Flujo laminar de fluidos de Herschel Bulkley: Modelización Matemática y simulación numérica
112
0
0
Texto completo
(2) %&$-"Ć$*¶/ :P 4*/5:" &4.&Ć-%" 4&3Ć/0 %& -" 5033& EFDMBSP CBKP KVSBNFOUP RVF FM USBCBKP BRVÎ EFTDSJUP FT EF NJ BVUPSÎB RVF OP IB TJEP QSFWJBNFOUF QSFTFOUBEP QBSB OJOHÙO HSBEP P DBMJėDBDJÓO QSPGFTJPOBM Z RVF IF DPOTVMUBEP MBT SFGFSFODJBT CJCMJPHSÃėDBT RVF TF JODMVZFO FO FTUF EPDVNFOUP -B &TDVFMB 1PMJUÊDOJDB /BDJPOBM QVFEF IBDFS VTP EF MPT EFSFDIPT DPSSFTQPOEJFOUFT B FTUF USBCBKP TFHÙO MP FTUBCMFDJEP QPS MB -FZ EF 1SPQJFEBE *OUFMFDUVBM QPS TV 3FHMBNFOUP Z QPS MB OPSNBUJWJEBE JOTUJUVDJPOBM WJHFOUF. 4JOUZB &TNFSBMEB 4FSSBOP EF MB 5PSSF. J.
(3) %S 4FSHJP "MFKBOESP (PO[ÃMF[ "OESBEF %*3&$503 %& 130:&$50. JJ.
(4) "(Ć%&$*.*&/504 2VJFSP BHSBEFDFS B NJ GBNJMJB QPS UPEP FM DBSJÒP Z BQPZP RVF NF CSJOEBSPO UPEPT BÒPT " 4BNZ Z &NJMZ QPS DBNCJBS NJ WJEB DPO TV BNPS JODPOEJDJPOBM " NJT BNJHPT QPS TJFNQSF TFS NJ TPQPSUF Z BMFHSÎB " .JHVF QPS DPOWFSUJS NJ WJEB FO VO TVFÒP : FO FTQFDJBM RVJFSP BHSBEFDFS B 4FSHJP QPS UPEB MB QBDJFODJB Z EFEJDBDJÓO RVF MF EJTUF B FTUF QSPZFDUP Z TPCSF UPEP QPS DPOėBS FO NÎ 4FSHJP QBSB NÎ IB TJEP VO QSJWJMFHJP USBCBKBS DPOUJHP. JJJ.
(5) %&%*$"503*" 4BNZ TJFNQSF FSFT MP NÃT JNQPSUBOUF QBSB NJ. .JHVF )BWF * UPME ZPV MBUFMZ UIBU * x2 + y 2 − 1. 3. − x2 y 3 = 0 ZPV 4JOUZB. JW.
(6) 5"#-" %& $0/5&/*%0. 3ĹňŊŁĹł. WJJ. "ĶňʼnŇĵķʼn. WJJJ . *łʼnŇŃĸŊķķĽł $ŃłķĹńʼnŃň 1ŇĹŀĽŁĽłĵŇĹň &TQBDJPT EF GVODJPOFT %JGFSFODJBCJMJEBE .ÊUPEPT EF BQSPYJNBDJÓO OVNÊSJDB .ÊUPEPT EF 0QUJNJ[BDJÓO TJO SFTUSJDDJPOFT. . . . . . . . . . . . . . . 1ŇŃĶŀĹŁĵň 2ŊĵňĽŀĽłĹĵŀĹň &DVBDJPOFT 2VBTJMJOFBMFT EF 4FHVOEP PSEFO 1SPCMFNB EFM Q-BQMBDJBOP .ÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QBSB FM QSPCMFNB EFM Q-BQMBDJBOP &YQFSJNFOUPT /VNÊSJDPT &ňʼnŊĸĽŃ ĸĹŀ ĺŀŊĽĸŃ ĸĹ )ĹŇňķļĹŀ#ŊŀĿŀĹŏ $BSBDUFSÎTUJDBT GÎTJDBT EF MPT ĚVJEPT .PEFMP EF )FSTDIFM #VMLMFZ 'MVJEP EF )FSTDIFM#VMLMFZ FO VOB UVCFSÎB 'PSNVMBDJÓO WBSJBDJPOBM 'PSNVMBDJÓO DPNP VO QSPCMFNB EF PQUJNJ[BDJÓO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4ŃŀŊķĽł /ŊŁ͐ŇĽķĵ ĸĹŀ 1ŇŃĶŀĹŁĵ ĸĹ )ĹŇňķļĹŀ#ŊŀĿŀĹŏ 3FHVMBSJ[BDJÓO EFM QSPCMFNB %JTDSFUJ[BDJÓO .ÊUPEP EFM EFTDFOTP NÃT QSPGVOEP W.
