• Aucun résultat trouvé

Ecole Polytechnique. Ne pas reproduire.

N/A
N/A
Protected

Academic year: 2022

Partager "Ecole Polytechnique. Ne pas reproduire."

Copied!
474
0
0

Texte intégral

(1)

Ecole Polytechnique. Ne pas reproduire.

(2)

Ecole Polytechnique. Ne pas reproduire.

"!$# &%('*),+.-0/1%(# #32 4532(-'6- !7328-041

9;:=<?>@:BADCFEHGJI=KLAD>MEN:POQSRTUEVGXWYQZ>@[R\A]>ME^:_T`RaAcb=WdTeR\AD>ZfBI=<

<LAgO*QhR*[D>@T*I=QhR\AD>ZE^:i:VIXTUWYCj>MfBI=<

kXlm nopqlZrJsJtutYswv@xHy

zV{h|~}qX€H|~}‚~ƒ„h{h…F†]‡‰ˆ0Š‹

Œ

swŽ‘c’

’’$“Z”Dp–•q•qr—V˜™™‘š

(3)

Ecole Polytechnique. Ne pas reproduire.

(4)

Ecole Polytechnique. Ne pas reproduire.

"!$#%&('*)+'*,-/.0)1&23'4&(,5'46.7,8'09&: !";;0

')+'<2=>,8!/)+'09& ?!/,8.7/&: !";

@BAC@EDGFBHJILKNMNOQPRHLSTKUF"VUW&FXWIZY\[T]A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @

@BA _`F$]>aN]bdcX[T]MN]^bdKNMNW[TSfeLY\HLSTKUFgA4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A

@BA h7OX][fijOX]&ebdKNMNk[T]&elP>[fYUeJemSfijOX]&e"A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A n

@UAXAT@ o1ijOpYqHJSCKBF"MX]^[TYdPZrpYU[C]&OXIA^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A n

@UAXA o1ijOpYqHJSCKBF"MX]&esKBFpMN]&e7A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @ut

@UAXA v=]^v3YUcX[fYUP>ST]FwA^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @S

@UAXAx o1ijOpYqHJSCKBF"MX]9yNPZrXIJz{MXSCFXVB]I0A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @S

@UAXA}| y{~NeHJkbd]MN]^v3YUbdW€A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @

@UAXA y{~NeHJkbd]MN]yjHJKU‚U]ue(A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @&x

@UAXA„ƒ o1ijOpYqHJSCKBFpesMN]uescX[fYUijOX]&e8A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @&x

@BAx …†Y\[fP>OX[=F{OXbdW&ILSfijOX]4cpY\IsMNSC‡ˆW&IL]&FpP>]ue‰pFXST]&eŠA^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @‹|

@UAxpAT@ ŒIJSCFpPSCcˆ]&elMN]4[TYdbdW>HJrXKNMN]ŽA4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A @‹|

@UAxpA sW&eLOX[CHJY\HJeF{OXbdWIJSTijOX]uecQKBOXIs[‘WuiBOQYqHLSTKUF$MN]^[fYdPZrpY\[T]OXI’A^A4A^A4A^A^A @u“

@UAxpA sW&eLOX[CHJY\HJeF{OXbdWIJSTijOX]uecQKBOXIs[‘WuiBOQYqHLSTKUF$M”YUMN•B]&P>HLSTKUF–A0A^A4A^A4A^A^Ai

@BA| s]b Y\IZijOX]&eseLOXIs[T]&ebdKNMNk[T]&eb Y\HLrXW&bdY\HLSfijOX]&eA^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AiU

@UA}|NAT@ —`KUHLSTKUF$MN]^cXIJKU˜X[Tkbd]4˜XST]F$cQKjemWA^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^Aijƒ

@UA}|NA …+[TYBeLeLSC‰QPY\HLSTKUF/MX]&esWuiBOQYqHLSTKUFQe†YUONa/MNW&ILST•UW&]&ecpYUImHJSC]&[C[T]&eA^A4A^A4A^A^AiUn

™E,8.76š/;›";42œ">ss.0€;0š#%;42žs>>;^2 Ÿ=

XAC@EDGFBHJILKNMNOQPRHLSTKUF A4A^A0A^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A p@

XA ¡lSC‡ˆW&IL]&FpP>]ue‰pFXST]&ecˆKUOpI`[¢W&ijOpY\HLSTKUF$MN]4[TY PZrQY\[T]OXI4A^A4A^A^A4A^A0A^A4A^A4A^A^A 0

NANAT@ ¡lST•U]&IJe]aN]bdcX[T]&esMN]eJPZrXWb YBe$A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A 0

NANA …+KUFpeLSfeHZY\FpP]7]H`cXILWuP>SfemSTKUF£A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A j|

NANA yjHJYU˜XST[CSCHLW^]HlY\FpYU[C~NeL]0MN]¤pKUOXIJST]IlA^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A {ƒ

NANAx …+KUF{•U]&ILVB]FpP]7MN]ueseLPZrpWb YUe<A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^Ax

NANA}| yNPZrXW&bdYBeb9OX[CHLSTFXST•U]uY\ONaA^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AxBx

NANA v=]^PYBeb9OX[CHLSfMNSTbd]FpeLSCKBFXFX]&[(A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^Axj

XA ¥`ONHJIL]uesbdK{MXk[T]&eA0A^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A¦|X@

NAXAT@ o1ijOpYqHJSCKBF"M=YBMN•U]uPRHLSTKUF"A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A¦|X@

NAXA o1ijOpYqHJSCKBF"MX]&esKBFpMN]&e7A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A¦|Un

S

(5)

Ecole Polytechnique. Ne pas reproduire.

Ÿ  €,8!/)+'09& l'/€>'4&;7)+)1;-/;^2^ )0,5;^2–;7)+)1!

u: !";42 "$#

pAC@&%0W&FXWIZY\[TSCHLW&e A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEB|

pAC@BAC@ DGFBHJILKNMNOQPRHLSTKUF A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEB|

pAC@BA ¤XKUIJb9Op[TY\HLSTKUF/P[TYBeLeLSfiBOp] A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEU

pAC@BA v=]^PYBesMN]^[TYdMNSTbd]FpeLSTKUF"OXFM”]uemcQYUP>]œA^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEjƒ

pA ¥lcXcXIJKNPZrX]4•‹YUILSfYqHJSCKBFXFX]&[C[T] A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEU“

pAXAC@ ¤XKUIJb9Op[C]uesMN]'%0IL]&]FA^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEU“

pAXA ¤XKUIJb9Op[TY\HLSTKUF€•qYUILSfYqHJSCKBFXFX][T[T]"A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A ƒ{@

pA (rpWKUIJST]4MX]v3Yqa*)+ST[CVBIJYUb A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A ƒ‹x

pApAC@ …†YUMNIJ]^Y\˜pemHLIZY\SCH A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A ƒ‹x

pApA ¥`cXcX[TSfPYqHJSCKBFY\O$v3Y\cp[TYBP>ST]F A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A ƒUƒ

,w;^2-+'#9;^2œ";*2 .d)1;/ 03

xQAC@DGFjHJILKNMNOpP>HLSTKUF"]>HlY‹•B]ILHLSfeLeL]bd]FjH A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AE“X@

xQA ¤pKUFpP>HLSTKUFpesMX]PY\IJIJW0eLKUbdb Y\˜X[T]4]>HlMNW&ILST•qYqHJSCKBF21YUSC˜X[T] A^A0A^A^A4A^A4A^A4A^A^AE“

xQAXAC@ h7OX][fijOX]&eIZY\cXcˆ][fesM”SCFjHJWVUIZYqHJSCKBFEA^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AE“

xQAXA ¡lW&ILST•qYqHLSTKUF31Y\ST˜X[T] A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AE“

xQA ¡7W>‰QFXSHJSCKBF]>HscXIJSCFQP>STcpY\[T]&ecXIJKUcpILSTW>HJW&e A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AE“U

xQApAC@ 41emcQYUP>]

H 1 (Ω)

A^A4A^A^A0A^A4A^A4A^A^A4A^A4A^A0A^A4A^A^A4A^A4A^A4A^A^AE“U

xQApA 41emcQYUP>]

H 0 1 (Ω)

A^A4A^A^A0A^A4A^A4A^A^A4A^A4A^A0A^A4A^A^A4A^A4A^A4A^A^AEnUt xQApA (3IZYUP]&e†]>H1KBILb%OX[C]uesMN]'%0IL]&]F A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEn

xQApAx _`F$ILWuemOp[HZYqH`MN]P>KBbdcpYUPSHJW A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AEnjƒ

xQApA| 41emcQYUP>]ue

H m (Ω)

A4A^A^A0A^A4A^A4A^A^A4A^A4A^A0A^A4A^A^A4A^A4A^A4A^A^AEnU“

xQAx h7Op][fiBOp]&esP>KBbdcX[CW&bd]FjHJeONHJSC[T]&e A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@&tX@

xQAxQAC@ ¡lW&b KBFpemHLIZYqHLSTKUF$MNO5(rXW&KUIJkbd]7xQApA|%MN]MN]&FpeLSHJW A^A^A4A^A4A^A4A^A^AŠ@&tX@

xQAxQA 41emcQYUP>]

H(div)

A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A4A^A4A^A0A^A^AŠ@ut xQAxQA 41emcQYUP>]ue

W m,p (Ω)

A^A^A0A^A4A^A4A^A^A4A^A4A^A0A^A4A^A^A4A^A4A^A4A^A^Aœ@utB|

xQAxQAx ¡lOpYU[CSCHLW A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@&tU

xQA| v3SC]&FY‹•U]&P7[T]&esMNSfeHJILST˜XONHJSCKBFpe A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@&tjƒ

#¦.0!"/; ,8'096.7,5'4u: !"; ";^26^ .%)7,8;42ž;0)1)17^&: !";^2 ˆQ

|XAC@DGFjHJILKNMNOpP>HLSTKUF A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@U@U@

|XAwoHLOpMN]%MNOv3Y\cp[TYBP>ST]F A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@U@

|XAXAC@ …+KUFpMNSCHLSTKUFQesY\ONa€[TSTb SCHL]ue`MN]¡7SCIJSfPZrX[C]HšA^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@U@

|XAXA …+KUFpMNSCHLSTKUFQesY\ONa€[TSTb SCHL]ue`MN]—l]OXb Y\FpFEA4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@U@&“

|XAXA …+Kj]78P>ST]FjHJe•qY\IJSfY\˜X[T]&e/A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@U|

|XAXAx ŒILKBcXIJSCWHLW&esijOpYU[CSCHJY\HLST•U]ue A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@\n

|XA `W&eLKU[TONHLSTKUF$M”Y\ONHJIL]ueb KNMNk&[C]ue A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@ Un

|XApAC@ y{~NeHJkbd]^MN]^[¢W[fYUemHLSfP>SCHLW^[TSTFXW&YUILSfemW&]lA^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@ Un

|XApA o1ijOpYqHJSCKBFpesMN]9yjHJKU‚U]ueA4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^AŠ@x{ƒ

(6)

Ecole Polytechnique. Ne pas reproduire.

"¦,8.76š/;›";42œ.0)1.0,5;0%2žs>&2 #”

pAC@¦¥`cXcXIJK‹aNSCb Y\HLSTKUF"•‹YUILSfYqHJSCKBFXFX]&[C[T] A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@u|X@

XAT@UAT@ DGFjHLIJK{MXOpPRHJSCKBF A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@u|X@

XAT@UA ¥`cXcpILK‹aNSTbdY\HLSTKUF/SCFjHL]&ILFp]^VUWFpWIZY\[T]"A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@u|

XAT@UA +W>HJrXKNMN]MN]'%4Y\[T]IJ‚{SCF–A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@u|B|

XAT@UAx +W>HJrXKNMN]MN]ueW[TWbd]FjHJes‰pFXSfe

cXIJSCFQP>STcQ]uesVUW&FXWIZY\ONa A0A^A4A^A4A^A^AŠ@u|B|

pA o[CW&bd]FjHJe‰QFXSTe`]FMNSTb ]&FpeLSCKBF

N = 1

A4A^A^A4A^A4A^A4A^A^A0A^A4A^A4A^A4A^A^AŠ@u|\

XANAT@ o[TWbd]FjHJes‰pFXSfe

P 1

A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@u|U

XANA …+KUF{•U]&ILVB]FpP]l]H`]&emHLSTbdY\HLSTKUF$M”]IJIL]&OXIšA^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&p@

XANA o[TWbd]FjHJes‰pFXSfe

P 2

A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&B

XANAx ŒIJKUcXIJSCWHLWuesiBOQY\[TSHZYqHLST•U]ue A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&B“

XANA}| o[TWbd]FjHJes‰pFXSfe`M”`]IJbdSHJ] A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@‹ƒ

pA o[CW&bd]FjHJe‰QFXSTe`]FMNSTb ]&FpeLSCKBF

N ≥ 2

A4A^A^A4A^A4A^A4A^A^A0A^A4A^A4A^A4A^A^AŠ@‹ƒ‹x

XAXAT@ o[TWbd]FjHJes‰pFXSfeHLIJSTYUFXVUOp[TYUSCIJ]&e A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@‹ƒqx

XAXA …+KUF{•U]&ILVB]FpP]l]H`]&emHLSTbdY\HLSTKUF$M”]IJIL]&OXIšA^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&“B

XAXA o[TWbd]FjHJes‰pFXSfesIL]uPRHZY\FXVBOX[TYUSCIJ]&e A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&nUx

XAXAx o[TWbd]FjHJes‰pFXSfescQKBOXIlyjHJKU‚U]ue0A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^AŠ@&nBn

XAXA}| lSfemOQY\[TSTeJYqHJSCKBF"MN]ueILWuemOp[HZYqHJeF{OXbdW&ILSfijOX]&e A^A4A^A^A4A^A0A^A4A^A4A^A^A,\tj|

^.%)7,5;^2œ'd!l'/)1;0!/$2 ^^/;42 ™

ƒNAC@&+K\HJSC•qYqHJSCKBF$]>Hs]>aN]bdcX[T]&e†A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,\tBn

ƒ{AT@UAT@ DGFjHLIJK{MXOpPRHJSCKBF A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,\tBn

ƒ{AT@UA sW&eLKU[TONHJSCKBF"MN]uecXIJKU˜X[Tkbd]&eSTFpemHJY\HLSTKUFXFpYUSCIJ]&e7A4A^A^A4A^A0A^A4A^A4A^A^A,N@ut

ƒNA (rXWKBILST]^emcˆ]&P>HLIZY\[T] A^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,N@

ƒ{ANAT@ %0WFXW&IJYU[CSCHLWue A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,N@

ƒ{ANA ¡lWuP>KBb cˆKBeLSCHLSTKUF$emcˆ]&P>HLIZY\[T]4M=OXF$KBcQW&IJY\HL]OpIPKUbdcpYBPRHŠA0A^A4A^A4A^A^A,N@‹|

ƒNA Y\[T]OXIZecXIJKUcXIJ]&eM”OXF$cXIJKU˜X[Tkbd]4][T[CSTcNHJSTijOX] A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,N@u“

