .. o \IV\e f.n" e.o ~
F ~ X - ~ C -
00 J1))] - /\1-0 ~ ffi-1""(, w
l\-0\A..0eD~
-va~H ~
A) - t- e "'
/'},(,1,,
1).-u, ) ( IX , ha :
r~ ,
Xo
E (hB)
0~1tm i n-:f. tm , -
F
('>(o).... <:. F
(X)~=iVl+ r
(Y) •X -)Xo t'-'"lo+- X' E6r~)
X#'X'o
-
l:~~o, ~~LO , ,
'. r
,.X=I~
@ ~o~l;OV\~ V\~LVY)'òiì'e , Se
XoE X
e t.~ ~vv\Ako
0
\! W\-\ Vv~
VWootr: F
1~ Ù.OQ e" S ~ o ~"'· h~ ~
t ~ oUilll ~~è\
O'\I\~
Yve.UM ~e ol-:
llY\,·\A:
\N\~ h~"'.
Jr<, r~ Jtfùv\~ ~ ~va\Aolo ~ F ~;;4. ~~c.-k ~oLo,
~
V'vO\ai è-i~
WV\~ c.t ~~
t-a ( ~ L:~fl
tsll\-t. \1\1\.0~ \.Ao\e, ~, "'-". \a.)
f
o\ov ~" ~ v~~'<:l \o
1/\1\1'eD~lb-\"~ ole.\ -h' \.!.\
rf) b F (
X'o) :::. ':'.> ,-3-
\.c. F ~ f~
- ì1R ·<t o~ va h-;-ec,.
ì "1X'.o ~ f~ l
1~~ h'~
/ ~ r' ~·e-i..<
..J~ F
1rx" J ::: o.
T '}\ vot\a
(,V~ hi~ ~ a\t\vt.c. ~\.,.;è.'~
\'Yt:~'h"ie olt..l ~~ Dto\.-l~ J)Obt<> fo""""-'~
al: oh·~"r"-aè1'"~ oLe-1 h.~
Ò z
(=Z()(o)~O
...~
'-va...9it>X /'
1!tMOs r
ib'
o~ ~
"Ql\A~
,e' e~a 11
,flN--0<* J
;y{
v(;v;">V\1\.-i~1-h~ o~' t~ - La~p-
~" i
V\t~ ~ \;\; f-e. e.a ~re.
/k,~
VVYvV\l\f' V\;:
\NW ,T rov~
~~}:\~ ~w ~~'"" ; +: tn' c~V"neu.-+~ oh' R: (A·-ec. ..
e u.
\/\I~ tt [)
Ìl-t/n'~ eoviv~ cl~ )
,~~ . ~ F ; X
-">C-
'()O 1ti0]
eLF ( t
X+ ( !J .. -t) ~) < t F
(X)+ ( IJ -t- ) Fcq.)
~ X
:f-J
I \rtX
.e,-t
te
011ì
J 'a~-0~I (p~+o otl)
"
r( ~ er0f)te.) -
vn1
\/\A' 'vvvv-e
~ {..()I~r<-
~l'd
e-Ovvv~,,
~"-a
\I\vt..e..
~ I)h-itt
dn .
'fil t.c.~ ~ ~ ~
s ; -a.
vv0X
JX ed t- ; X -> (-
oo , 'DOJ frf: vf-e + l X = F ,
t'Jra i~ ~·~ i.a"''{(. ~ \MA/~ h>~v-:a ( olt. ~)
/\A-\;x
~ ~
cc:")-t-VVVl e;
OV\~
r(,,a
l\tv\-e. t-o w ot' \'"(;·lt.o ,
I . ,,... " I '
\e t r0v\ ~
\/\M \MA\Alt~
X f'X ~ F \!J'YY1-~
~ f'r\
1 '""'r-e-~d'"' x e. x ,
()~~
~\o cah-a ~ .. ~ ffn 1-l f1:o~• or.e.Uò- r-t1>lr-n'l-o":
11
\
wvi\;\;\·~ ~ t-o ;"""
11\MD'=>r~~o EJC: ~ à~ ~(,O
I
IV\ l/wvO
s~Ù~~ 0~ ~fr•:~
' i V\
-r-
si~ A e IR""'
VVvl iwn'rNVlca rh,
1:1(y{~ L ; -A'< I~
X 1Q ""'-;rR
\.NV\ •
~ ~·
lù\Ac. ~ p-..o r-v''
<.h" ol~ ok~(..V\; te,
'M! ,U;y,.1v\l\A7 k va. n'-a h,·-e; x
€4 e
11<~ ~tt G I~ e.. a G 1R
hl,L ~ f
lv\,l\tb. ~e, L e" ol
tbtdL'at\l"\Ar:i
\A3 •?()')to ~t. Md ~ O~vv~hi
1~1/oh°~ {I 1vvvit:-t0\1~
f ~ C
1CA) _, I~
F
{IN)=
l3 ,-
S-Ot"
'd ~\tu.ra
r~'i
\j\,t e,)A ~ .ee -
~ ~bQ/V\~ ;u,t~ty{t,
~
7;-\' l'Wl'1'Gc..
