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0pWRGRV 1XPpULFRV SDUD OD 5HVROXFLyQ GH (FXDFLRQHV 'LIHUHQFLDOHV

(FXDFLRQHV 'LIHUHQFLDOHV 2UGLQDULDV

&H&DO 

 3UREOHPD GH FRQGLFLyQ LQLFLDO  3ODQWHR \ UHVXOWDGRV VREUH SURSDJDFLyQ GH HUURUHV  3ODQWHR GHO SUREOHPD 6H GHQRPLQD HFXDFLyQ GLIHUHQFLDO RUGLQDULD ('2 FRQ FRQGLFLyQ LQLFLDO DO SUREOHPD GH KDOODU XQD IXQFLyQ \ ([ ) GHILQLGD HQ XQ LQWHUYDOR D, E  TXH FXPSOD

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 G\  = I ([, \ )∀[ ∈ [j, E ] ('2  G[  \ (D) = F 
([LVWHQ WpFQLFDV DQDOtWLFDV TXH SHUPLWHQ KDOODU OD VROXFLyQ H[DFWD

GH ('2 /RV PpWRGRV QXPpULFRV SDUD OD UHVROXFLyQ GH ('2 WUDWDQ GH DSUR[LPDU OD IXQFLyQ VROXFLyQ \ ([ ) SRU XQD HVWLPDFLyQ GH VXV YDORUHV HQ XQ FRQMXQWR ILQLWR GH SXQWRV /DV ('2 DSDUHFHQ HQ P~OWLSOHV SUREOHPDV GH OD FLHQFLD \ OD WHFQRORJtD (Q RFDVLRQHV OD YDULDEOHLQGHSHQGLHQWH HV HO WLHPSR \ OD HFXDFLyQ GLIHUHQFLDO H[SUHVD OD OH\ TXH JRELHUQD ORV FDPELRV GHO VLVWHPD

\ ([ ) HQ D ≤ [ ≤ E SDUD FLHUWRV WLSRV

(O FDVR JHQHUDO GHO SUREOHPD HV HO GH XQ VLVWHPD GH ('2 GH SULPHU RUGHQ FRQ YDULDV IXQFLRQHV LQFyJQLWD η1 , η2 ,....., η V  Gη = ϕ ( [ , η1 , η2 ,....., η )  L = 1, 2 ,...., V G[
L L V

η (D ) = γ
L

L

L = 1, 2 ,...., V

FRQGLFLRQHVLQLFLDOHV

(O VLVWHPD DQWHULRU VH SXHGH HVFULELU HQ IRUPD YHFWRULDO WRPDQGR

I = (ϕ 1 ,ϕ 2 ,....., ϕ V ) F = (γ 1 ,γ 2 ,....., γ V )
7

\ = ( 1 ,η 2 ,.....,η V ) η

7

7

FRQ OR FXDO HO SUREOHPD TXHGD GH OD IRUPD FDQyQLFD ('2  VLHQGR DKRUD \ XQ YHFWRU

G\ = I ([, \ ) FRQ G[

\ (D ) = F

2EVHUYDFLyQ  8QD ('2 GH RUGHQ PD\RU TXH  VLHPSUH VH SXHGH FRQYHUWLU HQ XQ VLVWHPD GH ('2 GHSULPHU RUGHQ 'DGD OD HFXDFLyQ GLIHUHQFLDO GH RUGHQ Q FRQ FRQGLFLRQHV LQLFLDOHV
\ (D

 GQ\ G\ G 2 \ G Q −1 \  = J  [, \ , , 2 ,......, Q −1   G[ G[ G[ Q G[   

Q −1 G\ (D ) = γ 2 , ......., G Q − \ 1 G[ G[ VH SXHGH HIHFWXDU HO FDPELR GH YDULDEOHV η1 = \ G\ η2 = G[ G2\ η3 = 2 G[ G Q −1 \ ηQ = Q −1 G[