ƒ{AXAT@ ŒIJKU˜X[Tkbd]4•qY\IJSTY\HLSTKUFXFX]&[ A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,N@u“

ƒ{AXA Y\[T]OXIZecXIJKUcXIJ]&eMNOv Y\cX[fYUPSC]&F?A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,

ƒ{AXA ¥`ONHJIL]uebdK{MXk[T]&e’A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,U

ƒNAx +W>HJrXKNMN]&esF{OXbdW&ILSfijOX]&e A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,Un

ƒ{AxpAT@ ¡lSfeJP>IJW>HLSfeJYqHLSTKUF"cpYUI†W&[CW&bd]FjHJe‰QFXSTeA^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,Un

ƒ{AxpA …+KUF{•U]&ILVB]FpP]l]H`]&emHLSTbdY\HLSTKUFpesM=]&ILIJ]OXI A4A^A4A^A^A4A^A0A^A4A^A4A^A^A, 0

0&^.%)7,5;^2œC./)1!"& ™QŸ #

“pAC@&+K\HJSC•qYqHJSCKBF$]>Hs]>aN]bdcX[T]&e†A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A, j|

“XAT@UAT@ DGFjHLIJK{MXOpPRHJSCKBF A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A, j|

“XAT@UA +KNMNW[TSfeLY\HLSTKUF"]>H`]>aN]&b cp[C]uesM”WuijOpYqHJSCKBFpecpY\IZY\˜ˆKU[TSfiBOp]&e A^A4A^A4A^A^A, B

“XAT@UA +KNMNW[TSfeLY\HLSTKUF"]>H`]>aN]&b cp[C]uesM”WuijOpYqHJSCKBFper{~jcˆ]IJ˜ˆKU[TSTijOX]ueA4A^A4A^A^A, {ƒ

“pA 4 aNSTemHL]&FpP>]^]H`OXFXSfP>SCHLWMXYUFpe[C]P&YUecpYUIJYU˜QKB[CSfijOX] A4A^A4A^A^A4A^A0A^A4A^A4A^A^A, B“

“XANAT@ ¤XKBILb%OX[TY\HLSTKUF/•‹YUILSfYqHJSCKBFXFX]&[C[T]$A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A, B“

“XANA _`F$IJW&eLOX[HZYqHsVUW&FXWIZY\[†A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,qxjt

“XANA ¥`cXcp[CSfPY\HLSTKUFpe A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A,qx{|

“pA 4 aNSTemHL]&FpP>]^]H`OXFXSfP>SCHLWMXYUFpe[C]P&YUer{~{cQ]&IL˜ˆKU[TSfiBOp] A^A4A^A^A4A^A0A^A4A^A4A^A^A,U|Ut

(7)

Ecole Polytechnique. Ne pas reproduire.

“pApAC@ ¤XKUIJb9Op[TY\HLSTKUF€•qYUILSfYqHJSCKBFXFX][T[T]"A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,U|\t

“pApA _`F$ILWuemOp[HZYqHsVUW&FXWIZY\[†A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,U|N@

“pApA ¥`cXcX[TSfPYqHJSCKBFpe A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,U|qx

“pAx Œ1ILKBcXILSTW>HJW&esijOpYU[CSCHJY\HLST•U]&eMXYUFpes[T]^PYUecQY\IZY\˜ˆKU[TSTijOX] A4A^A0A^A^A4A^A4A^A4A^A^A,U|Bƒ

“pAxQAC@ …+KUbdcQKBImHJ]bd]FjH`YUeL~{bdcNHLKUHLSfiBOp] A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,U|Bƒ

“pAxQA ŒILSTFpPSCcˆ]MNO$b YqaNSCb%OXb A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,U|\n

“pAxQA ŒILKBcpY\VjYqHJSCKBF •{SHJ]&eJem]4STFN‰pFXST]%A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\Ut

“pAxQAx sWVBOX[TYUILSCHLW0]>H`]‡ˆ]H`ILW&VUOX[fY\IJSfeLYUFBH A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\X@

“pAxQA| o1ijOpYqHJSCKBF"MN]^[fYdPZrpY\[T]OpI`MXY\Fpe†HJKUONH`[‘]&eLcpYUP] A^A0A^A^A4A^A4A^A4A^A^A,\

“pA|wŒ1ILKBcXILSTW>HJW&esijOpYU[CSCHJY\HLST•U]&eMXYUFpes[T]^PYUer{~{cˆ]IJ˜QKB[CSfijOX]’A4A^A0A^A^A4A^A4A^A4A^A^A,\B|

“pA|XAC@ sW•B]IZemST˜XST[CSCHLW4]F/HL]&bdcpe A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\B|

“pA|XA …+KUbdcQKBImHJ]bd]FjH`YUeL~{bdcNHLKUHLSfiBOp]0]H`W&ijOXSTcpY\ILHLSCHLSTKUF$MN]4[‘W&FX]IJVUST](A^A^A,\U

“pA|XA lSCHL]&eJeL]4MX]^cXILKBcpY\VjYqHJSCKBF ‰pFXST]A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\jƒ

“pA + W>HLrpK{MX]&esF{OXbdWIJSfiBOp]&esMXYUFpe[C]^P&YUecpYUIJYU˜QKB[CSfijOX]gA^A4A^A0A^A^A4A^A4A^A4A^A^A,\Un

“pApAC@ y{]bdS) MNSfeJP>IJW>HLSfeJYqHLSTKUF/]F$]&eLcpYBP>] A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\Un

“pApA ¡lSfeLPILWHLSfeLY\HLSTKUF/HLKUHJY\[T]4]F"]&eLcpYUP])‘HL]&b cQeA4A^A4A^A0A^A^A4A^A4A^A4A^A^A,Bƒ{@

“pA}ƒ + W>HLrpK{MX]&esF{OXbdWIJSfiBOp]&esMXYUFpe[C]^P&YUer{~{cQ]&IL˜ˆKU[TSTijOX]A4A^A0A^A^A4A^A4A^A4A^A^A,Bƒ‹x

“pA}ƒNAC@ y{]bdS) MNSfeJP>IJW>HLSfeJYqHLSTKUF/]F$]&eLcpYBP>] A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,Bƒ\|

“pA}ƒNA ¡lSfeLPILWHLSfeLY\HLSTKUF/HLKUHJY\[T]4]F"]&eLcpYUP])‘HL]&b cQeA4A^A4A^A0A^A^A4A^A4A^A4A^A^A,Bƒq

>9 /!#%&) ^,5&23'09& ™0=

npAC@ + K\HLST•qYqHJSCKBF"]Hs]>aN]bdcX[T]&e†A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\“X@

npAC@BAC@ DGFBHJILKNMNOQPRHLSTKUF A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\“X@

npAC@BA 4 aN]bdcX[T]&e(A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\“

npAC@BA ¡lW‰pFXSCHLSTKUFpes]H`FXK\HZYqHJSCKBFpe?A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\“U“

npAC@BAx 7cNHLSTbdSTeJYqHJSCKBF/]&F$MXSCbd]FQemSTKUF/‰pFXST]9A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\“Un

npA 4a{SfemHL]FQP>]M”OXF$bdSCFXSTb9Opb*]FMNSTbd]FpeLSTKUF"SCFN‰QFXSC]-A^A4A^A0A^A^A4A^A4A^A4A^A^A,\nX@

npAXAC@ 4 aN]bdcX[T]&e`MN]4FXKBF )]>aNSTemHL]&FpP>] A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\nX@

npAXA ¥`FpYU[C~NeL]4PKUF{•U]aN] A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\n\x

npAXA sW&eLOX[CHJYqHZesM”]>aNSTemHL]&FpP>] A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A,\njƒ

š#d ">u š2œf^9>,5'€)+>. ;0'€)€>96,5;42 Ÿ =

@utXAT@ %0W&FXWIZY\[TSCHLW&e A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A UtX@

@utXAT@UAT@ŽDGFBHJILKNMNOQPRHLSTKUF A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A UtX@

@utXAT@UA ¡lSC‡ˆW&IL]&FjHLSfY\˜XST[CSCHLW–A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A Ut

@utXAŠ…+KBFpMNSCHLSTKUFpesM=KBcNHLSTb Y\[TSHJW A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A Utjƒ

@utXANAT@ŽDGFXW&ijOpY\HLSTKUFpesM”4OX[T]Il]>H`P>KBFBHJIJYUSCFjHJ]&eP>KBFj•B]>aN]&e A0A^A^A4A^A4A^A4A^A^A Utjƒ

@utXANA +OX[CHLSTcX[CSfPY\HL]&OXIJesMX]v3Y\VBIJYUFXVU] A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A X@&t

@utXAžŒ KBSCFjH)GeL][T[C]{HLrXW&KUIJkbd]^MN] 4OXrXF"]>H( OpPZ‚U]&INMNOQY\[TSHJW A^A0A^A^A4A^A4A^A4A^A^A N@

@utXAXAT@ŽŒ KBSCFjH )eL][T[C]$A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A

@utXAXA (rXWKBILk&b ]^MN] 4OXrpF"]H(3OQPZ‚U]IŠA^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A

@utXAXA›¡lOpYU[CSCHLW A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A qx

@utXAxŠ¥lcXcX[TSTP&YqHJSCKBFpelA^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A \“

@utXAxpAT@ŽoFX]IJVUST]^MNOpYU[C]4KUOPKUbdcX[TWbd]FjHJYUSCIJ] A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A \“

(8)

Ecole Polytechnique. Ne pas reproduire.

@&tXAxpA …+KUbdb Y\FpMX]0KBcNHLSTb Y\[T] A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A Bt

@&tXAxpA 7cNHLSTbdSTeJYqHJSCKBF"MX]&eseL~{emHLk&bd]&esMNSfeHJILST˜XOXWueA4A^A4A^A^A4A^A0A^A4A^A4A^A^A j|

@utXA}|œ¥`[TVUKUIJSCHLrXbd]&eF{OXbdW&ILSfijOX]&e A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A {ƒ

@&tXA}|NAT@ŽDGFjHLIJK{MXOpPRHJSCKBF A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A {ƒ

@&tXA}|NA ¥`[TVUKBILSCHLrXbd]uesMN]0H~jcˆ]^VUIZYUMXSC]&FBH

P&YUe`eJY\FQe†PKUFjHLIZY\STFjHL]ue A4A^A4A^A^A B“

@&tXA}|NA›¥`[TVUKBILSCHLrXbd]uesMN]0H~jcˆ]^VUIZYUMXSC]&FBH

P&YUe`Y‹•B]&P0P>KBFBHJIJYUSCFjHJ]&e A4A^A4A^A^A \xQ@

@&tXA}|NAx +W>HJrXKNMN]MN]^—`]

HLKBF A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A \xNƒ

Q€,8.76š/;^2ž/; )+'€;^#96;0#%6;›.^.7/'09& ;0)1)+;

 2 !#"$% &%'$(() Ÿ$#”

@B@UAT@ DGFBHJILKNMNOQPRHLSTKUF A4A^A0A^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A B|X@

@B@UAžŒILKBVUIZY\bdb YqHJSCKBF€[TSCFXWuY\STIL] A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A B|

@U@UANAT@ ¡lW‰pFXSCHLSTKUFpe`]>H`cXIJKUcXIJSTW>HLWueŠA4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A B|

@U@UANA ¥`[TVUKBILSCHLrXbd]^MNOeLSCbdcX[T]>aN]A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A B|jƒ

@U@UANA›¥`[TVUKBILSCHLrXbd]uesMN]4cQKBSCFjHJe`SCFjHLW&ILST]OpIJe A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A U0

@U@UANAx ¡lOQY\[TSHJW A0A^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A U0

@B@UAœŒ KB[C~Bk&MNIJ]&e†]&FBHJSC]&IJeA^A4A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A UB

@U@UAXAT@ŽŒ KUSTFBHZe]>a{HLIJWb YUONa/MN]^P>KBbdcpYUP>HJesP>KBF{•U]>aN]ue A4A^A^A4A^A0A^A4A^A4A^A^A UB

@U@UAXA + YqHJILSfP>]ue†HLKUHJY\[T]bd]&FBHsOXFpSCbdKNMNOX[fY\STIL]ue A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A UBn

@U@UAXA ŒIJKU˜X[Tkbd]&esMX]+*pK\HZe A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A jƒ

@B@UAxŠŒILKBVUIZY\bdb YqHJSCKBF/MN~{FpYUb SfijOX] A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A jƒ\

@U@UAxpAT@ŽŒIJSCFpPSCcˆ]M”KUcNHJSCb YU[CSCHLWMN]-,†][T[TbdYUF-A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A jƒ\

@U@UAxpA-ŒIJKU˜X[Tkbd]4]F$rXKBILS/.KBF‰pFXS7A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A jƒBƒ

@U@UAxpA ŒIJKU˜X[Tkbd]^MNOPZrX]bdSTFMN]^P>K10XH`b STFXSTb9OXb XKBO"M=YUILI32>HsKUcXHLSTbdYU[=A^A^A U“Bt

@B@UA}|œ¥`[TVUKUIJSCHLrXbd]&eVB[CKBONHLKBFpe^A^A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A U“j|

@U@UA}|NAT@ %0WFXW&IJYU[CSCHLWuesemOpIs[C]ueb WHLrXKNMN]uesVU[TKUOXHLKUFpFX]&e A4A^A^A4A^A0A^A4A^A4A^A^A U“j|

@U@UA}|NA ¥`[TVUKBILSCHLrXbd]lMX] 4IJOpeL‚‹YU[XcˆKUOXI†[C]7cXIJKU˜X[Tkbd]7MX]l[‘YUIL˜XIJ]lPKUOX•{IZY\FjH1MN]

P>K0NHsb STFXSTb9OXb A4A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A U“B

@B@UAgy{WcpYUIJY\HLSTKUF"]>HsIJ][fYqaXYqHJSCKBF<A^A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A U“B“

@U@UAXAT@5y{WcQY\IZYqHLSTKUF/]>H`W&•‹YU[COQYqHLSTKUF

˜pIJYUFpPZr"YUFpM/˜QKBOXFpM*A4A^A0A^A4A^A4A^A^A U“B“

@U@UAXA s][fYqaXY\HLSTKUF/MX]^cXILKB˜X[Tkbd]&esPKUb9˜pSCFpY\HLKBSCIJ]&e A4A^A^A4A^A0A^A4A^A4A^A^A Un

'€; €;54 ;^21'#%;42œ"; 6>) ;00 , QŸ

'€; €;54 '/'€)6$23; !",5.7/&: !";›,5'4€u#9;7)+)1; ,

@ XAT@’sW&eLKU[TONHLSTKUF$MN]ue`em~NemHLk&b ]ue[CSTFXWuY\STIL]ue›A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xBtBn

@ XAT@UAT@ `Y\cpcQ]&[TeseLOXIs[T]&esFXKBILbd]ue+b Y\HLIJSTPSC]&[C[T]&e$A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xp@ut

@ XAT@UA …+KUFpMXSHJSCKBFXFX]bd]&FBHs]HleHZY\˜XST[CSCHLW A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xp@

@ XAT@UA +W>HJrXKNMN]&elMNSCIJ]&P>HL]ue A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xp@‹|

@ XAT@UAx +W>HJrXKNMN]&e`SHJWIZYqHLST•U]ue A4A^A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A x0U“

@ XAT@UA}| +W>HJrXKNMN]MNO$VUIZYUMXSC]&FBHsP>KBF7OXVBOXW A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A x 0

@ XAg…†Y\[fP>OX[3MX]0•qYU[C]&OXIJe]H`•U]uPRHL]&OXIZe+cpILKBcXIL]ue A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xUxjt

@ XANAT@ +W>HJrXKNMN]MN]4[fYdcXOXSfeLeJY\FpP]A4A^A0A^A^A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xUxjt

@ XANA +W>HJrXKNMN]MN]'%0ST•U]FQe ) `KBOpem]&rXKU[fMN]&I?A4A^A4A^A4A^A^A4A^A0A^A4A^A4A^A^A xUx

(9)

Ecole Polytechnique. Ne pas reproduire.