e,l ~, ~
DleÀ),'-"" ~ r9w\1-
w.,J)~ va no. h Ol\A:
U>VVìr
1btt.
ol.tU
'Cl1o
r VVì "(}(a-\ t&io oLc.l4
f(U;) ~ Féu+ ey)
1
~) ;h o~' e.e,
Va n ~"'DVII( .
t-i ~~ ~'ì'wvO ~ Cr
Ee e (A )
e~ .ole--n'
'aivvv~
i : 1R - ') 1R
= ffLl,.f. tlf)
I I
\;\.'V I \IV\ VV\-"
- ç; -
'
j l (x
1U.tJ<).,..tCp<xl ,
\7u()()+~vc:r<x>J dx A
"ba i:w.hf:care,
= j -! ( L (X,
t.1.tx)+ hf M , 'i71..1 ()() +I. V 4 rxl ~ olx
A
M-t tt~olo t. : o r-/i f-rov ~ .e} ~a b' ~ ol: Et.... 4-ro - L '4 ~~
~"" :fo
fVV) 'dote.,.h.ole. :
(Elci ) o "' s ( L,.
()(I1.1.1~1, V'tt !)()) lf
!)<)+e~ I_(
JC',I.(.()(\ IVu1~). Y'<j>O<)))cl~
A
'I,f
~"
f,"v~ '0' ta ra V Cf € cc CA ì, G (x)
""
).e Gr ~ A -> ~
6-6')= \7 L
()(I U(X')V'~(>())
.e'"'ci-t'
p )
Jcl,~~ e~CA~ ~n.) !h --ha
-+-
1 j
Cf t cc. (A) e G E e (A; IRk.) l~
J d,.i
V (Cf
(j()&
()<)}eh·
A =
t:>,
te.o~ ote.Utct oV:vei--~;d a ?-""'~
-tn'~
~~ohi
~~tel\M~
~l~V'f' W-1~
,. eh Fu.-h,\ .. ~-: - I QIY\~ e
d1l+w~a
1llN\tlQ~
,.e, ' e~~
b'o.-vvo~ f'~ - L~ ~~ CFlcl ) ol~v-tAA-h
o = s A Cp(ll) t L..., (X' U.1"1, vl.t(Y>) - cHv (~ L rx,
U1,()Iv .. )~ dx
r' 9 r~ Cf E e .., e (A) -
}e, S Cy (X) :ç
<>e)ohe' =
o, A
O~\
i=
o -- 8 -
1}
WY\~)
/\.(.,X
j-")t
n /~
f..>i~ h
\.\,lA.o fy; h-ovò
t' e'r"'"~ è;~ ol~ EV\.<.e.-ro - LayrrV\.~
(~L)
xe A
a,..~ il ~I.i.;""""' '1-,' "' 'fv"" at: "' t: e
2r A > .
L
1&\V ari
\11.IV\ e, (f" /... ) ;,
\l\N\1
e.o(V" ~ h ·~ ') Lle. o~ ~ l-e,
~'a r b -a~ ol.e;\ )( ~ o roh
\/\;C. : \A}orW\ ~ ol:
0L:vur1a,
- t I ,
U)
t-lNV\ t\ &'V\\a r~ vt.e. .. ~ .... VvOk>
~ ~Cl
LAI=
o-e '1 L = 3 ,
F' ( t,.(..1 =-
"IJ J
\7 \.(.I f dx
I- A
- 5 -
eL _,
L(~) = ~ 1§1 2
2
=o
1"' Q-(p-@ F
Il.-"-' e,ovt. v' {i'V)-0 "L' e°Y,_.~/d 1:\ ~
• t
u.
~obOl\N\; V\
©
-i_V~\/lt ~{"e
1! \'\ (i) - @ ~ ~ .e· \.'VC-a.r: .
l\r1 ohwi(Nl,..;ov,v n.