(7) 3ĹňŊŀʼnĵĸŃň /ŊŁ͐ŇĽķŃň .ÊUPEP EFM EFTDFOTP QSFDPOEJDJPOBEP &YQFSJNFOUPT DPO FM .ÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QPOEFSBEP . $ŃłķŀŊňĽŃłĹň. . 3ĹĺĹŇĹłķĽĵň. WJ.
(8) 3&46.&/ &M PCKFUJWP QSJODJQBM EF FTUF USBCBKP FT EFTBSSPMMBS VOB UÊDOJDB FėDJFOUF EF TJNVMBDJÓO OVNÊSJDB EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ 1BSB FTUP BOBMJ[BNPT FM DBTP QBSUJDVMBS EFM ĚVKP MBNJOBS QPS DBNCJP EF QSFTJÓO FO VOB UVCFSÎB EF TFDDJÓO USBOTWFSTBM QBSB PCUFOFS VOB JOFDVBDJÓO WBSJBDJPOBM RVF DBSBD UFSJ[B MB FDVBDJÓO EF NPWJNJFOUP EF FTUPT ĚVJEPT -VFHP QSFTFOUBNPT MB FDVBDJÓO EF NPWJNJFOUP DPNP VO QSPCMFNB EF NJOJNJ[BDJÓO EF VO GVODJPOBM FMÎQUJDP EF TFHVOEP PSEFO OP EJGFSFODJBCMF Z MP USBOTGPSNBNPT FO VO GVODJPOBM SFHVMBSJ[BEP NFEJBOUF MB BQSPYJNBDJÓO EF )VCFS QBSB DPOTFHVJS MB EJGFSFODJBMJEBE EFM QSPCMFNB PSJHJOBM -VFHP EJTDSFUJ[BNPT FM QSPCMFNB SFHVMBSJ[BEP NFEJBOUF FMFNFOUPT ėOJUPT Z FODPOUSBNPT TV TPMVDJÓO BQMJDBOEP FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP 'J OBMNFOUF QSFTFOUBNPT BMHVOPT FKFNQMPT OVNÊSJDPT PCUFOJEPT DPO MPT BMHPSJUNPT EFTBSSPMMBEPT 1BMBCSBT DMBWFT 'MVJEPT EF )FSTDIFM#VMLMFZ .ÊUPEP EFM EFTDFOTP NÃT QSPGVOEP Q-BQMBDJBOP SFHVMBSJ[BDJÓO EF )VCFS. WJJ.
(9) "#45Ć$5 ĉF NBJO PCKFDUJWF PG UIJT XPSL JT UP EFWFMPQ BO FđDJFOU OVNFSJDBM TJNVMBUJPO UFDIOJRVF GPS )FSTDIFM#VMLMFZ ĚVJET 'PS UIJT XF BOBMZ[FE UIF QBSUJDVMBS DBTF PG B QSFTTVSF DIBOHF PO B MBNJOBS ĚPX PWFS B DSPTTTFDUJPO QJQF UP PCUBJO B WBSJBUJPOBM JOFRVBMJUZ FRVBUJPO UIBU DIBSBDUFSJ[FT UIF NP WFNFOU PG TVDI ĚVJET ĉFO XF TIPX UIF FRVBUJPO PG NPUJPO BT B NJOJNJ[BUJPO QSPCMFN PG B TFDPOE PSEFS FMMJQUJDBM OPU EJĎFSFOUJBCMF GVODUJPOBM BOE XF USBOTGPSN JU JOUP B SFHVMBSJ[FE GVODUJPOBM CZ UIF )VCCFS BQSPYJNBUJPO UP HFU UIF EJGFSFODJBMJUZ PG UIF PSJHJOBM QSPCMFN ĉFO XF EJTDSFUJ[F UIF SFHV MBSJ[FE QSPCMFN CZ UIF ėOJUF FMFNFOUT NFUIPE BOE ėOE B TPMVUJPO VTJOH UIF EFFQFTU EFTDFOU NFUIPE 'JOBMMZ XF QSFTFOU TPNF OVNFSJDBM FYBNQMFT PCUBJOFE XJUI UIF EFWFMPQFE BMHPSJUINT ,FZXPSET)FSTDIFM#VMLMFZ ĚVJET %FFQFTU EFTDFOU NFUIPE Q-BQMBDJBO )VCFS SFHVMBSJ[BUJPO. WJJJ.
(10) i/PUIJOH IBQQFOT VOMFTT ėSTU B ESFBNu. $BSM 4BOECVSH.