@ XANA +W>HJrXKNMN]MN]^v3YUFpP .&KBe A4A^A0A^A4A^A^A4A^A4A^A4A^A0A^A^A4A^A4A^A4A^A^A xUxB

(10)

Ecole Polytechnique. Ne pas reproduire.

! " #$% "&('*),+.-0/21342537662/8

9 0:#;<>=?@ (A 9 (CB(D5FE#GH/17IIJ;K/147I

(11)

Ecole Polytechnique. Ne pas reproduire.

"!#$

…+] P>KUOpIJeHJIJYUSHJ]9MN] MN]ONaeLO7]>HZel]ueLeL]FjHLST][fepb Y\Sfe`cpYUILbdS3HJY\FjH7M=YUONHLIJ]&ep]&F b YqHLrpWb Y )

HJSTijOX]ue9YUcXcX[TSTijOXW&]&e% [¢Y\FpYU[C~NeL] F{OXbdWIJSTijOX] ]H%[‘KBcNHLSTbdSTeJYqHJSCKBF=A3¥ •‹YUFjH9b 2bd]€MN]cpILWuem]&FBHJ]I

P]&e`MX]ONaMNSfeLPSCcX[TSTFX]&eQMNSfeLKUFpeHJKUONH0MN]9eLOXSHJ]9ijO= HLIZY‹•U]IZe†[C]&OXIl]&FpeL]STVUFX]&b ]&FjH`[¢KU˜7]uPRHJS1MN]

P]7PKUOXIZe ]&emH†M=STFjHLIJK{MXOXSCIJ]`[T]l[T]&P>HL]OpI†YUO b KBFpMN]0MN]l[fY$& ' & " & ( '$ ]>H

MX]l[fY '& '% ' & ( '$ iBOpSQKBFBH+cXILSfe+OXFX]lSTbdcQKBImHZY\FpP]lP>KBFpeLSTMNW&IJYU˜X[T]`P>]&e+MN]IJFXSTkIJ]&e

MXW&P>]&FXFXST]&e%MXY\Fpe^HLKUOQe[T]&e9MXKUb Y\STFX]&e9MN][fY eLPSC]&FpP>]€]>H%MN]ue9YUcXcX[TSTP&YqHLSTKUFQe^SCFQMNOpemHLIJSC]&[C[T]&e

KUO

eJP>ST]FQP>]&esMX]^[¢SCFpVUWFpSC]&OXI >Apv3YdbdKNMNW&[CSfeLY\HLSTKUF"b YqHLrpWb YqHJSTijOX]^]ueH`[‘YUImH

KUO$[TY eJP>ST]FQP>] XeL][TKUF

[T] cQKBSCFjH0MX]%•{OX] lMN] IJ]cXIJW&eL]FjHL]&I

KBO MN]%HLIZY\FQe1KBILbd]I OXFp]9IJW&Y\[TSCHLW9cprj~NeLSTijOX]%]F–MN]&e7bdK )

MXk[T]&e`YU˜peHJIJYUSHZeYUPP]&eJemST˜X[T]&e [‘YUFpY\[T~Nem]4]>HlYUO"P&Y\[fP>OX[‘Apv Y eLSTb9OX[fYqHJSCKBF/F{OXbdWIJSfiBOp]^]&emHX˜XST]F

e 0XI Q[T] cXILKNP]&eJemOpe0ijOXS cQ]&ILbd]H4MN]dPYU[TPOX[T]I4emOXI4KBIJMNSTFpY\HL]OpIl[T]&e4emKB[COXHLSTKUFpe0MN]dP>]&e0bdKNMNk[T]&e

]HlMNKUFQP0MX]emSTb9OX[T]Is[fY ILWuY\[TSHJW4cXr{~{eLSfiBOp]UA

+ YUSTeXHJKUONH0M”YU˜QKBIJM QijOX]9eLKUFjH`[T]&e`b Y\HLrXW&bdY\HLSfijOX]&e`YUcXcX[TSTijOXW&]&e)9¡7SCIJ]9ijO=SC[ eY\VBSHlMX]&e

b Y\HLrXW&bdY\HLSfijOX]&elHLKBOXILFpW]&e7•U]&IJel[C]ue7YUcXcX[TSTP&YqHJSCKBFpe7eL]IZY\SCHlOXFp]9HZY\ONHJKU[TKUVUST]]H0OXFp] 1YUOpeLeL]%P&Y )

IZYUP>HLW&ILSfeLY\HLSTKUF=A 41F ]>‡ ]>H +MN]HJKUONH/HL]&b cQe[T]&eb Y\HLrXW&bdY\HLSfP>ST]FpeKUFjH€W>HJW STFpemcpSCIJW&e€cpYUeMX]&e

cpILKB˜X[Ck&bd]&escXIZYqHJSTijOX]ue`ijO=SC[fe`KBFBH`]ueLeJY‹~UW^MN]9ILWuemKBOpMNIJ] N]>H0P>]cˆ]FQMXY\FjHl[‘W&bd]IJVU]FQP>]MN]uesb Y )

HJrXWb Y\HLSfiBOp]&e0Y\cXcX[TSfiBOpW]&e0P>KBbdb ]dMNSfeLPSCcp[CSTFX]9STFpMNW&cQ]&FpMXYUFBHJ]%]ueH4IJ][fYqHLST•U]&bd]FjHlIJW&P>]&FjHL]UA 4F

1YUSH HJKUONH$Y’PZrpY\FXVBW Y‹•U]uP$[¢Y\cXcQY\IJSHJSCKBFŠMX]&ecpIL]&b ST]IZe KBIJMNSTFpY\HL]OpIJe YUOg[C]&FpMN]b YUSCF MN]š[TY

eL]&PKUFpMX]%VBOX]IJIL]%bdKUFpMNSfY\[T]UA”Œ1[COpe^ijOX]dcˆKUOXI0HJKUONHY\OXHLIJ]dMXSTeJP>STcX[TSCFX]d[‘KBIJMXSCFpY\HL]&OXI0Y"WHLWdOXFX]

IJW•BKU[TONHLSTKUFœcˆKUOXI [T]&e b Y\HLrXW&bdY\HLSfijOX]&e*%1SC[lY ]&FŠ]>‡ ]>H€KBOX•U]&ImH OXFgPZrpYUb cŠFXKBOX•U]uY\O P][TOXS

MX]/[fYšbdKNMNW&[CSfeLY\HLSTKUF ]HMN]/[fY–emSTb9OX[fYqHJSCKBF=A vKUIZMNSCFQYqHL]&OXIdY1YUSH MN]&e b YqHJrXWb Y\HLSfiBOp]&e OXFX]

eJP>ST]FQP>]0]>aNcQW&ILSTbd]FjHJYU[C]

KUF31Y\SCHsMX]&e +m]>aNcQW&ILST]FQP>]&e+F{OXbdWIJSTijOX]ue,4P>KUbdbd]^M”Y\ONHJIL]ue1KBFjHsMX]&e

]aNcQW&ILST]FpP]&e^cprj~NeLSTijOX]ue 3]H%[fY P>KBFpP>]&cNHLSTKUF YUSCFpeLS+ijOX]€[¢Y\FpYU[C~NeL]MN]&e9bdW>HLrpK{MX]&e MN]€PYU[TPOX[

eLOXIKBIJMXSCFpY\HL]&OXIemKBFjH9MN]•B]F{OX]&e4OpFX]FXKBOX•U]&[C[T]d˜XIJYUFpPZrX] MN]ueb YqHJrXWb YqHJSTijOX]ue-% PU]ueH[fYemS)

b%OX[fYqHLSTKUF$F{OXbdWIJSfiBOp]UA …+]ue`cXIJKUVBILkue†KUFjH7Y\OQeLeLS=cˆ]IJb SfelYUONa$bdY\HLrXW&b YqHLSfijOX]&elMN]9e&Y\HmHJYBijOX]I

–MN]ue%cpILKB˜X[Ck&bd]&e ˜Q]uY\OpPKUOXcžcX[TOpe P>KBb cp[C]a{]ue ]>HP>KBFpP>IJ]>HZe STeJeLOpedMN]$b KUHLST•qYqHLSTKUFQe%STbdbdW)

MXSTY\HL]&elSCFQMNOpemHLIJSC]&[C[T]&esKBO eLPSC]&FjHLSC‰QijOX]&eQYUONaXiBOp][fesKUF$cˆ]ONH4Y\cXcˆKUILHL]&IlMN]&elILW&cQKBFpeL]&e d[fY 1KUSfe

ijOpYU[CSCHJY\HLST•U]ue+b YUSTe`YUOpeJemS”ijOpY\FjHJSHZYqHLST•U]ue.%pP\]&emHs[TY bdKNMNW[TSfeLY\HLSTKUF"bdY\HLrXW&b YqHLSfijOX]UA

7Fgcˆ]ONHMNKBFpPPYUIJYBPRHLW&ILSfeL]I [T]&eb Y\HLrXW&bdY\HLSfijOX]&e/Y\cXcX[TSfiBOpW]&e/P>KBb bd][T]&e/b YqHLrpWb Y )

HJSTijOX]ue`MN][fYdbdKNMNW[TSTeJYqHJSCKBF$]>HlMN][fY eLSCb%OX[TY\HLSTKUF$F{OXbdWIJSTijOX]BAp¡7]P>]cˆKUSTFjHlMN]•{OX] p[C]uesb Y )

HJrXWb Y\HLSfiBOp]&e Y\cXcX[TSfiBOpW]&e em]†emSCHLOp]FjH `[¢SCFjHJ]IZem]uPRHLSTKUF%MN]1cX[TOpeLSC]&OXIZe3MNSfeLPSCcp[CSTFX]&e eJP>ST]FjHLSC‰QijOX]&e/%

b Y\HLrXW&bdY\HLSfijOX]&epPYU[TPOX[”STF 1KUIJb YqHLSfijOX] QeLPSC]&FpP>]uecXr{~{eLSfiBOp]&eXPZrXSCbdSfijOX]&eXbdW&P&Y\FXSfijOX]&eN˜XSTK )

[TKUVBSTijOX]ue BW&PKUFXKBb SfijOX]&ej]>H†eJP>ST]FQP>]&e†MN]0[¢STFXVUW&FXSC]&OXI

eLKUOpe†P>]0MN]IJFXST]I†•UKNPYU˜X[C]lKUFIJ]VBILKBOXcQ]

OQemOX]&[C[T]bd]&FBH0[C]ue7MXS‡ WIJ]FjHJe0MNKBbdYUSCFp]&e0M”YUcXcX[TSTP&YqHJSCKBFpe`STFpMNOpemHLIJST][fe7PKUbdbd]9[‘YUWIJKUFpYUONHLSfijOX]

[fYcpILKNMNOpP>HLSTKUFšM=W&FX]IJVUST] Q[TY‰pFpY\FQP>] ˆ]>HZP\ARA v=]%bdY\HLrXW&b YqHLSfP>ST]F–Y\bdWIJSfPY\STF10BKBeL]cXr 4]&[C[T]I

Y 8 ILb Y\SCH%eLKUOpe 1KUIJbd]€MN]"˜QKBONHJYBMN]/ijOX]/[C]ue9b Y\HLrXW&bdY\HLSfijOX]&e Y\cXcp[CSfijOXW]ue9eLKUFjH.+m[TYšeLPSC]&FpP>]

MXKUFjH1[T]&e b YqHJrXWb Y\HLSfiBOp]&ecpOXIL]ueeLKUFjH 7OpemHL]lOpFX]`˜XIZY\FpPZrp],RA\DG[p•BKUOX[fY\SCHbd]>HmHJIL]7Y\STFpemSp]F IJ][TSC]71

[T]dPY\IZYUP>HLk&IL]cp[COXIJSfMNSTeJP>STcX[TSCFQY\STIL] MN]&e0b YqHJrXWb YqHJSTijOX]ue4Y\cXcX[TSfiBOpW]&e

b YUSTe0ST[ F3]ueH^cpYBel]aNP[CO

ijO=ST[ˆYUSH†•UKUOp[CO€YUOpeJemSQIL]&FpMNIJ]l[fY^bdKUFXFpYUSC]0MN]l[T]OpI+cXSTk&P] %P>]ILHJYUSCFQeb YqHJrXWb YqHJSTPSC]&Fpe2+cXOpIJe3,

(12)

Ecole Polytechnique. Ne pas reproduire.

iBOpS3Yq‡ ]&P>HL]FjH`MN]^bdWcpILSfem]&I[C]uesb YqHLrpWb YqHJSTijOX]uesY\cXcX[TSfiBOpW]&e RA

41F9cpYUIJYUcXrXIZYUeJY\FjH”[C]1HLSCHLIJ]+M”OXF‰Q[Cb P>W&[Ck&˜XIJ] ub KBFP>KB[C[TkVBOX]ŒST]IJIL]7)v3KUOXSfe3v3SCKBFpe=cpILWHL]FQM

iBOp]0[T]&e`bdY\HLrXW&b YqHLSfijOX]&esYUcXcX[TSTijOXW&]&eeLKUFjH`PY\IZYUP>HLW&ILSfemW&]&e+cpY\IHJILKBSTePZrXKjem]ue.%

+

H

G

H2 D A=v=]&e0PYBeLeL]>HLHL]ue`•{STMXWK/eLKUFjHl˜pSC]&F e0XI7[T]%eL~{b9˜ˆKU[T]%MN]%[TY/emSTb9Op[TY\HLSTKUFF{OXbdWIJSTijOX]

]>H"MN]&e+7KB[CSfed‰p[Tb eijO=][T[C]cXIJKNMNOXSCH 1[T]&e bd]FQemKBFXVU]ue P>KUIJIJ]&eLcQKBFpMN]FjH Y\ONaœbdK{MXk[T]&e

cpYUe

HLKUO 7KUOXIZe‰ˆMNk[T]&e ž[TYžILWuY\[TSHJW †]H"[T] em]a{] P\]&emH$W•{STMX]bdbd]FjH"[‘Y\FpYU[C~NeL]šb YqHJrXWb YqHJSTijOX]

bdK\HJ]OXIsSTFXWcpOXSTeJY\˜p[C]MN]uecpYUeJeLSCKBFpe†rjOpbdYUSCFp]&e]>HleLKUOpIJP]0MN]4HZY\FjHlMN]4cX[fY\SfemSTIZe >ACATA

¥lcXIJk&e€P]