='IJ t' ~niovie ol: tu~ -Lo'}'"'l-"'~
~ p\.AJo'"' :
vtteP("a
\ei M~C.: ra me-wt-ei IArYJa Votfa
>a
p~b\"o tM t~ L~"f)'-V'-~~V)a ~'a ~ ~u,,\-0\/\0W\'3
"/ . OWer-ro\/\OVl ol~ ~ dal plAMlh> X G A
e0v1A e: 1R
; v\,te;y~
I·-') \R a-eWl~
tGl~;n.e, CtCUZ
2)s;~ L : 1R
X1R
Ol1lii li;
lfa'
3
,
w
bci e tt)) A- e rR ~\lltel'~
1 %~~t\I.~
.e,,
rlldoltll
1~-V\;~~ O\;\~ o\~ ~u~- La~o""-~
I 'I ) I
L
I( Lj ( [;\,
lu ) -
L(. (u
Iu., ) =
D ;%A,
Lci
(\.{Ai)~~\d
-0v
#
I IH' - - \ili " h(Lt,uf) +- Lv ( L l tu. ) u' 1)
3
Lift. ( ~
1u/ )
!Al I- L3 r
Le ljI
L"' ru,w'i ~
u,' le ~3~~,Ji) - =o
-
- Il -
D
S ~a _ A e IR.
n, tA/\11a \.e/,,·h> ~·W1{hrf-o ~ ~,\e>vvh·~
.,. ·~ (yr< : \ Teo
WW1 <Iclc,Ll;~ ol-: V«rt~ ) ' 'tk:l ~ \-r'O
,.foc oVvvì.ev...re °ò'eò f\U> o t~
VVV\d ~" èt"~ L ,· p r:t ~: t-t .
$ ~ d L
JA " IK
XIR
t\,> lR
LVV\ é)flNV\ ~
O'V\.e,t}'{.r, ~ :
, 1 L e e i (-A >< 1~ "' 1~"') ~
2) 11< " 1R
\'\I~
({..{;,i)
i----,-)L
(X J INI3)
o~{
X6 A,
r~~~~ :1 ~bl>'\A.~ F : C 1 (.A) -"> l'K
r
~F{t,A,)
= J LCx,u.cxl
1 \7lk'>e)J o\x,
... rt
S {~ e.o : GJA _,, IR
VVY\d~~ O'V\"" ~K """ò
i~-a.ra
,
..
':.ì10 Lv E;
e,, CA)
WY'tdlt "
, 1) u. ve-r: -f..
è,;.i (111<~ ok,loW { '~ièt'~ o~
Fu.f..etro - L -a~\AF /
o= ~ l+<xi. L.,,rx,u.,V'u) +<'.~Lfx,u,Vu.-\ \7-/;)lotx A
~ o~ ~ f e
1 -(A )t~ ~ 4
=o/\u 'àA -
-13-
'Ttoft~r'\A N~ll., ~ys>~ ~rt~h' la ~c-.-~-e,
"1 - li , I
u. e C ( A )
i! WV1 Wl ' \A.Ai \;\IVOole.t bro ~ ')
1 -
~ vGCCA)
)
~
(.te )-= F ( "" + ~ ( V ... \N)) ,
~ . i
{I\) ::::F (V) .w\ -j.Coì = F
(u,) ,Voll.f'~ ~ c.-\-el i- <11) ? i.Ci:>J •
)-t { 6 C~(!'i() VY>fltt -6* G (01 IJ b~ ~
1
(I\)-:: i ( ") + t
I(o) + .::!_ .+Il ( t- ~) ,
2..
l~Ò\V\'\O
,ti
1<-b'l = \
v~ d o f - ) I
(xu, + h
t. v-"") ') vu..-t ~ v ( \'-""" )-\J cl.x
A
- \IL(,,,) rv-w)+(.~L(1,,) V<v-w))\c*x
-
..: \ l "'l ) , )
A·
I
- o
,I-
)rf
1-
\.,
f
°\~avt-to - y_u., t e (A 1 e <òA -
I V\i
- :o
i\U+,'Col= o ,
{'h:) = ) / H L (
111Ì (
ll-V,\k-V11) (u-v ,Vu-W )'-
-A~ ci..i.,~1 , y
dwe. l-1 L -e"' ~
\/V)(}fn' w -Hetn·io-vv~ gl,f . ' L , L
d0 ~'va- ~
·. fu, S)
0
\,wkro ~
1ìu.r~ .e'"' ~C.:~ . . Per -ea ~V-e;)1-; Ja
v ; {~a:
( t.J L -( x,
U,t1:: ( v-u.), v.. + h
\7 IV-14l ) (et-V, Vu.-\711
)J(u-v, \7L<-'7v1)'0J
(u,,1) .