(11) i4DJFODF JT B DPPQFSBUJWF FOUFSQSJTF TQBOOJOH UIF HFOFSB UJPOT *UT UIF QBTTJOH PG B UPSDI ěPN UFBDIFS UP TUVEFOU UP UFBDIFS " DPNNVOJUZ PG NJOET SFBDIJOH CBDL UP BOUJRVJUZ BOE GPSXBSE UP UIF TUBSTu /FJM EF(SBTTF 5ZTPO. 1 *OUSPEVDDJÓO &M FTUVEJP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ FT JNQPSUBOUF EFCJEP B MB HSBO DBOUJEBE EF ĚVJEPT RVF QSFTFOUBO FTUF DPNQPSUBNJFOUP &KFNQMPT EFTUBDBEPT TPO MPT DSVEPT QFTBEPT DJFSUPT BMJNFO UPT QMÃTUJDPT GBSNBDÊVUJDPT <"MFYBOESPV FU BM .FTTFMNJ FU BM /JLJUJO FU BM > 6OB NPEFMJ[BDJÓO QSFDJTB Z VOB FėDB[ BQSPYJNBDJÓO OVNÊSJDB TPO GVOEBNFOUBMFT FO MB SFEVDDJÓO EF HBTUPT DPNQVUBDJPOBMFT BTÎ DPNP QBSB PCUFOFS VOB NFKPSB FO FM UJFNQP EF DÃMDVMP &M NPEFMP EF )FSTDIFM#VMLMFZ GVF QSFTFOUBEP QPS 8JOTMPX )FSTDIFM Z 3POBME #VMLMFZ FO <)FSTDIFM BOE #VMLMFZ > &TUF NPEFMP DBSBDUFSJ[B B MPT ĚVJEPT OP/FXUPOJBOPT RVF QPTFFO VO DPNQPSUBNJFOUP EVBM TÓMJEP Z MÎRVJEP EFCJEP B MB FYJTUFODJB EF VO MÎNJUF FMÃTUJDP &M MÎNJUF FMÃTUJDP FT FM FTUSÊT NÎOJNP RVF VO NBUFSJBM OFDFTJUB QBSB QBTBS EF VO NPWJNJFOUP EF VO TÓMJEP SÎHJEP B VO NPWJNJFOUP UÎQJDP EF ĚVJEP FTUF MÎNJUF QFSNJUF DBSBDUFSJ[BS MBT ÃSFBT SÎHJEBT EFM ĚVJEP FOUSF NBZPS TFB FTUF MÎNJUF IBCSÃ NÃT [POBT SÎHJEBT &M DPNQPSUBNJFOUP MÎRVJEP EF FTUF UJQP EF NBUFSJBMFT TF UJFOF DVBOEP MB GVFS[B BQMJDBEB TPCSFQBTB FM MÎNJUF FMÃTUJDP &M NPEFMP EF )FSTDIFM #VMLMFZ FT BCPSEBEP FO WBSJPT UFYUPT DPNP <#PHFS BOE )BMNPT )VJMHPM BOE :PV .BKEB .FTTFMNJ FU BM .JUTPVMJT /JLJUJO FU BM 2VPD )VOH BOE /HPD%JFQ 4BOLBE BOE 3BEIB LSJTIOBNBDIBSZB 4BSBNJUP 4BZFE"INFE FU BM > &O FTUF EPDVNFOUP USBUBSFNPT FM QSPCMFNB QBSUJDVMBS EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ FO VOB UVCFSÎB EF TFDDJÓO DJSDVMBS DPO NP WJNJFOUP MBNJOBS QPS DBNCJP EF QSFTJÓO Z TJO MB JOĚVFODJB EF GVFS[BT EF GSJDDJÓO 1SFWJP BM BOÃMJTJT EFM NPEFMP EF )FSTDIFM#VMLMFZ QSFTFOUBSFNPT BM PQFSBEPS RVBTJMJOFBM EFM Q -BQMBDJBOP EBEP FO <"MCFSUZ FU BM #BSSFĨ BOE -JV #FSNFKP BOE *OGBOUF $BTBT .