[TKUFpV MNWHLKBOXI€FXKBOpecˆKUOp•UKUFQe b Y\STFBHJ]FpYUFjH€IL]&•U]&FXSCIYUOŠHLSCHLIJ] MN] P>]šPKUOXIZeA

vY\FpYU[C~NeL]/F{OXbdWIJSfiBOp]/]ueH€MNKBFpP"[TY–MNSfeLPSCcp[CSTFX] ijOXS`P>KBF>KBSH ]H YUFpY\[T~Nem]"[T]&edbdW>HJrXKNMN]&edKBO

Y\[TVUKUIJSCHLrXbd]&e MN]PYU[TPOX[sF{OXbdWIJSTijOX]BA Œ Y\I Y\ST[C[T]OpIJe [‘KBcNHLSTbdSTeJYqHJSCKBF ]&emH [TY HJrXWKBILST]"MX]&edbdW)

HLrXKNMN]ueijOXS`cˆ]IJb ]HmHJ]FjH€M”Y\bdW[TSCKBIL]&I [C] 1KBFpPRHJSCKBFXFX]&b ]&FjH[T]IL]&FpMN]&b ]&FjHKBOž[fY ILW&cQKBFpeL]

M”OpFœeL~NeHJkbd]/]&F b YqaNSTbdSTeJY\FjHdKUO bdSTFXSTb SfeJY\FjHdMX]&e 1KUFpP>HLSTKUFpedYBeLeLKNP>STW]&e&A…7]ueHMNKBFpP€OpF

KUONHJSC[=]ueLeL]FjHJSC]&[”cQKBOXIs[TY bdKNMNW[TSfeLY\HLSTKUF=A

v3]&e ' $ 1'$ ' emKBFjH†MN]!1Y\bdST[CSfY\IJSfem]&I+[T]7[T]&P>HL]&OXIsY‹•U]&Pl[C]uecXIJSCFpPSCcQY\ONa bdK )

MNk[T]&e

ijOXS emKBFBH†emKBOX•U]&FBH1MX]&eWuijOpYqHJSCKBFpe1YUONadMXWIJSC•BW]&ecpY\ILHLST][T[C]ue U[T]OXIZebdWHLrXKNMN]ue1MN]7ILWuemK )

[CONHJSCKBF/F{OXbdW&ILSfijOX]7]H`[C]&OXIKUcNHJSCbdSfeLY\HLSTKUF=A,†ST]F$e 0XI N[‘YUb9˜XSCHLSTKUF"MN]^P>]^P>KBOXIJe†]ueH`MN]^MNKBFXFX]I

[C]ue4˜pYUeL]&e4ijOXS cˆ]IJb ]HmHJILKBFBH^Y\OXa1ONHLOpIJe4STFXVUW&FXST]OXIZe4MN]d˜XOXIJ]&Y\O M”W>HJOpMN]&e4KBO–MX] IJ]&PZrp]IZPZrX]

]>H MXW•U]&[CKBcXcˆ]bd]FjH%MN]"P>IJW]&I9MN] ' ' & ' ]H MN] $' %' % (" & '

' & ( ' ' cˆKUOXI1MX]&e+cXILKB˜X[Tkbd]&e cp[COpe+P>KBb cp[CSfijOXW&e1FXKUFMNSfeJP>ONHJW&eSfP>S‘A{…+]&cQ]&FpMXYUFBH Bb 2&b ]

P>]OXa’ijOXS†FX]"em]"MN]&emHLSTFX]FjHdcpYUe OXFp]€HJ][T[C]"PY\IJIJSCk&IL] KBFBH SCFjHJWI32>H š˜XST]F PKUbdcXIJ]FpMNIJ][T]&e

]F7]&ONa€MN]0[TY emSTb9Op[TY\HLSTKUF€F{OXbdW&ILSfijOX]UA 4F"]>‡ ]>HNMN]4FXKBb9˜XIJ]OQem]ue1MNWuP>SfemSTKUFpe+STFpMNOpemHLIJSC]&[C[T]&e†KBO

cQKB[CSCHLSfijOX]&e+em]`cpIL]&FXFX]FjH+MXW&eLKUIJbdYUSTe1emOXI+[TY 1KUSQMX]7PYU[TPOX[fe KUOMN]0emSTb9Op[TY\HLSTKUFpe F{OXbdW&ILSfijOX]&e&AUDG[

SCbdcˆKUILHL]`MNKBFpPsijOX]s[C]ueMNWuP>SfMN]OXIZe ]&emH1[fY4PY\cQYUP>SCHLW`MX]#7OXVU]&I1MN]s[fY ( '$% ]>H1MX]s[TY #

MN]&esP&Y\[fP>OX[feiBOpSˆ[T]OpI`emKBFBHcXIJW&eL]FjHLWueAp…+]^P>KBOXIZe1[T]OXIscˆ]IJbd]>HLHLIZY%MN]^PKUFXFpYTHJIL]7[C]uecXIJ]bdSC]&IJe

P>IJSHJkIJ]&eijOXS=VBYUIJYUFjHLSfeLeL]FjH+[fYd•qY\[TSTMXSHJW0]H`[TY cˆ]ILHLSTFX]FQP>]MN]ue`emSTb9Op[TY\HLSTKUFpe†F{OXbdWIJSTijOX]ueA

v3] cX[fY\F MN] P] P>KUOpIJe]ueH [T] eLOXSC•qYUFBHuA7¥lcXILkue$OXF cXIJ]bdST]IšPZrpYUcXSCHLIJ] M=STFjHLIJK{MXOpPRHJSCKBF

Y\ONaœcXILSTFpPSCcpYUONažbdKNMNk&[C]ue+LP>[fYUeJemSfijOX]&e3,"]H [C]&OXIIJW&eLKU[TONHLSTKUFŠF{OXbdWIJSfiBOp] [T]–…+rpYUcXSHJIL]

]&emH PKUFpeJYUPILW [‘W>HLOQMN]MN]$[TY bdW>HLrpK{MX]"F{OXbdW&ILSfijOX]"MX]&e $ ' $ ' A1…+]&edMN]&ONa

cXIL]&bdSC]&IJe4PZrpYUcXSHJIL]uelcˆ]IJbd]>HLHL]FjH9M”Y\[T[T]I0HLIJk&e0•{SHJ]d•U]IZe0MN]&e^ijOX]ueHJSCKBFpe0FjOpb W&ILSfijOX]&e0]ueLeL]F )

HLST][T[C]ueijOXS bdK\HLST•U]&FjH4[T]&e9MNW•B][TKUcXcˆ]bd]FjHZelHJrXWKBILSfijOX]&e^ijOXS+emOXST•{ILKBFjH&A”v=]ue9…+rpY\cXSCHLIJ]&e ”x

]>H7| emKBFjHsPKUFpeJYUPILWue d[fY ILWuemKB[COXHLSTKUFHJrXWKBILSfijOX]4cpY\I !!( " %%$ MN]4bdK )

MNk[T]&e+eHZYqHLSTKUFpFpY\STIL]ue

STFpMNW&cQ]&FpMXY\FjHZe+MNOdHJ]bdcpe RAjDG[Te1cQKjem]&FjH+Y\OpeJeLSX[C]ue˜pYBem]ueM”OXFX]7b WHLrXKNMN]

FjOpb W&ILSfijOX] HLIJk&e^STbdcQKBImHZY\FjHL]=MNSCHL]MN]ue & ' ' 3ijOXS]ueH^cXIJW&eL]FjHJW]d]F’MNWHJY\ST[1YUO

…+rpY\cXSCHLIJ] XAv Y bdW>HLrpK{MX] MN]&e W&[CW&bd]FjHJed‰QFXSTe€]&emH –[TY ˜pYBem] MX]FXKUb%˜XIL]&ONa [TKUVBSTPSC]&[Te MN]

PY\[fP>Op[TelSCFpMXOpeHJILST][fe`KBO YUP&YUMNW&b SfijOX]&e&Apv3]&e0…+rQY\cXSCHLIJ]&e4ƒ ]>H7“cQKBImHJ]FjH0eLOXIl[fY IJW&eLKU[TONHLSTKUFMN]

!$ & ' "'%$% '

KUO M”W&•UKB[CONHJSCKBF]FHJ]bdcpe QHJYUFBH4MNO cˆKUSTFjH0MN]%•jOp]^HLrXW&K )

ILSfijOX] iBOp] FjOpb W&ILSfijOX]UA3yNS[C]ue“"cXIJ]bdSC]&IJe^PZrQY\cXSCHLIJ]&e^eLKUFjHMNW&MXSCWue$[¢Y\FpYU[C~NeL]%F{OXbdW&ILSfijOX]

[C]ue MN]&ILFpSC]&IJeHJIJYUSHJ]FjH !$ & ' ANv3] …+rpY\cpSHJIL]9ndcpILWuem]&FBHJ]OXFX]%emW&ILST]M”]>aN]bdcX[T]&e

P>KUFQP>IJ]>HJeMN]^cpILKB˜X[Ck&bd]&eM”KUcNHJSCbdSfeLY\HLSTKUF"]>HlMNKBFXFX]4OXFX]4HJrXWKBILST]^M”]aNSTemHL]&FpP>]^MN]eLKU[TONHLSTKUFQe

P>]&ecXILKB˜X[Tkbd]&e&A\v=]0…+rQY\cXSCHLIJ]^@&t4MNW&ILST•U]s[T]&ePKUFpMXSHJSCKBFpe

FpW&P>]ueLeJY\STIJ]&e KUO eLO 8eLYUFBHJ]&e M”KUcNHJS)

bdYU[CSCHLW^MN]uesemKB[COXHLSTKUFpe&Ap…+]&eP>KBFpMNSCHLSTKUFpeseLKUFjHSCbdcˆKUILHJY\FjHJ]&e+HJYUFBH`MNO"cˆKUSTFBH`MN]4•{OX]0HLrpWKUIJSfiBOp]

iBOp]/F{OXbdWIJSfiBOp]UA4[C[T]&e cˆ]IJbd]>HmHJ]FjH MN]"P&Y\IZYUP>HLWIJSfem]&I[C]ue9KBcNHLSTb Y ]>H ][T[C]uedemKBFBH [fYš˜QYUeL]

MN]&e4Y\[TVUKBILSCHLrpb ]uelF{OXbdW&ILSfijOX]&e4ijOX] FXKUOpe4MNWuP>IJSC•BKUFpe&A ¤ STFpYU[C]&b ]&FjH”[T]…+rQY\cXSCHLIJ]/@B@9]ueH^OXFX]

SCFjHLIJKNMNOpP>HLSTKUF [fY ("$("$ ! Ap¥lcXILkuelY‹•BKUSTI`W>HJOpMNSTW%[fY cXIJKUVBIJYUb b Y\HLSTKUF

(13)

Ecole Polytechnique. Ne pas reproduire.

[TSTFXW&YUSCIJ] BFXKUOpe†MNKBFXFXKUFQeOXF"Y\cˆ]I>OMN]ue1bdW>HJrXKNMN]&eMN]7[¢KUcXHLSTb SfeJYqHLSTKUF/P>KUb%˜XSTFpYqHJKUSTIL]

P\]&emH )

)MXSCIJ]7MN]0[‘KBcNHLSTbdSTeJYqHJSCKBF]F/•‹YUILSfY\˜p[C]ue+MNSfeJP>IJk>HL]ue 1ijOXS ]&emH†]ueLeL]FjHLST][T[T]lcˆKUOXI[fY9cX[fY\FpS‰QP&YqHJSCKBF

KBcNHLSTb Y\[T] MN]&e€IL]ueLeLKUOpIJP]&ed]>H"MN]&eHUPZrp]&eMXYUFpedHJKUONHJ]&e[T]&e€VUIZY\FpMX]&ed]FjHJIL]&cXILSfeL]&e&A+…+rpYUijOX]

PZrQY\cXSCHLIJ]lP>KBbdb ]&FpP>]7cpY\I†OXFX]7SCFjHJILKNMNOpP>HLSTKUF€ijOXSˆ]F"MNKUFpFX]l[T]lcX[fY\F€]>H[C]ue1SfMNW]ue1cpILSTFpP>STcpYU[C]ueA

v1W&cpY\SfeLeL]OpI MN]†P>]+PKUOXIZe=Fp]+MNKBSH cpYUe SCFQiBOpSCWHL]I [T]1[T]&PRHJ]OXI/%‹]FcX[TOpe MN]&e cˆKUSTFBHZe3]ueLeL]FjHJSC]&[Te

ijOXS eL]IJKUFjH`HJIJYUSHJW&e0MXY\Fpe7[T] PKUOXIZe`KBIJYU[ˆ[C] P>KBOXIJe7W&PILSCH0PKUFjHLST]FjH4MN]%FpKUb9˜pIL]&ONa MNW•B][TKUcXcˆ])

bd]&FBHZeP>KBbdcX[CW&bd]FjHJYUSCIJ]&e ijOXSXcˆ]IJbd]>HLHL]FjH1YUO [T]&P>HL]&OXI1P>OpILST]ONa +mM=YU[C[T]I OXFdcˆ]O cX[TOpe [TKUSTF ,s]>H

MX]/1YUSCIJ]0[C]4[TSC]&F"Y‹•B]&P0M”YUONHLIJ]&e†KBOX•{IJYUVU]&e+KUO"M”Y\ONHJIL]uesMNSTeJP>STcX[TSCFp]&e&AjDG[ eY\VBSHsMNKUFQPlcX[TOpesM”OXF

KBOX•{IJYUVU]`MN]0IJW1WIJ]FQP>]7ijOX]0MN]0[TYHLIZY\FpeJP>IJSCcXHLSTKUF ]>aXYBPRHL]0MNO/PKUFjHL]&F{O€MN]&e†P>KBOXIZe1b Y\VBSTemHLIZY\OXa A

Œ KUOpI4HL]IJbdSCFp]IP>]HmHJ] STFjHLIJKNMNOpPRHJSCKBF FpKUOpe4MNKBFXFXKUFQe^iBOp][fiBOp]&e0IL]&FpeL]STVUFX]&b ]&FjHJe4M”KUIZMNIL]

cpIJY\HLSfiBOp]UA&¡0Y\FQe3[fYsbd]&eLOXIJ]1MNOcˆKBeJemST˜X[T]1P>]†P>KBOXIJe3e]&emH •BKUOX[TO +mYUONHLK ) PKUFjHL]&F{O , cˆKUOXI W•{SCHL]&I MN]

HJILKBc1IJW&ijOX]FjHZeIJ]F{•UKBSTe M”Y\ONHJIL]ue1KBOX•{IJYUVU]&e&Aj…+]&[TY9]&emH+cpYUImHJSTPOX[CSTkIJ]bd]&FBH†em]&FpemST˜X[T]lcˆKUOXIMN]

FpKUb9˜pIL]&ONa%IJW&eLOX[CHJYqHZeM”Y\FpYU[C~NeL]`ijOXSpFX]7emKBFBHSfP>SˆijOX]lMN]ueKUOXHLST[TeHL]&PZrpFXSTijOX]ueONHJSC[T]&eUb YUSTe cQYUe