ì"" o~ r~~.. x.
éA ~ ~ ~ ·té C.0.11, L:i
t~ ~-
o
E~EM P
lo '( L~%ft' QU,u.! olf-; ((}_ ~ ~ca )
S..:a 'f ~ L "°(o••) ~ -fwn~~ t~ c..t~
o" wv ~ i'. r .. n (, 11) r or:
J( Eeo,,) _
f e"' u. é L:~ (Co, 1)) rh r;~.~ ~
F <
w)= J 'i r
>( )fì
fv.' rK/ cl x
[ o. O A
1...__--v ,
r ~~f'b ol~
\
- I
~~1:1:9 0%~!-a" olf~e-a
vk
oL:\v'NOl,-c,
ot~Uo s r. o-~
o {ì'HA' 1\1~
)-15-
l'il ~~~ i 1----> {1 f ~2.
~ "~d ~ l a
1)-u/aoltrrf va ~
t"'e,,., !lf-~f:I-~ v-n~te 3 /f 1Ha '
~to
( \IV\I~ c.:~ v\..e,
et't'i' f\t
'f.,Vw\i ~hvw~ e e 'R
~ <-4
(~~Rct~lo f )
i O<) u} lx l =- e ~
~1D~xe(o) 1)
,ft, +"(A.i~)(
ìi. ..
~t~ ..{o<l)'o
d.~Lu.~ OvvY\..o~
U-I IX)~a~~
e.m ra.,-wtc, ('1.01)
II~e
..f
2. '
2..C Z. ( 1 t v.f
f'X:)Z)
-t <xl
w
hO=
tN
10 ol. ( icx> 2 - Gz.) =
'1)~ i- on 1 ) (, z..
JJ~
co ' j l,.' ' ~.< . \, \I ' •
1 vv
o~\1\1\h va :
I . --1,-
(o,tto1eu<
'2.e
1AAA,, f
L(M.
~~\-o
L<Nb~
f
V\"){.&, ( ,. I u. I 1G 1R
'I. 'f
Uf'M-a-rc. ~
f I
tOlu
/\4~~'hYYl-9
e,~YvC>t:
111\/ '
I
u, I
I\A.,t~
~"l'YYW
'.\~ +
ej{-) u (l\l -= u, \
WJ
i= ~j:> _ ~ À.
E:1e i i-
f)()~
')u\e" t1-'),0,
Xe[eil] J
l
Po,,... /
11) I - - ~{ ~hvvvw ;"" ~OYe VY\v'V)èi"'~ ol~ i-, v• \, r
f"VY1a~l,.;'hmO m
) o ,DwCI ~
' ' '\ J- - ' >
Io
1w)
• \ t )
~(S)
.
- ' !-
fl \J
~ h
\'V\;\Jd _e,/)t\df
Ò VYW.M.JtC; (/y.ef>l0v<.Jt(! V-11--
[j - ~ (s) .
~'\tG
l/\M-a ~l~e E r o
- J\t\tl) tt{c,
~l l
~~~o'v: r
\Atvv\-c-b
'\,-\){.e. (*)/Yia
I\)~~,~~ -
X :: t
(NELi~( [01 O} :
U {o)-= Vio
e, Ll( 1) ::l-(11 ,
~ ;\ ~hOV\~ F ~ x - > 1R
1)1M,
i- (
!.{,)= J i ( )() I t1 + u.' ()(, l d
l(~0 1 l]
< '
-18 -
o
Pr<"'t:~" o\! t=~
t'\IYlà't..
J\)~1'\/\IW-. "= é Cc,() e.. l\<kr~w~
~
\'V I" Eo ( c>,-b)
-{6' )
:::h2.
)<ECt,1)
•
- - - · - - + - - - - 4 - -
o -t
I
('À
I~
'1.!V\i)oL _J j-•A,,_
I '-r. .. r -
""l«-
'I tf'll>Wo
IVWI~'(
VVH\I\.,, 1-l •
1l~n-e l-a ~~ ci: r: "fa h
O'\f\,e :- - ,
- '19 -
Xo' xa -" u.. >
lNi ,I
(O\fl'l'c. Lo l-n-1 d.to l'"i. k \llM
lr >r'h c..elt, ~
Ù)f"f"(/\ ~ +ttm \-'> m\\IV\W\A> la~l.D 1\-obto la 1P<t:9
O~ Vl v: ~·
Ir H ~ 01-,.J r
IMA;h. ( J<4
I lt<> )6 (K2.