)=

γ

1

,

(D ) =

γ

Q

\ GHULYDQGR

Gη1 = η2 G[ Gη2 = η3 G[ η1 (D ) = γ 1  η 2 (D ) = γ 2

Gη Q G Q \ = Q = J ( 1 ,η 2 ,.....,η Q ) η Q (D ) = γ Q η G[ G[
(QWRQFHV OODPDQGR

\ = (η 1 , η 2 ,....., η Q ) , I ( [ , \ ) = (η 2 , η 3 ,... , J( [ , \ )) 
7

OD HFXDFLyQ VH SXHGH HVFULELU FRPR

G\ = I ( [ , \) = \( 2), \( 3),......., J ( [, \) G[

(

)

7

2EVHUYDFLyQ  8Q VLVWHPD HVWi HQ OD IRUPD DXWyQRPD VL I QR GHSHQGH H[SOtFLWDPHQWHGH [

&XDOTXLHU VLVWHPD QR DXWyQRPR VH OOHYD D XQR DXWyQRPR DJUHJDQGR OD HFXDFLyQ WULYLDO

G\ = I (\ ) G[

Gη V +1 = 1, η V +1 (D ) = D  G[

TXH WLHQH VROXFLyQ

η V +1 ([ ) = [ 

7HRUHPD GH H[LVWHQFLD \ XQLFLGDG GH OD VROXFLyQ GH XQD ('2 V+1 V 6L VH DVXPH TXH OD IXQFLyQ I : ' ⊂ 5 → 5 ; HV GLIHUHQFLDEOH ∀[ ∈ D , E

\ SDUD WRGR YHFWRU

\ = (η 1 , ..., η V ) ∈ ' \ ⊂ 5 V VLHQGR '\ XQ FLHUWR GRPLQLR TXH FRQWLHQH D F FRPR SXQWR LQWHULRU HQWRQFHV OD VROXFLyQ H[LVWH \ HV ~QLFD GHSHQGLHQGR VROR GH OD FRQGLFLyQ LQLFLDO PLHQWUDV \ ([ ) SHUPDQH]FD HQ ' \ 
'HILQLFLyQ  I ( [ , \ ) YHULILFD OD FRQGLFLyQ GH /LVSFKLW] HQ HO SXQWR ( [ 0 , \ 0 ) ∈ '  UHVSHFWR GH OD VHJXQGD YDULDEOH VL ∃8 ⊆ ' HQWRUQR GH ( [ 0 , \ 0 )  \ /! WDO TXH 

∀( [ , \1 ),( [ , \2 ) ∈8 ⇒ I ( [ , \1 )− I ( [ , \2 ) ≤ / \1 − \2
7DPELpQ HV YiOLGR XQ WHRUHPD GH H[LVWHQFLD \ XQLFLGDG FRQ KLSyWHVLV XQ SRFR PiV GpELOHV TXH ODV DVXPLGDV 7HRUHPD V V +1 6HD I : ' → 5 ; FRQWLQXD, siendo D un abierto de 5  6L I FXPSOH OD FRQGLFLyQ GH /LVSFKLW] UHVSHFWR GH OD VHJXQGD YDULDEOH HQ ' HQWRQFHV HO SUREOHPD SODQWHDGR WLHQH VROXFLyQ ~QLFD 2EVHUYDFLyQ  6L I [, \ = I [  LQGHSHQGLHQWH GH \  HQWRQFHV OD ('2SDVD D VHU XQ SUREOHPD GH LQWHJUDFLyQ

(

)

()

G\ I ([, \ ) = I ([ ) ⇒ = I ([ ) ⇒ \ ([ ) = ∫ I (X )GX VL \ (D ) = 0 G[ D

[

 3UHVHQWDFLyQ GH ORV HOHPHQWRV GH XQ PpWRGR QXPpULFR GH UHVROXFLyQ GH ('2 D WUDYpV GHO PpWRGR GH (XOHU 6L

(O YDORU K VH GHQRPLQD SDVR \ HQ HVWH PpWRGR VH DGRSWD FRQVWDQWH

\ ([ ) HV OD VROXFLyQ GH OD HFXDFLyQ GLIHUHQFLDO HO REMHWLYR VHUi KDOODU...
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