]ueLeL]FjHJSC]&[Te&AUv=]ue WFpKUFpP]IeJY\FpeMXWbdKUFpemHLIZYqHJSCKBFdIL]&•jST]FQMNIJYUSH 4[T]&e OXHLST[CSfem]&I1]&F +m˜ˆKUSCHL]`FXKBSCIJ],sP>]

ijOXSB[C]&OXI MNKUFXFp]OXF9YBemcˆ]&P>H+IJ]&P]>HLHL]1MN]+P>OXSfemSTFX] , HJILKBcY\ILHLSC‰QP>ST][‘A&¡0Y\FQe=[fY`bd]&eLOXIJ]MNOcˆKBeJemST˜X[T]

FpKUOpeY‹•BKUFpe MXKUFpPSTFpP>[TOpe [T]OpI+MNW&b KBFpemHLIZYqHLSTKUF qbdYUSTecX[COQe 0HLSCHLIJ]`M”SCF 1KUIJbdY\HLSTKUFd]>H+cQKBOXI[T]&e

+LMNW&b9~NeHJS‰Q]I , ijOX]dcQKBOXI^[¢STFBHJWI32>H^HJrXWKBILSfijOX] MN]&e^YUILVBOXbd]FjHJe0b YqHJrXWb YqHJSTijOX]ueA

"!#$%&%(')%*)+-, './021345'% 4 36)%-.#

'7)+6)+'89:$-:' ;'</=>'@?Nv=]0[C]uPRHL]&OXIscˆKUOXIJIJY%MNKUFQP0PKUFpeLSfMNWIJ]IP>]ue+cQYUeJeLYUVU]&e+]F

cˆ]>HJSHZe+P&Y\IZYUP>HLkIJ]&e1P>KUbdbd]/+mrXKUIZe cXIJKUVBIJYUbdb ] ,ZA

G (G F ,#DDH 2 >A GDH (G

GD G.F2D*BDCH<F < 2>>CD3E%FGE"H#EJILK#MK"E+FF6NPO%QREJOSK+TJILT3K"QU/ILE+FFVTO#FGFVW/I>MX

K+N2YZYGERK"E/[%H#\K#M^]

v3Y0‰pFM=OXFp]lMNWbdKBFpeHJIJY\HLSTKUF ]&emH1STFpMNSfiBOpW]scpYUI[C]7PY\IZYUP>HLk&IL]

\HJYUFpMNSfeijOX]

[fY ‰QFšM”OXFX]9IJ]b YUIJijOX]KBOšM”OXF ]a{]&bdcX[C]%]&emH7STFpMNSfijOXW]%cpY\I0[C] PYUIJYBPRHLk&IL]

AQ_lFšSTFpMN]>a]ueH MXSTeLcQKBFXST˜X[C] [fY%‰pFMN]^[¢KUOp•jIZY\VB]UA

v3]&e P>KBILIJSCVBW&e MN]&e ]>aN]IZP>SfP>]ue eL]IJKUFjH–cXILKNPZrpYUSCFp]bd]FjH cpOX˜X[TSCWueA^v3YcX[TOXcpY\ILH’MN]&e–cXIJK )

VBIJYUbdb ]ue`STF 1KUIJb YqHJSTijOX]uelijOXS b ]HmHJ]FjH0]&F K{]OX•{IJ]9[T]&e7bdW>HLrpK{MX]&e7FjOpb W&ILSfijOX]&elW>HJOpMNSTW]&e ]>H

ijOXS3KUFjHscQ]&ILbdSfesMN]^ILWuY\[TSTeL]I[T]&e†‰pVBOXIJ]&esMN]^P>]H`KUOX•{IZY\VB] NemKBFjHsMXSTeLcQKBFXST˜X[C]uesemOpIs[C]eLSHJ]

]&˜

_$``

baVcc3dddde@d/f

`

2e

_

‘#gh‹iJjPcPkh00#jc$e Jhjl/m2l ~n‹

_$`

‘

Ko€[T]4[C]uPRHL]&OXIscˆKUOXIJIJY9[C]ue†HLW&[CWuPZrpY\IJVU]&I+[TSC˜pIL]&b ]&FjH&ANv=]ueseLPZrXW&b YUe+FjOpb W&ILSfijOX]&e†]&F"MNSC‡ WIJ]FpP]&e

‰QFXSC]ue Y\STFpeLS ijOX]%[TY bdWHLrXKNMN] MN]&e7W[TWbd]FjHZes‰pFXSfe7]FšMNSTbd]FpeLSTKUF OXF QKBFBH7W>HLW%cXIJKUVUIZY\bdbdW&e

MpY\Fpe7[C]%[TYUFXVBYUVU]MNOš[CKBVUSfP>ST][ yXP>ST[TYU˜šMNW•B][TKUcXcˆW9cQY\Il[‘D—lsD¥-]>H0[¢41—`Œ… MXSTeLcQKBFXST˜X[C] VUIZY )

HJOXSCHL]bd]&FBHleLOXIs[T]emSCHL]

_$``

baVcc3dddd e0 ‰Jj

HZY\FpMXSTe4ijOX]d[C]ue0ILWuemOX[CHJY\HJe0MN]d[TY/bdW>HJrXKNMN] MN]&e0W[TWbd]&FBHZe7‰pFXSfe4]F MXSCbd]FQemSTKUF–MN]ONa KUFjH0WHLW

KB˜NHL]&F{Ope€[‘Y\SfMN] MNO [TKUVBSTPSC]&[¤XIJ]]u¤X]bqp p›MXW•U]&[CKBcXcˆW%cpYUI¤†A `]&PZrjH^]>H %A”ŒSTIJKUFXFX]uY\O–]>H

YUOpeJemS=MNSfeLcQKBFXSC˜p[C]4VUIZYqHJOXSCHL]bd]&FBH`eLOXIs[T]emSCHL]

_$``

baVcc3ddddiji\ j0

Œ YUI YUSC[T[T]OXIZe u[fYlcX[TOXcpYUImHMN]ue ‰pVUOXIJ]&e ˜XSfMNSTbd]FpeLSCKBFXFX]&[C[T]&e ]>H [fY`HJK\HZY\[TSHJW†MN]ue ‰pVUOXIJ]&e HJILSfMNSTb ]&F )

eLSTKUFXFX]&[C[T]&elKUFjH7WHLW%HLIZYUPW]&e [‘YUSTMX]%MNO [CKBVUSfP>ST][3VBIJYUcXrXSfijOX]^aXM UMšMNW•B][TKUcXcˆW9cpYUI7¤pIJYUF>KBSTe

0BKUOp•U] %[‘o1P>KB[C]^Œ KU[T~jHL]uPZrXFXSfiBOp]0]HlY\OpeJeLS=MNSTeLcˆKUFXST˜X[T]0VBIJY\HLOXSCHL]&bd]FjH`emOXIs[T]eLSHJ]

]&˜

_$``

baVcc3dddde@d/f

`

2e

_

‘#gh‹iJjPcPk3r0/h$s$cJtuvu

DGFQMNSTijOXKBFpeOpFX]7YUONHLIJ]lYUMXIL]ueLeL]

]&˜cˆKUOXI+[C]7[C]uPRHL]&OXI+POXIJSC]&ONadM=]&F€eLY‹•BKUSTIcX[TOpe1eLOXI†[¢rXSfeHJKUSTIL]

MX]&eb YqHJrXWb YqHJSTijOX]uesKUO"[fY%•{ST]^MN]P>]&ImHZY\STFpeb YqHLrpWb YqHJSTPSC]&FpesP>SCHLWue`MXY\FQesP>]^P>KBOXIJe

_$``

baVcc3ddd‘

_

0

`

jf (e0c `

h$u‰$eDhw2c

_

`

Jjf

(14)

Ecole Polytechnique. Ne pas reproduire.

v=]0[C]uPRHL]&OXIiBOpSˆ•BKUOpMXIJYUSHem]lHL]FpSCIsY\O"P>KBOXIJYUFjH+MN]&e†cXIJKUVBILkue]HMN]ue+Y‹•qYUFpP>W&]&e†MN]&e†b YqHJrXWb Y )

HLSfiBOp]&eY\cXcp[CSfijOXW]ue+cˆ]OXHsPKUFpeLOX[CHL]&I†Y‹•B]&P`˜ˆWFpW>‰QP]7[T]^emSCHL]4MN]0[TYdy{KNP>STW>HJW0MN] + Y\HLrXW&bdY\HLSfijOX]&e

¥`cXcX[TSfiBOpW]&e`]>HsDGFpMNOpemHLIJST][T[C]ue

_``

baVcc@c‰

`_

*ij

KUO/P][TOXS”MN]4eLY P>KBFpemK{]&OXI†YUbdWIJSTP&Y\STFX] BHLrX]^y{KNP>ST]>H~ 1KBIDGFpMNOpemHLIJSTYU[”Y\FpM/¥`cXcX[TST]&M+ YqHJrX]b Y )

HLSfPe

_``

baVcc/dddd hS\‰j0

v3]/FpSC•B]&Y\OžMN]/P]"PKUOXIZe]ueHdSCFjHJILKNMNOpP>HLS1`]>HdSC[F=]>aNSTVU]"Y\OpPOXF Y\OXHLIJ]€cXIJWIJ]&ijOXSfe9ijOX]"[C]

FXSC•B]&YUO"MN]^PKUFXFpYUSTeJeJY\FpP]&eYUP&iBOpSTe†]FP[TYBeLeL]&e†cXIJWcpYUIJY\HLKBSCIJ]&e+KUO"]F"cXIJ]bdSC]&I`P>~NP>[T]4OXFXST•U]&IJeLS)

HJY\STIJ]UANs]&PKUFXFQY\SfeLeLKUFpe+ijO=ST[ ]&emHMXS8P>ST[T]4MX]1YUSCIJ]7cpIL]&OX•U]0MN]4˜Q]uY\OpPKUOXc€M=KBILSTVUSTFpY\[TSCHLW7eLOXIsP>]

emO7]H†MNW7 ^˜XST]FP[TYBeLeLSTijOX]`MpY\Fpe [fY^[TSHLHLW&IJY\HLOXIJ]UA 4FcpYUImHJSTPOX[CST]I \FXKUHLIJ]`P>KBOXIJe1MNKUSCH1˜ˆ]&Y\OQP>KUOpc

P]&e†cXIJW&MNWuP>]ueLeL]OXIZe+]>H`FXKUHJY\bdbd]FjH`Y\OXa€P>KUOpIJeMN]-,0AXv3YUILIJKUONHJOXIJKUO jŒ A)GvAXv3SCKBFpe N]HsŒ A)G¥A

`Y‹•{STYUImH4Y\ONaXijOX][fe7SC[1Y\SCH4cpY\I1KUSfe7MX] [fY\IJVU]ue`]&b cpILOXFjHZeA”vY\OXHL]OpI7IJ]bd]IZP>ST]HJKUOpe4P]ONašijOXS

KUFjHIL]&[COdP>]&ImHZY\STFX]&e cQY\ILHLST]&e MNOdb Y\F{OpeJP>IJSH \FXKUHJY\bdbd]FjH¤XIJW&MXWIJSTP ,†KUFpFpY\Fpe,†ILOXFpK¡l]uemcXIJW&e

]>H,†]ILHLIZY\FpM + Y\OpIL~BA\_`FX]sbd]FjHJSCKBFdeLcˆW&P>SfY\[T]]&emH1MNOX] yjHJWcXrpYUFX]%4Y\OX˜ˆ]ILHBiBOpSpY0cpY\ILHLSfP>STcQW

^[TY9ILWuMXYUP>HLSTKUFMNO$…+rpY\cXSCHLIJ]9@B@ jY\STFpemSˆijO= 7[CST•{SC]&I1Œ Y\FjH3. BijOXS YIJ][TO [‘STFjHLWVBIJYU[CSCHLW7MNOb Y )

FjOQeLPILSCH7Y‹•U]uP4˜Q]uY\OpPKUOXc M=Y\HmHL]&FjHLSTKUF]>H4ijOXS Y•UWIJSC‰pW^[C]ue`]a{]&IJPSTP]&el]>HlIJW&MNSTVUW9[C]&OXI7PKUIJILSTVUWBA

vY\ONHJ]OXI0IL]&bd]IZP>ST] /[¢Y‹•qY\FpP]HLKBOpe4P>]&ONašijOXS •UKUOQMNILKBFjHl˜XST]F–[COXS eLSCVBFpY\[T]I7[C]ue0SCFXW&•{SHZY\˜X[T]&e

]IJIL]&OXIJedKBOŠSCbdcˆ]I 1]&PRHJSCKBFpe€MX] P>]>HLHL] WuMNSCHLSTKUF +cpY\I]aN]bdcX[T]cpY\I€PKUOXIJIJSC]&I W[T]&PRHJILKBFXSfiBOp]

[¢YUMNIJ]&eJeL]

j j‰00#j/f

`

$e

_

3gJh~]ij

% Ap¥l[C[fY\STIJ]

Œ YUILSfe {[T]%ƒ0UOXST[T[C]Ha\tUtj|

(15)

Ecole Polytechnique. Ne pas reproduire.

(16)

Ecole Polytechnique. Ne pas reproduire.