1rri\1~ 11 r·~ ( X'a,
Lc..t)G l'K '2
j v-~1:.ra ~èt"OV\ei
9 ,Uo \ ~
'l:\
: \' \.
'
f,.t I
I .,
I
~
I
I
-1t>
Xo
)(x.
XWi:
W\1oSS3~ = ~~t, ~~ ~hi-v\~
v = "~""' ~~V < v = o ~e-' r \N\..V~
(Xo.u. ) )
(.(., lX l = 'if-eA~a a.«.' i~Ma X é [)(o,<•]
~V}~QM~ ~'r'l:
- io -
d.s s ::. ~feèia
v= - ci .f:.
)~rv.'t:~a
e. gF~~ dt= ds/v , r1 +evwr h,Ji<,, -e "
=(5 ~
p~l\WW j\-t.e.~rrc. 2.~ = '1 ~ L ~ ~1-i ~
/ t+-j2.
I ltt-o -
LA,( "11
3)
1 - - - D'I = I Uo -
U.. ".:>.; O. Gr
(yi)=
- 21 -
L~ ~,,,'~
(V,)11,____;~ V I J... +432
\112
,
ill'V Iu~ , f p\M.Nh ~nblW\~ ( ~1r1-~
0 t~ ~~ - La~ ) of.e..\ ~,,.:~~
F l~) = r "'
j Xo
CQN\ ulet) -= Uo -e- ltl 1) =
U1oW ~~-
mlV\ \, ç: ( u.) ~ U. E C ([
Xo,x-J) Ì <ì G
1
\
x.o, '>C1) ~ ,
(Ai (o)
= ll\.o
U. Cn
= u,J\ u,< o ·
L 'd. L$~~,\N,
~ e~v~i-t.
J ' . ~ i .'
L ( u., i ) = l(rW / /u., -v., ha
L'
~a.è\~, ot: ;~-Li~~ "'
"'
I,
I
I (1 +
lll'z.
tA, 4
(
{ I+. '1'
2.{v.,-~ ) - - 2' (~ -(A;J312
I
V 171 . f V ft
V\.
reAf;, \..· ~~ hovve
l'y\~
1 \A,~re , ~
~\~ Uv
<301-&·~·~oud; ~-"+<- \/w\~ ecnr°'V\--te
t~ ~
~ I 2.
(1-1-u.•1.
l/,.,_~,i. v~-U.,
{~
-IA-..
~(,
ol~v~ta
- 1
I
""
e 1,~(Al'l.
/ IAo -l,\,e z.
(Uo-Lv) Il - e
2(Uo-tN)
-i3-
"
J
- 1
- -
'
e
1~+~~ ,
l-r-.svv~
f'y,
X 2.
J
lfo' (-b)
e, (!A.o -Wt> )
- ( X-X'o)
d~- -
1 - G
2. (Lii,o -Ult)Ì
Xo
l
'Z. (
l,f,o -'T )dT :.
X -Xo.t1 -
CL(tAo ..-r)
{a
llo
J
Xl -Xo•
olwe,,
'L
I / )D~G
< : - - - .
Uo -U.1
X~ - Xo <
IR;àl'(rwm,''d.vno ea DL'I}~'~ r re~rc, ~
;}et~~(; tc,or~a .
TFo "çt) A S ; ~
V\-0 Xof X
tfe-
\À!o~
IA.-1re-.;;. ~ v~ ç:c
O-\M>~ ~)
•S ; a ~
x = ~u. é CC[xo,x,J)n C 2 CXo,x,> ~ ucxo) =~ '(
dx ,
A1N X ,
1A fXi) =lt/f
j
u,I .(. o
L-a "~~h-a'.21.0\Ae
1?,lJ\;\~ ra~·w ~~UD~
Il"" ..e.-\~l*~ ,\ 'V°~
Goo~
wv\\A\\M.c L-V~C/y\'v-e,
lNv\
~ r
0co~ v\ etc~ ole ,.
, A. e 1R
n. 'i)~.(.,("'h;,
;:,,: VV\:t-a +o)
I
LA, ; <a A. > 1'K iL-vy'\ è\'°"' e ~ss~et~ ( L:pn~ r~ )
-
~
\11,,\,(,a >• c.rSJ ,., i t4 E Up LA ) : [,\\M :; \A. j
Lt~~ ot! il.lvV\ b
O\A (a
\iV\ W\ -\'ine
J~;ho~~
llwro~e fwvY~D-vla~ F : A-> fR
J .f ( l7l,\,
< x1) dx .
F tUJ) A
V~~ ol~\ ~ ~ot:eAA,te .
\"\,
Pe~ '1\ l~t~d olA' R9~a~(I'" '7u.cx) E 1'12 -e,y)~e
~ °I' o.
XG A ~ e. !\,.J.h-e. I v~oo I :$ U~.<IA.)
J{~ ~Ìò.vv~ Ql) L\~n~+-t oh' u,,
IV\rltrc,
.lxl--> f (\7~tx)) -e" ~vv
~} \vt,,t~ UYv\V~r _
V o t' ~ !\bC: o.
"f"C.,t' ~lì t~ 1'd
-2,-
~~E M p I o ( FWY1-=t~~ ~ c*LLL' a.. fed) l ~ ~è{~~
J (~) = / !\ +- \?; 12. .e(., 11trt.tta.m~te. ~ v~a .e,
F cu,,) = 5 / 11 + 'Vu.<~,,z.. ol>e
A
f . .'"
~e f
\/W\t-;
(iV\~ o\..c,ll' a
t"e,a 'Neof- Spab ol: Sol.o.fui {
1<11VVIb.:w.tt lii~~~
l),~<'L~'°" w"·" e.A 1 U>Yv f> = "' "' '°)~~ __
Ae;~~dra
a a.r r(,O
~yQ\.o
',\U. ~ Lir (?JA ) Vo~~ +-rova-ce
o~ 'i\-re~ vvv\~VV\d ~ ~~ ~e
~ -ç..
\A>o\-'. I.A. , rJ
rMl\flwl ~ ~.;
rit-e •
( ~~ 'òto\-f- ~~ h'""" ~ P~~a i·1-rnfal<i) ( A
1
\A,) ~ Ll: fdA --:>IR> v~1·tc..a
"..te,\"" ~
Jh'~ ( 13SC) M ;
t~~te G >
0taM vt..o ~ o~
Xof '6A f!h'l~
~ ,R '"' - ., 1R ia ff; vv~ ta-t ~.e,
:- +
w
Xo )w
')(~:) w
•<
-U ~ w+
XoXo
'') l\
, ..
Hl)
\;\
(~o) - w~
(xo) ;- Z1- -
A
\
ì
!
'i w;
!
!
/
ES"tRat;10
R.\;;\-1A\<K Se, r-aA t o~ "l~
GZ. e -ie e.u.~ ~
p- 0 -.,.~r~ o~ 'dA
fWvv•~hvt (>o) {.., o~
pvvwt"o 'atlo<d ( A.
1\A;) V~ b'tél Bsc., ~ o~
u f e}· e~ A ) . I iio "1/vv\a o t~ ~
11>Vl.ci"b
JVeo~ er-
'111h
l'\~~o~
7) ; teibt~ gW Cd V <),e t
a'1 , Z. ,
1JOl\t(tt> Jhelf\Vo ; ot-: rroV(}ft- ,'i ~~ f-fl
r{~ah>.
-2&-·
/ l 'wi
1'lo lo
Tt9oRE"MA1 Su.-~~~~ww ~ CA,U) V~f~ (3Sc 1
~ ~a 1 ~ 1Q.
\Il, - >tR. e 5 tre tf-ch'V\ ~te) ~\Jetv'\~
,~ ~e VVYl'\A_,~~
NoT-A~\oNI •
L: PA r \.\,)
~pJlA..
(,C) -U.l~) J
Lt' p {
u,)- - -
x,"t-A
IX-~\X
f 'J
- < 00}
L[ f (A) - - ( v. ~ A -) 1R
Li~(~)L: r. ( A ) -= ~te, ~ Li p t A) L: p
(L<,)~ K J
) t(;u..
ll'òA = L-L'}
Upf(. (A) !A,) -: ( l(." L:plA) U) : lii-
ff.N)~
K~.
= s 1 ('71.tf><)) ollf
. 'A
-29-
'
•k)O
,.
1) : ~ , I rv ~ r\· ~ k~ · L: p ( A j lt) -:;i ~
~ )
!k~ ol~C TeJJ~ al: IU'lfe..Nh~ dt,,; "1ac s-t.avie •
•->)wr~~w"'o ~ ~~·C.:fciL 'f G c1(r'KM-).
'S'ia L :: fil\;~ t ~rw) i ._.. i0 L; p~ (A; UJ f ~
I (_ I I I I J..
~~Wl1~V'YY"7 LN\1~ ~CUtV'h ~ vvi1'~vvi1 tt-~wie
Gli;:.. G L:p~ LA 'i V..)