! "

2 ! $$#!% &% ('!)"*

…+] PZrpY\cpSHJIL]^]ueHlOpFX]STFBHJILKNMNOQPRHLSTKUF €MN]ONaYUeLcˆ]&PRHZelMNSfemHLSTFpPRHZe QbdYUSTe`HLIJk&es[TSTW&eˆMN]uesb Y )

HLrXW&bdY\HLSfijOX]&edY\cpcX[CSfijOXW]ue% [TY & ' & "$ & %( ' ]>H [TY ' & '$ '-

& (( '$ A†_`FbdKNMNk&[C] bdY\HLrXW&b YqHLSfijOX]–]&emH/OXFX]–IJ]cXIJW&eL]FjHJY\HLSTKUFŠKUOOXFX]–SCFjHL]&ILcpILWHJYqHJSCKBF

Y\˜pemHLIZY\SCHL]MN]9[TYdIJW&YU[CSCHLW^cXr{~NeLSTijOX]9ijOXS ]&emHlYUP&P>]&eJeLSC˜X[T] [‘YUFpY\[T~Nem]4]H7Y\O P&Y\[fP>OX[‘Apv3Y€emSTb9OX[fY )

HLSTKUF FjOpb W&ILSfijOX]cQ]&ILbd]H1MN]sPY\[fP>Op[C]&I eLOXI KBIJMNSTFpY\HL]OpI [T]&e eLKU[TONHJSCKBFpe MN]`P>]ue b KNMNk&[C]ue q]HMNKUFpP

MN]^emSTb9OX[T]Is[fY ILWuY\[TSHJW4cXr{~{eLSfiBOp]UAX¡0Y\FpesP]4PKUOXIZeN[C]ueb KNMNk&[C]uesijOX]4FXKUOpeWHLOpMXSC]&ILKBFpesem]&ILKBFjH

MN]&esWuijOpYqHJSCKBFpe`Y\OXa"MNW&ILST•UW&]&escpYUImHJSC]&[C[T]&e

KUO]BAM”Ac=AQ]F Y\˜pILW&VUW XP\]&emH ) )GMNSTIL]^MN]ue`W&ijOpY\HLSTKUFpe

MNS‡ WIJ]FjHJSC]&[C[T]&e cX[TOpeLSC]&OXIJe•qYUILSfY\˜X[T]&e

[C]4HJ]bdcpe]H`[¢]&eLcpYUP] NcpYUIs]>aN]bdcX[T] >A

Œ KBOXI4[¢SCFQeHZY\FjH7FXKBOpe7[TYUSTeJemKBFpelMX] PK\HLW OXFHJILKBSTeLSTkbd]%YBemcˆ]&P>H/1KBFpMXY\bd]&FBHZY\[ MN]&e7b YqHJrXW)

bdY\HLSfijOX]&e%Y\cXcX[TSfiBOpW]&e eLY‹•BKUSTI^[¢Y\FQY\[T~{eL]b YqHJrXWb Y\HLSfiBOp]€MN]&e%bdKNMNk[T]&e eLOXI%[C]uiBOp][1FpKUOpe

IL]&•jST]FQMNILKBFpeOpF cˆ]OœcX[COQe%[TKUFpVUOX]&b ]&FjH MXYUFpe9[T]&edPZrpYUcXSCHLIJ]&e emOXST•qY\FjHJe&A4FžcpIJY\HLSfiBOQY\FjH MN]

[TY eLKUILHL]jFpKUOpes•BKUOX[TKUFQe+]&FiBOp][fiBOp]emKBImHJ] NbdK\HJSC•B]Is]>H 7OQeHJS‰p]&IlP>]HmHJ]^FXW&P]&eJeLYUSCIJ]4SCFjHLIJOpeLSCKBF

MN]/[‘Y\FpYU[C~NeL]€b YqHJrXWb YqHJSTijOX]BA —`KUOQe YU[C[TKUFQe9•BKUSTI ]F ]‡ˆ]HijOX]"[C]$PYU[TPOX[F{OXbdWIJSTijOX]"MN]&e

emKB[CONHJSCKBFpeMN]4P>]&e†bdKNMNk[T]&e†cXr{~NeLSTijOX]ue+IJW&eL]IJ•U]lcpY\I1KUSfeMN]&eeLOXILcpILSfem]ue

MNW&eJY\VBILWuY\˜X[T]&e ijOXS”FX]

cQ]&OX•U]&FBH0e]>aNcX[TSTijOX]&I`]>H4eW•{SCHL]&IlijOX]cpYUI`OXFX]%˜QKBFXFX]9PKUbdcXIJWrX]&FpemSTKUFMN]9[C]&OXIJe`cXILKBcXIJSCWHLW&e

bdY\HLrXW&b YqHLSfijOX]&e&A=`YUcXcQ]&[CKBFpe4]FQP>KUIJ]dOXFX] 1KUSfe^[T] PY\IZYUP>HLk&IL] 1KUFpMpY\bd]FjHJYU[C]&bd]FjH^b9OX[CHLSfMNSfe )

P>STcX[CSTFpYUSCIJ]šMN]ueb YqHJrXWb YqHJSTijOX]ue€YUcXcX[TSTijOXW&]&e]>H$MNKUFpPšMN] [fY emSTb9Op[TY\HLSTKUFgF{OXbdWIJSTijOX]+ijOXS

b 2&[C]&FjH`bdY\HLrXW&b YqHLSfijOX]&eXPYU[TPOX[”STF 1KUIJb YqHLSfijOX] N]HleLPSC]&FpP>]uesMN]^[‘STFXVUW&FXST]OXIuA

,†ST]FŠijOX]$[TY cX[TOXcpY\ILHMN]&edcpILKB˜X[Ck&bd]&e ]>HMN]ue YUcXcX[TSTP&YqHJSCKBFpediBOpSsb KUHLST•U]&FBH [T]&e b YqHJrXW)

@

(17)

Ecole Polytechnique. Ne pas reproduire.

b Y\HLSfiBOp]&e Y\cXcX[TSfiBOpW]&edeLKUFjH 1KBFpMXYUb ]&FjHJY\[T]bd]&FBH

$ 3$% '

•BKUSTI%cpYUI%]aN]bdcX[T]T@

\n =FXKUOQe7FpKUOpe4IJ]&emHLIJ]STVUFXKBFpe0MXY\Fpe^P]>H^KUOp•jIZY\VB]%YUONašcXIJKU˜X[Tkbd]ue7[TSTFXW&YUSCIJ]&e0cpYUI4eLKUOQP>SMN]

eLSTb cp[CSfP>SCHLWBA ¡7][fYb 2bd]1Y KUF FXKUOQeF=]F{•{STeJY\VB]KBFpe^ijOX]€MN]uecXIJKU˜X[Tkbd]ueMNW>HJ]IJbdSCFXSfemHL]&e

PU]ueH ) )MXSCIJ]+eJY\Fpe STFjHLIJKNMNOpPRHJSCKBF%M”Y\[TW&Y\HLKBSCIJ]+KBO MX]emHLKNPZrpYBeHJSTijOX]BA 4FN‰pF BP>]sPZrpY\cpSHJIL]em]•BKUO )

[fY\FjH4SCFjHJILKNMNOpP>HLS11]>H^YqHLHLIZYUP>HLS1 ˆFpKUOpe0IJ]&emHL]IJKUFQeleLKUOX•B]FjH7OXF cQ]&O *QKUOšMXYUFpe0[¢Y\IJVUOpb ]&FjHJY\STIJ]

b Y\HLrXW&bdY\HLSfijOX]^cQKBOXIsFX]^cpYBe`Y\[TKUOpIJMNSTISCF{ONHJSC[T]bd]&FBH`[‘]>aNcQKjemWBAˆh7OX]^[T]4[C]uPRHL]&OXI`IJSCVBKUOXIJ]OXa€em]

IZYUeJeLOXIL] % FXKUOpe%IJ]cXIJ]FpMXILKBFpeHJKUOpe [T]&edP>KBFpP>]&cNHJe STFjHLIJK{MXOXSHZedMN]/b YUFXSCk&IL]/cX[TOpedcXIJW&P>SfeL]/Y\O

cpILKNPZrpYUSCF"PZrpY\cXSCHLIJ]UA

v3] cp[TYUF MN] P>]dPZrpYUcXSHJIL] ]&emH^[C]deLOXSC•qYUFBHuA=v3Y$y{]&P>HLSTKUFž@UA]ueH^P>KBFpeLYBP>IJW] €OXF–]>aN]&b cp[C]

W&[CW&bd]FjHJYUSCIJ]7MN]7bdK{MXW[TSTeJYqHJSCKBF/ijOXS P>KUFQMNOXSCH / ( ' $ "%'$ AXv3Y y{]&P>HLSTKUFš@BA

]ueH OXFX]/IL]&•jOp]IJYUcXSfMN]€MN]ue9cXIJSTFpP>STcpY\[T]&e9W&ijOpY\HLSTKUFpe%Y\ONa’MNWIJST•UW]uecpYUImHJSC]&[C[T]&e%ijOX]€[‘KBF IJ]F )

PKUFjHLIJ]/MpY\Fpe [T]&e bdK{MXk[T]&edOpeLOX][fe ]Fžb WuPYUFXSTijOX] cXr{~{eLSfiBOp] KUOœeLPSC]&FpP>]ue MX]/[‘STFXVBWFXST]OXIuA

v Y€y{]uPRHLSTKUF’@BAx ]ueH7OXFX]STFjHLIJK{MXOpPRHJSCKBF YUeJem].^SCF 1KBILbd]&[C[T]9Y\O P&Y\[fP>OX[ F{OXbdWIJSTijOX]^]H d[fY bdW)

HJrXKNMN] MN]&e

$ '

$ '

A$4FN‰QF ”FXKUOQe0MXKUFXFXKBFpe4MXY\FQe7[fY$y{]&P>HLSTKUFœ@UA}| [fY"MXW>‰pFXSCHLSTKUF

M=OXF

! #

&

&#!'

Y\STFpeLS3ijO=OpFX]P>[fYUeJemSC‰QP&YqHLSTKUF

eLKUbdb Y\STIJ] MN]uesW&ijOpY\HLSTKUFpesYUONa

MXWIJSC•BW]&e†cpYUImHJSC]&[C[T]&e&A

* *&) * &* % ) ! '."!#$

v Y’bdK{MXW[TSTeJYqHJSCKBFœIJ]cpILWuem]&FBHJ]OXFX] cQY\ILH€P>KUFQemSfMNWIZY\˜p[C] MNOŠHJIJY‹•qYUSC[lMXOgbdY\HLrXW&b YqHLSfP>ST]F

YUcXcX[TSTijOXW+]>H FXW&P]&eJemSCHL]OpFX]†PKUFXFQY\SfeLeJY\FpP]Y\cXcpILK 1KUFpMNST] &FXKUF%em]&OX[C]&bd]FjH MN]ue3b YqHJrXWb Y\HLSfiBOp]&e

YUcXcX[TSTijOXW&]&e b Y\Sfe YUOpeJemS`MN]$[fY MNSTeJP>STcX[TSCFp]$eJP>ST]FjHJS‰QijOX] –[fYUijOX][T[T]/]&[C[T]&e e&YUcXcX[TSTijOX]&FjH&A 4F

]‡ˆ]HXMXYUFpesMN]4FXKBb9˜XIJ]ONa€P&YUe†[T]0bdKNMNk&[C]0b YqHLrpWb YqHJSTijOX]4F=]&emHscpYUe†]FQP>KUIJ]0W>HJYU˜X[TSNKBO/˜XST]F

ST[/1 Y\ONH$]FeLW[T]&P>HLSTKUFXFp]I€OpFcˆ]ILHLSTFX]FjH/cQY\IJb S0cX[TOpeLSC]&OXIJe/MNSfemcˆKUFXST˜X[T]&e+KBO ]FQP>KUIJ] ST[!1Y\ONH

eLSTb cp[CSC‰p]I MN]ue b KNMNk&[C]uedP>KUFpFjOQe%b YUSTe%HLIJKUcœP>KBb cp[C]a{]ueA —`WuY\FXbdKBSCFpe SC[sFp]/FXKBOpe ]&emH cQYUe

cˆKBeJeLSC˜X[T]`MXYUFpe OXFX]scpIL]&b STkIJ]scXIJW&eL]FjHJY\HLSTKUFdMN]`[fY^MNSfeLPSCcX[TSTFX]`MN]`IJ]FpMXIL]sP>KBbdcNHL]lY‹•B]&P 7OpemHLSfP>]

MX] [‘STbdcQKBImHZY\FpP]–MX]–P]>HmHJ]’MNWb Y\IZPZrX]–MN] b KNMNW&[CSfeJYqHLSTKUF %sSC[!1Y\OXH$˜pSC]&FP>KBbdb ]&FpP>]&I$cpY\I

YUcXcXIJ]FpMXIL][T]&eFpK\HLSTKUFQeMN]`˜pYBem]scpILKBcXIL]ue Y\ONa b YqHLrpWb YqHJSTijOX]ueY\cXcp[CSfijOXW]ue"!j…7]&emH1cˆKUOXIZijOXKUS

FpKUOpe FpKUOpe [TSTb SCHLKBFpe ’MNWuP>IJSCIJ]OXFg]a{]&bdcX[C] MX] MNWIJSC•qY\HLSTKUFgM”OXFgbdK{MXk[T] cXr{~NemSfijOX]"HLIJk&e

P[TYBeLeLSfiBOp] ]>H%FpKUOpe IJ]F{•UKq~BKUFpe4[T]/[T]&P>HL]&OXI MXW&eLSCIJ]ONa M”]FœeLY‹•BKUSTIcX[TOpe šMN]ue KUOX•{IZY\VU]ue^KUO

PKUOXIZe†cX[TOpe`eLcQWuP>SfY\[TSTeLW&e&A

v3]0bdKNMNk&[C]^ijOX]4FXKUOQesY\[T[CKBFpeMNW&PILSTIL]0]&emH`P>KBFXF{O/eLKUOpe†[T]0FpKUb ( ' ("*

' NKUOM”W&ijOpY\HLSTKUF$MN]^MNSC‡ˆOpeLSTKUF=A

…+KBFpemSfMNW&ILKBFpe1OpF"MNKBb Y\STFX]

MN]4[¢]&eLcpYUP]

N

MXSCbd]FQemSTKUFpe

FXK\HJW

R N

XY‹•U]uP`]F/VUW&FXWIZY\[

N = 1, 2

KBO dijOX]$[¢KUFgeLOXcXcˆKBeL]"KNP&P>OXcˆWcpY\I OpFœb Y\HLWIJSfY\O rpKUbdKUVBkFX] STeLK\HJILKBcQ] ]>H PKUFpMXOpPRHJ]OXIlMX][TY€PZrpY\[T]OXIuA 7FFXKUHL]

x

[TY •qYUILSfY\˜X[T]M”]&eLcpYUP] pPU]ueH ) )MXSCIJ]4OXF cˆKUSTFBH0MN]

]H

t

[TY7•qY\IJSTYU˜X[T]†MN]+HL]&bdcpeA\¡0Y\FQe

[C]ue eLKUOpIJP]&e MN]†PZrQY\[T]OXI

W&•U]&FBHJOX][T[T]bd]FjH FXKUF%OXFXS1KUIJb ]ue

]&F]&eLcpYBP>]]H •qY\IJSTYU˜X[T]&e3MpY\Fpe3[C]1HL]bdcpe emKBFBH IL]&cXILWuem]&FjHLW]ue”cpYUI OXFp] 1KUFQPRHLSTKUF%MNKUFXFpW]

f (x, t)

HZY\FpMXSTe^ijOX]d[fY€HJ]bdcˆWIZYqHLOpIL]d]&emH^OXFX] 1KUFQPRHLSTKUF STFpP>KBFXF{OX]

θ(x, t)

A v3Y$ijOpY\FjHLSCHLW MN] PZrpYU[C]&OXI ]ueHdcXIJKUcˆKUILHLSTKUFXFp][T[C]  [fYHL]&bdcQW&IJY\HLOXIJ]

θ

]>Hd•qY\ONH

Ko

c

]ueHdOXFX]$P>KBFpemHJY\FjHJ]€cXr{~NemSfijOX]

ijOXS0MXWcˆ]FpMMNOgH~{cQ]–MN]šb Y\HLWIJSfY\O YUcXcˆ][TW] PZrQY\[T]OXI"emcˆW&PS‰QijOX]BA+Œ KBOXI/MNWHL]&ILbdSTFX]I/[TY

HJ]bdcˆWIZYqHLOpIL]

θ

jFpKUOpe1W&P>IJST•UKUFQe [TY #$ 1"' %% KUO€MN]7[TY9iBOQY\FjHLSCHLW MX]4PZrQY\[T]OXIuAj¡0Y\FQe+OpF/•BKU[TOXbd]7W&[CW&b ]&FjHJY\STIJ]

V

SCFQP>[TOpesMXY\FQe

N[fY%•‹YUILSfYqHJSCKBF]F/HL]&bdcpesMN]0[TY

(18)

Ecole Polytechnique. Ne pas reproduire.