>{é- iN
1~\M F (~ ì =
~-~
00L € .. (-cc)~) . ( L G I~)
4~~1wW I ( Xo E f(iA e r\·a~)
i) li~ ((A1v) V'~
' I< /
(I) I u,,;, li<.) I ~ I (;{ t e
X') ...4fv ( X:o )
J+ }~(~))
' k I X - xc I + I U { x'o) )
' K o~aVVl (A) +
JU fxoJ) Y?'v 'ix-é tr { A N
1~00~ ( 1_ ( f_{ ... (
\I\~
If ~
OU!'.A Ieo~d ol--t f\11 ~ - Arr:-eea \,,,
~~+e, ~3 Aolt?tt<Atc~~ - ~amarn
(~) tf:tN - ul< ~vtfl ~ ~~r\/v\fNn~te
1VVV\~ ~~ U E L l Pi ( A j U) :
K
\;)'\a~ j ~XJ
'J.u <>'J
J= o .
X E A
- 3o -
...
~e Vu.,,fv (() ) = ~e
(7µ lx') -!-V1 ( u~
(X},u lXJ)) ~
;~~-t-G
9,0,
F(u,) ~ F(w) -l- J zvf(\7~(Xl) V(Ur;CX)-UllO))olx
A '
~,
Vo~~
Wl'h"Gla e-Ov!V.t'1"fMèa W.V:.~cWle. L{, ~u.. ~
I! J,. • (J_ I L__
~
1)1
~o~'èJ ' l,,"f ~
-1\..ei~
V&-re,S:~ c.COW\e, t'f ( \7tA on) G L
00(A ; ~ IA.) , V t:
/ o~ 1)- ~ & € G
04r 1K, ~ ~ 1Q ~ ) t~ ~
1)~~
J I
G.(X) -V'.f ( Vu
!JO)I olx ~ L
A
[ (V'f cvu.1x1), "\7(u4 -tN)) ol)(' = J <. G-, V'< ut -IN ))obc +
A A
+ J ( V_f (\?u ) - G- > \7({,Jf.. _ u,)) dx
_ A
~ J ( fr
1g (\A~ -
L-V) ')o\.,c + 2 K f: ,
A
-31-
~ WY\3 tLA..-+t-~o-~e ~ ~arh' e, M,~ IA~-u..=c
ltu aA
J < G-, V'<Uf.. _ u. )) otx = - J o1;v ( G-) (u,.e. - w) dx
A 0 r J
~ {, _,""
~ tlc..c.o\,.( ~ ~ o
L = ..e:
V'V1F ( (t~) ~ F ( l-V) - 2. K €. Jif
€. ':> D{ _') «>
F cu.-) ~ L e ~~
fFru) = L,
Cl
~~~A R K 5 ~a u, ~ l VV\{ V\~
Vv"vOolilla p
Cor b
Q/\t\<,2
I)-c. ~ L~ ~ cuJ < K rot-~V\i\.-0 pro~
heA
fr(,-47l(.A,~A.:h:,
'VV\oçw .
S;a v 6 ' Lfp lA j U.) ~ca~te eov, L; P t
y) ' Q1 ,U + t cv-u,) E l;p (A;u)
I<
Fi"") :S- Fl tv +-(Il-~) lA-) ~ 1:
t=(v)+ ('1-k) fcCN}
~ c:t-70 >
ç: (~) ' F(V)
'D~J t.v ll~-<eb~G ~~~ m1~V\l\.o ~~\'TE""~. 1,
CJ
1)ff\ N\t\ otJ~
L\;v-\3~ èi~ w E L;r\( e A) ~ o~(.(, /}lv\-er< _
IMtV\\\'"'0
~ ~è10v1oa F
I).e,~ o~; 9
ELp~ (A
Jw)
w ~ g ;"' A => F
(W)~ F(9)
f1LArAl;Ì~ Vf: Llp
(A) Iol~
u:,!\-lv b -
\ry) (\,,q' WvDUM4
/h!(,
~ f
/\e,~ oyvv~ g E LI~ (A, V) ~{a
lv
e
~ V I , V\A
= )F(v>
~F (9).
Co
MMftvTO1 W1\\A~~ d; r ( ~ ebro ~ ~or-ok> ~'IY\~\-o)
!).ovvo
~ ~ ~~
-v\vG !\U\r;, -
Wl,·u,·V\A.-[
TtoK.~ MA ( P r;\f\ ~ y,,;o ole,,~ ~Jal\')-~vvvo)
J ~ l'R
lrv - >I~ /\ treltò
Mel\..d-e
Covlvvn~ - 5 [
b \M)w
E L~r. \G(A )
l\Ut \u<'
\N) \"""i
Vvv<>ol ~ r ((';
VE L~ ~
(A) lNV\ fYU;b -VV\1'\A..1\~
"'
Ot~ F ' /~<ò;
V * w /\tA.
<òA =)
1)1
M.
Li
XE6
/}-(,
X fA\ t3
Po\ vlZ
s jtVw) d'I( = t-(
w} ::;F (e ì -
A S f r tJe ) dx ~
A
)
J trvw) cix ~ J tr\lv) dx,
8
(3L av6>'<.C)~ ~ e= ~··"" l v,w1 tr-: oltl™e {:J
oh ~ lt1 ~a f ~"'
't do J'P
01ta
j t ( \7 v) olx ~·
B
J f <vv) o\x
G
... 34 ...
J J (\7w) olx
B
JS J rvw) o\x ..
B
' I
/
Sule' '~ e'lvv,.c.. 13 tn
1oLw~ -a verr e- \lv ~
Vw M
lNv1:%~(J D~ \fv\~~\,\.{~ ro,;hV~
J'àttn'\ri/)~h' Sarek~
V= w (Vl 12 ( {/\e;\~bo ì . D U;yy"ì~) \,fr"f' -e:
6l\t'fVt\'J (,ov\v~ ~" o~ J ~
A
V -
v+w
z
V
Il
J ~cv .. /)dx .
G
A
y
=
V/\V- <dA
e, A V~ V ,A
t\/\
o
CoRoU.A~'o Silh-o A e 1e"" o(r<t"o f/\·m;~t-.o
11 :11<"'-
-11\<.Ah<Ai~VV\wte. t<Mv~~
.e;~)O, Si}V\o v,wE Li~ (A)
K.
ob
\/Y\~
V\ \ Wv \ol~ f e_; a ~LU~
.oh(;LL
"b j\iu.~ eJ
éYYWo \A' ol ir\-o
5u,f> I
V ..w I =- ~ p \ V -
\NI
IA 'òA
L ~MM A ( R\ciuhoV\~ '3ita frOV\lbera)
i ~to ~\IV) ~ ta to) ~a f : t'K
ri, . _ '>l'R.""
tov\v~a. (, ~ ~ u, s Lf P1e ( A )
Vvv15~~ A e 1Rn,
Wn/}tre tta Wl-evvte,
'D :V() , 5 :~\1\0
't : X'I -
'Xz.
0Jlv\ Ltic 11
.A-r:
I I
W\ \ V\\ VV\ \ ~°è-~ YVO
- -
X1J X2.
6 A e..ow
A~ " - -
tz:+- A
- ' )
1R , s:
j t (\lu.
fX)Jd
XA <ì-A<"t
I LA ( )( ) - lk C!:P I
I X -
jI
X1
::f: ><z.
.e,o\.e, 'b \,"[ a
vYtO(..t,~ ()()
- - u,
(X-'"t)
>
{,a :
.A"1:
I I l I
I
( C..-:'òJ)(...W'l''d
~ :e ~ cl'lf~ ~ h.oTo\_o) ,
~;~\
If
u,f X11
-t(, fti)l
i
)(1 - Xi.'Ju<x-11 - tt~(X11) ~
c..
I'! I
J t,t
rx)-ul~l I
'ii.c,,t-'
:t, ,A I i-~ I
J~ f: ~A
I I f A'~ ~•
e, 1"' UOVVl-t X1,
Xi. ~ ~ ~ , "l~to ~ ~
ti rt"""'
f IQ
)) I l'\OSTR-At:tofV T;'
'Dt
LT-;-oR.~H A 1 . 5: a Q > o ea.
~~te ole.Ll:a BSG e ~'M~l\'VYO K>Q.
s ~ 1 u, G Li p (A j tA,)
U/\A. (:t, ~ tYt~~ ~~:t-a" )
l •
KI I .... ....,
V\'\ l V\\ vvv~ O '-
"Mi w l ~A f (Vw) olll 1 u.
EL't-/A..i U.)},
- 31'-
s:~ Xo G~A
i5,'c~
U..CK)= Ut<J ~
XE ~A
Gl~I ?r\\1\-cJ~o oW Mi~
\N\tOM~
M<,
A
f,,...,. : l L!IVV1 mo ~ y{ otv.,t\(W.,G alta 1'-Q.1A h·~ ;
} U
f X) -(),(~o) J I)( -
X'olJ
~ Wxo - oc) - w~ (Kc) ~ -Q 1~-~I
f
U,()() -UlXcr) I s Q \ X'-Xo I ow~
)