iBOQY\FjHLSCHLW^MN]PZrpYU[C]&OXI]&emHs[C]4˜XST[TYUFMN]^P>]^ijOXS=]&emHscXIJK{MXOXSH`cQY\I[C]uesemKBOXIZP>]&e†]HlMN]^P>]^ijOXS3eLKUILH

KUO"IL]&FBHJIL]%HLIZY‹•U]&IJe+[T]&escpYUILKBSTe&AN¥`ONHJIL]&bd]FjHlMNSCH

d dt

Z

V

cθ dx

= Z

V

f dx − Z

∂V

q · n ds

@UAT@

KJo

∂V

]ueH+[C]`˜ˆKUIZMdMN]

V

M”W&[CW&bd]FjH+MN]leLOXI 1YUP]

ds

n

]ueH1[fY4FXKBILb Y\[T]]>a{HLW&ILST]OpIL]sOXFpSHJWlMN]

V

j]>H

q

]ueH†[C]7•U]&P>HL]&OXI *pONaMN]0PZrpY\[T]OpI&A{y{SpKBF€Y\cXcp[CSfijOX]`[T]`HLrpWKUIJkbd]lMX]!%4Y\OQeLeUKBF KU˜NHJSC]&FBH

Z

∂V

q · n ds = Z

V

divq dx.

s]VBILKBOXcpY\FjH[T]&e+MNS‡ WIJ]FjHZeHJ]IJb ]ue1MN]

@BAC@ ]>H†ONHLST[CSfeJY\FjH1[T]1Y\SCH†ijOX]l[C]7•UKU[TOXbd]`W&[CW&b ]&FjHJY\STIJ]

V

]ueH`ijOX]&[TPKUFpijOX]jSTFpMNW&cQ]&FpMXYUFBH`MNO/HL]&bdcpe NKBF/]&F"MXW&MNOXSCHs[¢W&ijOpY\HLSTKUF$MN]^P>KBFpem]&IL•qY\HLSTKUF€MN]

[¢WFX]&ILVBSC]

c ∂θ

∂t + divq = f

@UA

iBOpSpY0[CST]Od]F HJKUONHcˆKUSTFjH

x ∈ Ω

]>H 0HJKUONH HJ]bdcpe

t

AUlY\cXcˆ][TKUFpe ijOX]s[‘KUcˆWIZYqHL]&OXI MNST•U]IJVU]&FpP>]

]&emHlMNW>‰pFpS”cpY\I

divq =

N

X

i=1

∂q i

∂x i

Y‹•U]uP

q = (q 1 , ..., q N ) t .

DG[1YUONHdbdYUSCFjHJ]FpYUFBH IL]&[CST]I [T] *pONa MX]"PZrpYU[C]&OXI [fYHL]&bdcQW&IJY\HLOXIJ] ]>HdKUF 1Y\SCHdYUcXcQ]&[ šP>]

iBO3KBF"YUcXcQ]&[C[T]4OXFX]

1"''$

AX¡7YUFpe[C]^P&YUecXIJW&eL]FjH{ST[3e&YUVUSCHMN]^[fY [CKBS=MN]^¤XKBOXILST]I

iBOpS=IL]&[CST]4[C] *pOXa"MN]PZrpYU[C]&OXIsMN]^b Y\FXSTkIJ]4cXIJKUcˆKUILHLSTKUFXFX]&[C[T]4Y\O"VUIZYUMNST]FjHsMN]4HL]&b cˆWIZYqHJOXIJ]

q = − k ∇ θ,

@UA

KJo

k

]&emH OXFp]†PKUFpemHJYUFBHJ]1cˆKBeLSHJSC•B]

ijOXSNMNWcˆ]FpM MNO9H~jcˆ]MN]+b Y\HLWIJSfY\O3Y\cXcˆ][TW]P>KUFQMNOpPRHJSC•{SCHLW

HLrX]&ILbdSfiBOp]UAX`YUcXcQ]&[CKBFpe`ijOX]4[‘KBcQW&IJY\HL]OpI†VBIJYBMNST]FjHs]&emHsMXW>‰pFXS3cpY\I

∇ θ = ∂θ

∂x 1

, ..., ∂θ

∂x N

t

.

4F PKUb9˜pSCFpYUFjHl[TY€[CKBS MN] P>KBFpem]&IL•qY\HLSTKUF

@BA s]H7[fY[TKUS PKUFpemHLSCHLOXHLST•U]

@UA QKBF KU˜NHJSC]&FjH7OXFX]

W&ijOpYqHJSCKBF/cˆKUOpIs[TY%HL]&bdcQW&IJY\HLOXIJ]

θ c ∂θ

∂t − k∆θ = f,

KJo

∆ = div ∇

]&emH`[¢KUcˆWIZYqHJ]OXI[fY\cX[fYUPSC]&F"MNKBFXFXW4cpYUI

∆θ =

N

X

i=1

2 θ

∂x 2 i .

DG[ 1Y\ONH0Y7KUOXHL]I dP]>HmHJ]W&ijOpY\HLSTKUF$ijOXS=]&emH`•qY\[fY\˜X[T]MXYUFpe†HLKBONHl[T]MNKUb Y\STFX]

pOpFX]^IL]&[TY\HLSTKUF Y\cXcˆ][TW] $ %' & ' ijOXSlSCFpMXSTijOX]šP] ijOXS7eL]cpYUeJeL] [TY 1IJKUFjHLSTkIJ]KUOgYUO

(19)

Ecole Polytechnique. Ne pas reproduire.

n

∂Ω

@UAT@ ]&PRHJ]OXIsFpKUIJbdYU[”OXFXSCHLW^KBILST]FjHLW0•U]&IJe[‘]>a{HLW&ILST]OXIuA

˜ˆKUIZM

∂Ω

MNOžMXKUb Y\STFX] ]>HdOXFX]"Y\ONHJIL]/IL]&[TY\HLSTKUF ijOXSSCFQMNSTijOX]"ijOX][]&emH%[‘W>HJY\H%STFXSCHLSfY\[sMN]€[fY HJ]bdcˆWIZYqHLOpIL]BAQŒ Y\I0P>KBFj•B]FjHLSTKUF pKUF PZrXKUSfemSCH7[‘SCFpemHJYUFjH

t = 0

cˆKUOXI 2>HJIL]%[C]9HL]&b cQe7STFXSCHLSfY\[ ]>H

KBF/STbdcQKjem]^OpFX] 1$ $%

θ(t = 0, x) = θ 0 (x),

@UAx

Ko

θ 0

]&emH^[TY31KUFQPRHLSTKUF–MN] MNSfeHJILST˜XONHJSCKBF STFXSCHLSfY\[T]dMX]%HJ]bdcˆWIZYqHLOpIL]dMXY\FQe7[T] MNKUb YUSCFX]

A 4F

P]šijOXS7PKUFpP]IJFX][fY P>KBFpMNSCHLSTKUF Y\OXaœ[TSCbdSCHL]ue †P>][fY MNW&cQ]&FpM MNO PKUFjHL]a{HL] cXr{~NeLSTijOX]BA+y{S0[C]

MXKUb Y\STFX]š]ueHeLOXcXcˆKBeLWš˜pYUSCVBFX]I$MXY\FQe€OXFHLrX]&ILbdKjeHZYqH HL]&b cˆWIZYqHJOXIJ]–PKUFpemHJYUFBHJ] †Y\[TKUIZe

ijOXSCHmHJ] "b KNMNSC‰p]&I^[¢W&PZrX]&[C[T] MN]&e0HJ]bdcˆWIZYqHLOpIL]ue ”[fY/HL]bdcˆWIZYqHJOXIL]d•BWIJS‰p] [fY"PKUFpMXSHJSCKBF–YUONa

[TSTb SCHL]ue`MN]¡7SCIJSfPZrX[C]H

θ(t, x) = 0

cQKBOXIHLKBONH

x ∈ ∂Ω

]>H

t > 0.

@UA}|

yNS†[T]/MXKUb Y\STFX]€]ueHdemOpcXcQKjemW"YUMNSfY\˜QYqHLSfijOX]€KBO’HLrp]IJb SfijOX]bd]&FBHdSfemKB[CW€MX]/[‘]a{HLWIJST]OXI YU[CKBIJe

[T] *pONašMN] PZrpYU[C]&OXI4emKBImHZY\FjH4YUOš˜ˆKUIZM ]ueH^F{OX[ ]>H^[fYHJ]bdcQW&IJY\HLOXIJ] •UWIJSC‰p]%[fY"P>KUFQMNSHJSCKBF YUONa

[TSTb SCHL]ue`MN]—l]OXb Y\FpF

∂θ

∂n (t, x) ≡ n(x) · ∇ θ(t, x) = 0

cˆKUOXIHJKUONH

x ∈ ∂Ω

]H

t > 0,

@UA

KJo

n

]&emH0[fY€FpKUIJbdYU[C]%]>a{HLW&ILST]OpIL]%OXFXSCHLW MN]

•BKUSTI7[fY/¤ STVUOXIJ]/@BAC@ RA _`FX] eLSHJOpYqHJSCKBFšSTFjHL]I) bdWuMNSTYUSCIJ] cQ]&ONH9Y\OpeJeLS Y‹•BKUSTIl[TSC]&O %”[T] *pONa MX]dPZrQY\[T]OXI4emKBImHZY\FjH^Y\Oš˜ˆKUIZM ]ueH^cXIJKUcˆKUILHLSTKUFpFX][

YUOœeJY\OXH MX]/HJ]bdcˆWIZYqHLOpIL]"]FjHLIJ]$[¢]>a{HJWIJSC]&OXId]>H[‘STFjHLWIJST]OXI ]>H[fY HJ]bdcQW&IJY\HLOXIJ]$•UW&ILSC‰p]"[TY

PKUFpMXSHJSCKBF"YUONa€[TSCbdSCHL]&elMN]¤XKBOXILST]I

∂θ

∂n (t, x) + αθ(t, x) = 0

cˆKUOpIHLKUOXH

x ∈ ∂Ω,

]>H

t > 0

@UA„ƒ

Ko

α

]ueHOXFX]^PKUFpemHJYUFBHJ]7cˆKBeLSCHLST•U]UA{ŒOpSTeJijO=ST[ 1YUONH`PZrXKUSfemSTI

P\]&emHsOXFX]^MN]ueW>HJYUcQ]ueMN]4[TY%bdK )

MXW[TSTeJYqHJSCKBF FXKBOpe7YU[C[TKUFQe7eLW[T]&P>HLSTKUFXFX]&I7[fY"P>KUFQMNSHJSCKBF YUONa[CSTbdSHJ]&e^MN]d¡lSTIJSTPZrX[T]>H

@BA| RA”`YBe)

eL]b%˜X[fY\FjH`]FN‰QF [¢W&ijOpY\HLSTKUF Q[TY€P>KUFQMNSHJSCKBFSCFpSHJSTYU[C]p]>H0[TY PKUFpMXSHJSCKBF Y\ONa$[CSTbdSHJ]&e0eLY\HLSfe 1YUSHJ]&e

(20)

Ecole Polytechnique. Ne pas reproduire.

cpY\Is[fY%HL]bdcˆWIZYqHJOXIL]NKUF"KU˜NHJSC]&FjH`[¢W&ijOpY\HLSTKUF$MN]^[TYdPZrpYU[C]&OXI

 

 

 c ∂θ

∂t − k∆θ = f

cQKBOXI

(x, t) ∈ Ω × R +

θ(t, x) = 0

cQKBOXI

(x, t) ∈ ∂Ω × R + θ(t = 0, x) = θ 0 (x)

cˆKUOXI

x ∈ Ω

@UA“

v=]0cXILKB˜X[Tkbd]

@UA“ +]&emH`MNKUFQP7PKUFpemHLSCHLOXW^M”OXFX]4WuiBOQYqHLSTKUF"Y\ONa€MXWIJSC•BW]&e+cpYUImHJSC]&[C[T]&e†b9OpFXSC]^MN]

P>KUFQMNSHJSCKBFpe Y\OXa9[CSTbdSHJ]&e ]H1M”OpFX]`P>KBFpMNSCHLSTKUF SCFXSCHLSfY\[T]UAB¥œP&Y\OpeL]sMN][TY0cXIJW&eL]FQP>]sMN]sP>KBFpMNSCHLSTKUFpe

Y\ONaš[TSTb SCHL]ue 3KUF MNSH9ijOX]

@UA“ 7]&emHOXF !(# & %' & ' ˆb Y\Sfe4KUF MNSCH9Y\OpeJemSijOX]

P\]&emHOpF ! # & š# %'$(" dPYUOpeL]4MX]^[TYdMNKBFXFXW]4STFXSCHLSfY\[T]^]F/HL]bdcpe&A

&

( ' ™¢

¡0Y\Fpe†P>]7b KNMNk&[C]0MN]lcpILKBcpY\VjYqHLSTKUF MN]7[TY%PZrpY\[T]OpIBSC[ˆFXKUOQe 1YUONH+cXIJW&PSTeL]I

[C]ue9OXFpSHJW&e%KUO MNSTbd]FpeLSTKUFpe%cXr{~NemSfijOX]&e % [fYHJ]bdcˆWIZYqHLOpIL]

θ

]>aNcXIJSTb ]/]F 4]&[C•{STF

K

[TY

PZrpY\[T]OXIeLcˆW&P>SC‰QijOX]

c

]F0BKUOX[T]cpY\I ‚{ST[CKBVUIZY\bdbd]+cpYUI 4][T•jSTF

J/(kg × K)

‹[TY^PKUFpMNOQPRHLST•{SHJW HLrX]&ILbdSfiBOp]

cpYUIOXFXSCHLWMN]€bdYBeLeL]

k

]F 0BKUOX[T] bdk>HJIL]P&Y\IJILW cpY\I9‚jST[TKUVUIZY\bdbd]%cQY\I 4][T•{SCF cpY\I`eL]&PKUFpMN]

Jm 2 /(kg × K × s)

RAQ¡%OXF$cQKBSCFjHlMN]^•{OX]^b YqHJrXWb Y\HLSfiBOp] XFXKBOpesY\[T[TKUFpe†HJILkue emKBOX•U]&FBHsKBOX˜X[TSC]&IlP>]uesOXFXSCHLWue Q]>Hlb 2bd]P]&e`PKUFpemHJYUFBHJ]&eX]FeLOXcXcˆKBeJY\FjHlijOX]

c

]>H

k

•qY\[T]FjH@

P>]&[TY IJ]•{ST]FjH YBMNSTb ]&FpeLSCKBFXFX]Is[T]&eVBIJYUFpMN]&OXIJe†cXr{~NeLSTijOX]ue>A

& ( ' ™™ —`KBOpesY‹•UKBFpe+b ]&FjHLSTKUFXFXW0HJILKBSTe+H~{cQ]uesMN]^P>KBFpMNSCHLSTKUFpeY\OXa€[CSTbdSHJ]&eX¡lSTIJS)

PZrX[C]H{—`]&OXb Y\FXF N¤pKUOXIJSC]&I

bdYUSTe+ST[ˆ]&F]>aNSfeHJ]7M”Y\OXHLIJ]&e ijOXSˆKUFjH+[TSC]&O/eLOXI†[¢SCFjHJWVUIZY\[TSCHLWlMN]7[TY

1ILKBFBHJSCk&IL]

∂Ω

A1,†SC]&Fe0pI\KBFdcQ]&ONH+Y\SfeLWbd]FjHSTbdYUVUSTFX]IMX]&eeLSHJOpYqHJSCKBFpe Ko [T]&e+P>KBFpMNSCHLSTKUFpe YUONa [CSTbdSHJ]&e0emKBFBH0bdW[fY\FXVBW]&e %ˆ¡7STILSfPZrX[T]>H0eLOXI

∂Ω D

ˆ—l]OXb Y\FpFšemOpI

∂Ω N

Q]H0¤XKBOXIJSC]&I7eLOXI

∂Ω F

Y‹•U]&P

∂Ω D , ∂Ω N , ∂Ω F

1KBILb Y\FjHsOXFX]4cpYUImHJSHJSCKBFMN]4[TY 1IJKUFjHLSTkIJ]

∂Ω

A

& ( ' ™fŸ vW&ijOpY\HLSTKUFMN][fY PZrpYU[C]&OXI

@BA“ †]&emH $% Y\O eL]Fpe`KJoeLY eLKU[TONHJSCKBF

θ

MNWcˆ]FpM [CSTFXW&YUSCIJ]bd]&FBH9MX]&e9MXKUFXFXW&]&e

(f, θ 0 )

A4F’cXr{~NemSfijOX]€P>]HmHJ]cXIJKUcXIJSCWHLWMN]€[CSTFXW&YUILSCHLW ]&emH0eLKUOX•B]FjH`HLIZYUMXOXSHJ]9eLKUOpel[TY.1KUIJb ]%M”OpFšcXIJSCFQP>STcQ] MN]%eLOXcˆ]IJcQKjemSCHLSTKUF %ˆOXFX] P>KUb%˜XSTFpY\SfemKBF

[CSTFXW&YUSCIJ]$MN]&edMNKBFXFXW]ue

(f, θ 0 )

PKUFpMXOXSH  OXFX]eLKU[TONHLSTKUF

θ

ijOXSs]&emH [TY b 2bd]"PKUb%˜XSCFQY\SfemKBF [CSTFXW&YUSCIJ]lMN]ue1eLKU[TONHLSTKUFQe1P>KBILIJ]&eLcˆKUFpMXYUFjH 9PZrpYUijOX]HJ]IJb ]0MN]`[fYMNWuP>KUbdcˆKBeLSHJSCKBFMN]&e+MNKUFpFXW]ueA

¡9OXF cQKBSCFjH9MN]d•{OX]dcXr{~NemSfijOX] ”[fY"[CSTFXWuY\IJSHJW F3]ueHijO=OXFX] r{~{cQKUHLrXkuem]dcpYUILbdSM”Y\ONHJIL]ueA$4F

]>‡ ]>HjcˆKUOXI†[C]ue1cpILKB˜X[Ck&bd]&e 1KUILHL]l•qYUILSfYqHJSCKBFdMX]`HL]&bdcQW&IJY\HLOXIJ] j[TY[CKBSˆMN]0¤XKBOXILST]I+]&emH 1Y\OQeLeL]

]>HdSC[ 1Y\OXHd[fYšPKUIJILSTVU]&I9]&FžeLOXcXcˆKBeJY\FjH ijOX]"[TY–P>KBFpMNOpP>HLST•{SHJW€HJrX]IJbdSTijOX]

k

MNW&cQ]&FpM ]&F 1YUSH MN]/[fYHJ]bdcˆWIZYqHLOpIL]

θ

]HdMN]$eLKUF VBIJYBMNSC]&FjH

∇ θ

P]"ijOXSIJ]FpM [C]"cXIJKU˜X[Tkbd]/FXKUF )[TSCFXWuY\STIL] RA 4FpP>KBIL]dcXSTIJ] ”cˆKUOXIMN]ue4cXrXW&FXKUbdkFp]&e4]>a{HJI 2&b ]&bd]FjH^IJYUcXSTMX]&e

]aNcX[CKjemSTKUF ˆcQY\I^]>aN]&b cp[C] 0SC[

]&emH€FXW&P]&eJeLYUSCIJ] M”Y\˜pYUFpMNKBFXFX]I€[C] cXILSTFpPSCcˆ] b 2bd] MX]š[fY’[CKBS7MN] ¤XKUOpILST]I"iBOpS7eLOXcXcˆKBeL] [TY

cXILKBcQKBImHJSCKBFXFpYU[CSCHLWlMXO *pONa€MN]0PZrpY\[T]OpI

q

Y‹•B]&Ps[T]l[T]7VBIJYBMNST]FjH+MN]lHL]bdcˆWIZYqHJOXIL]

∇ θ

A 41F€]>‡ ]>H

P>]>HLHL]r{~{cˆK\HLrpk&eL]

+mFpYqHJOXIJ][T[C] , cpIL]&b STkIJ] •{OX] ^]FjHLIZY FX] OXFX]/cXIJKUcXIJSCWHLW cpYUIJYBMNK‹aXY\[T] % [TY

PZrpY\[T]OXI%em]cpILKBcpY\VB] OXFX]•{SCHL]ueLeL]STFN‰pFXST]/MXYUFpe[T]€MNKBb Y\STFX]

A —lKUOpe9•U]&ILIJKUFpecX[COQe[TKUSTF

•UKBSCI0[fY"`]b Y\IZijOX]€@BAXAn lPKUbdbd]FjH0WHJYU˜X[CSTI^P>]dcpYUIJYBMNK‹aN]UAQ`]>HL]&FXKUFQe7cˆKUOpI0[‘SCFpemHJYUFjH^ijOX]

b KNMNW&[CSfeL]I`P\]&emH1Y\STIL]^MN]uesr{~{cQKUHLrXkuem]ue]>HscXIJW&P>SfeL]Is[T]OXI`MNKBb Y\STFX]^MN]^•qY\[TSTMNSCHLWBACATA

& ( ' ™, v=]%cXIJKU˜X[Tkbd]

@UA“ F=]&emH7cQYUeleL]Op[C]&b ]&FjH7OXF bdKNMNk&[C]9MX]9cXIJKUcpYUVBY\HLSTKUF

MN]$[TY–PZrpY\[T]OpI&A DG[sY–]F 1YUSH OXFŠPYUIJYBPRHJkIJ]€OXFXST•U]&IJeL][ ]>H KUFž[C]$IJ]>HJILKBOX•U]"P>KUbdbd]"b KNMNk&[C]

(21)

Ecole Polytechnique. Ne pas reproduire.

MX]"FpKUb9˜pIL]&ONa’cXrXWFpKUbdkFX]uedeLYUFpedY\OpPOXF IZY\cXcˆKUILHd]FjHLIJ]/]&ONa

ST[1Y\OXH eLSTb cp[C]&b ]&FjH PZrQY\F )

VB]I0[C] FXKBb MN]&e4MNST•U]&IJeL]&el•‹YUILSfY\˜p[C]ue7MNO cXIJKU˜X[Tkbd] >A Œ YUI7]aN]bdcX[T]

@BA“ `]ueH^Y\OpeJeLS PKUFXF{OX]

eLKUOQe4[C]dFXKBb

( '$% $''

]>HbdKNMNW[TSTeL]d[TY$MNSC‡ OpemSTKUF KBO bdSTVUIZYqHLSTKUF M”OXFX]

PKUFpP]FjHLIZYqHJSCKBF KUO€MX]FpeLSHJW HLIZY‹•U]IZe[T]7MXKUb Y\STFX]

SCb YUVUSTFX]I†OXF€cˆKU[T[TOpY\FjH†MXS‡ OpeJY\FjH†MXYUFpe [‘YqHLbdKjemcXrpkIJ] jKUO˜XST]F"OXFX]0]&eLcˆk&P>]4PZrXSTbdSTijOX]7b STVUIZY\FjHMXY\Fpe†OXF"emOp˜peHJIJY\H >A{¡0Y\FQe+P]0P&YUe

θ

]ueH`[fY P>KBFpP>]&FjHLIZYqHLSTKUF"KUO$[fY MN]FQemSCHLW^]&F$ijOX]ueHJSCKBF

q

]&emH`[T]+*pONaMN]^b YUeJem]

k

]ueHl[fY MNSC‡ˆOQemS)

•{SCHLWQ]H

c

]&emHl[fYMN]&FpemSCHLW%•UKB[COXbdSfijOX]MN]9[‘]&eLcQkuP>]UAˆ¡l]%b 2bd] ˆ[TY[CKBS MN]9PKUFpeL]IJ•qYqHJSCKBF

@UA

]ueH`OpF/˜XST[fY\FMN]^b YUeJem]jHJY\FQMNSTeliBOp]0[fY [CKBS3P>KBFpemHLSCHLONHJSC•B]

@UA +]&emHlY\cpcQ]&[CW&]0[TKUS MN]^¤ SfPZ‚ A

& ( '$– f™ # v=]7cXILKB˜X[Tkbd]

@BA“ STFBHJ]IJ•{SC]&FBH†Y\OpeJemSQ]Fd‰pFQY\FpP]lKJo ST[ˆcQKBImHJ]`[C]lFXKUb MN]

& %

š2 (" ' A”_`FX] •qY\IJSTYUFjHL]dMN]

@UA“ lcQ]&ILbd]H^MX]9HJILKBOX•U]&I7[T]dcXILSCašMN]

[‘KUcNHJSCKBFdM=YBPZrpYqH

KUOPYU[C[ M”OXFX]7YUPRHJSCKBF iBOpSX•qY\ONH1STFXSCHLSfY\[T]bd]FjH

x

]>H+ijO=KUF cˆKUOXIJIZY0YBPZrX]>HJ]I YUO/cpILSCa

k

MXY\FQeOXF"HL]bdcpe`OX[HJWIJSC]&OXI

T

AQ…+]4cXIJSa€]ueH`[fYdemKB[COXHLSTKUF

u

MN]

∂u

∂t − ru + 1/2rx ∂u

∂x + 1/2σ 2 x 22 u

∂x 2 = 0

cˆKUOpI

(x, t) ∈ R × (0, T )

u(t = T, x) = max(x − k, 0)

cQKBOXI

x ∈ R

@UAn

Œ1[COpe"cXIJW&PSTeLWbd]&FBH

u(0, x)

]&emH$[C]–cXIJSaYUOgHL]&b cQe

t = 0

MN]–[¢KUcNHJSCKBFM”YUPZrpY\H"MN]–cXIJSa M=]a{]&IJPSTP]

k

 [¢W&PZrXWuY\FpP]

T > 0

]>H€M”YUP>HLS1

x

]F

t = 0

A 7FœFXK\HJ]

σ

[fY •BKU[fYqHJSC[TSHJW"MX]

[‘YUPRHJSCKBF$]>H

r

[T]^HJYUONa"M”STFBHJWI32>HuAQs]&bdYUIJijOXKBFpesijOX]

@UAn †]&emHlOXFcpILKB˜X[Ck&bd]Y‹•U]&P4PKUFpMNSCHLSTKUF

‰QFpY\[T]+]H FXKUF%cpYUe STFXSHJSTYU[C]‹bdYUSTe ijOX][T]†eLSCVBFX]MN]+[fY0MNWIJSC•BW]em]uP>KBFpMN]1]&F%]uemcQYUP>]+]&emH KUcXcˆKBeLW

P][TOXSQMpY\Fpe

@UA“ >AUŒ Y\I1PKUFpeLW&ijOX]&FjHUYUcXILkue STFj•B]IZemSTKUFdMNOdHJ]bdcpe

@UAn ]ueH˜pSC]&F OXFX]lWuijOpYqHJSCKBF

cQY\IZY\˜ˆKU[TSTijOX]BA

DG[X]aNSTemHL]sMN]sFXKBb9˜XIJ]OQem]ue •qYUILSfY\FjHL]ue MN]s[¢W&ijOpY\HLSTKUFdMN]s[fY0PZrQY\[T]OXI

@BA“ MNKUFjHFXKBOpe ]>aNcX[TK )

IJKUFQe+P]ILHJY\STFX]ue+b YUSCFjHL]&FpY\FjHuA 0BOpeJijO=SfP>S FXKBOpeY‹•UKBFpe1eLOXcXcˆKBeLW0ijOX]0[fY%PZrpYU[C]&OXIsem]0cXIJKUcpYUVU]uY\SCH

MpY\Fpe4OXF’bdST[CST]O STbdb KB˜XST[C] KBO YUO IJ]cˆKBe&A3y{OXcpcQKjemKBFpe"cXIJW&eL]FjH9ijO=]&[C[T]eL] cpILKBcpY\VB] MXYUFpe

OpF bdSC[TSC]&O ]FbdKBOX•U]&b ]&FjH7P>KBb bd]pcpYUIl]>aN]&b cp[C]pOXF*QOXSTMX]%Y\FpSCbdW9M”OXFX]%•{SHJ]&eJem]

V (x, t)

OXFX]1KUFpP>HLSTKUF ^•qY\[T]OpIJe1•U]&P>HLKBILST][T[C]ue MXY\FQe

R N

RA{¥`[TKUIZe BSC[ 1Y\ONHPZrpY\FXVB]I[fY^[TKUSˆP>KUFQeHJSHJONHLST•U]

P&Y\I`[T]-*pOXa$MN]%PZrpY\[T]OXIl]&emH`[TYeLKUbdbd]M”OXF*pOXa$MN]%MNS‡ OpeLSCKBF

PKUbdbd]cXIJW&P>WuMN]bdbd]FjH ]>H

M=OXF *QONadMX]`P>KUF{•B]&PRHJSCKBF

cXIJKUcˆKUILHLSTKUFXFX]&[ 4[fY4•jSCHL]ueLeL]

V

U]H+MN]&e1P>KUFQemSfMNWIZYqHJSCKBFpe eLSCbdST[TYUSCIJ]&e dP][T[C]uesiBOpS=cXIJW&P>kuMN]FjHsFXKBOpesP>KBFpMNOXSfem]&FjH OXF$cXIJKU˜p[Ck&b ]XMNSCHlMN] 1 - $''

 

 

 c ∂θ

∂t + cV · ∇ θ − k∆θ = f

MXYUFpe

Ω × R +

θ = 0

emOXI

∂Ω × R +

θ(t = 0, x) = θ 0 (x)

MXY\Fpe

@UAT@&t

v Y`MNSC‡ˆW&IL]&FpP>]+]FjHLIJ]

@UA“ ]H

@BAC@ut ]ueH [¢Y\cpcpY\IJSHJSCKBF^M”OpF^HL]&ILbd]+MX]+P>KBF{•U]&P>HLSTKUF=A 7F^b ]uemOpIL]

[fYs˜pYU[TYUFpP>]1]FjHLIJ]+P]FXKUOp•U]&YUO4HL]&ILbd]1MN]†P>KBFj•B]&P>HLSTKUF4]>H [C]HL]IJbd]+MN]†MNSC‡ˆOpeLSTKUFcpYUI3OXF9FXKUb%˜XIL]

eJY\FQesMNSCbd]&FpemSTKUF XY\cXcˆ][TW $ & $ $ 1 pMNW‰pFXS=cpYUI

Pe = cV L

k ,

@UAT@U@

Références