(12) BOE 'FSOÃOEF[ (MPXJOTLJ BOE .BSSPDDP )VBOH FU BM ;IPV FU BM > &T UF PQFSBEPS FT DMBWF FO MB FDVBDJÓO RVF EFTDSJCF FM NPWJNJFOUP EF FTUPT ĚVJEPT .PTUSBSFNPT RVF FM QSPCMFNB EFM Q-BQMBDJBOP QPTFF TPMVDJÓO ÙOJDB <1FSBM > -VFHP DBSBDUFSJ[BSFNPT FM PQFSB EPS DPNP VO QSPCMFNB EF NJOJNJ[BDJÓO Z MP EJTDSFUJ[BSFNPT VUJMJ[BOEP FM NÊUPEP EF FMFNFOUPT ėOJUPT FO EPT UJQPT EF NBMMBT "QMJDBSFNPT FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP DPO EJGFSFOUFT EJ SFDDJPOFT EF EFTDFOTP <)VBOH FU BM ;IPV FU BM > QBSB DBMDVMBS TV TPMVDJÓO EFM QSPCMFNB EJTDSFUJ[BEP : EFTQVÊT QSFTFOUBSFNPT EPT FKFNQMPT Z NPTUSBSFNPT MPT SFTVMUBEPT PCUFOJEPT QPS FM BMHPSJUNP FO MB NBMMB SFDUBOHVMBS Z DJSDVMBS DPO EJSFDDJPOFT EJTUJOUBT FO DBEB DBTP 3FWJTBSFNPT BMHVOPT DPODFQUPT Z UFPSFNBT EF NFDÃOJDB EF ĚVJEPT QBSB MB NFKPS DPNQSFOTJÓO EFM QSPCMFNB B USBUBS : QSFTFOUBSFNPT FM NPEFMP NBUFNÃUJDP RVF EFTDSJCF FM NPWJNJFOUP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ 1BSB FM DBTP QBSUJDVMBS EFM ĚVKP FO VOB UVCFSÎB EF TFDDJÓO USBOTWFSTBM NPTUSB SFNPT FM QSPCMFNB DPO VOB JOFDVBDJÓO WBSJBDJPOBM EF TFHVOEP PSEFO -VFHP DBSBDUFSJ[BSFNPT FM QSPCMFNB WBSJBDJPOBM NFEJBOUF VO QSPCMFNB EF NJOJNJ[BDJÓO Z QSPCBSFNPT MB FYJTUFODJB Z VOJDJ EBE EF TV TPMVDJÓO 1BSB FWJUBS QSPCMFNBT EF FTUBCJMJEBE BQMJDBSFNPT VOB SFHVMBSJ[BDJÓO EF )VCFS QBSB USBOTGPSNBS FM GVODJPOBM B NJOJNJ[BS FO VOP EJGFSFODJBCMF 1BSB FM QSPCMFNB EF NJOJNJ[BDJÓO EFM GVODJPOBM SFHVMBSJ[BEP NPTUSBSFNPT MB FYJTUFODJB Z VOJDJ EBE EF TV TPMVDJÓO Z EJTDSFUJ[BSFNPT FM QSPCMFNB DPO FM NÊUPEP EF FMFNFOUPT ėOJUPT FO VOB NB MMB DJSDVMBS DPO SBEJP Z DFOUSP FO DPO VOB QBSUJDJÓO VOJGPSNF FO FM TFOUJEP EF $JBSMFU -VFHP BQMJDBSFNPT FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QSFEFUFSNJOBEP Z QPOEFSBEP QBSB FODPOUSBS TV TPMVDJÓO 'JOBMNFOUF NPTUSBSFNPT FKFNQMPT OVNÊSJDPT EF MB BQMJDBDJÓO EF MPT BMHPSJUNPT EFTBSSPMMBEPT FO MPT RVF EFUBMMBSFNPT FM UJFNQP EF FKFDVDJÓO EFM QSPHSBNB Z FM OÙNFSP EF JUFSBDJPOFT SFBMJ[BEBT BOUFT EF PCUFOFS VO SFTVMUBEP -PT EPT QSJNFSPT FKFNQMPT NVFTUSBO MPT SFTVMUBEPT PCUFOJEPT DPO FM BMHP SJUNP EFM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QSFDPOEJDJPOBEP QBSB MPT DBTPT QBSUJDVMBSFT n = 0.75 Z n = 3 FO FM QSPCMFNB EF NJOJNJ[BDJÓO 5BNCJÊO NPTUSBSFNPT MBT HSÃėDBT DPSSFTQPOEJFOUFT B MB WFMPDJEBE MBT [POBT BDUJWBT F JOBDUJWBT Z FM FSSPS FO DBEB JUFSBDJÓO EFM BMHPSJUNP &O MPT TJHVJFOUFT EPT FKFNQMPT NPTUSBSFNPT MPT SFTVMUBEPT PCUFOJEPT BM BQMJDBS FM BMHPSJUNP EFM EFTDFOTP NÃT QSPGVO EP QPOEFSBEP QBSB n = 0.5 Z n = 9 FO FM QSPCMFNB EF NJOJNJ[BDJÓO Z FYIJCJSFNPT MPT HSÃėDPT DPSSFTQPOEJFOUFT.
(13) i:FT * BN B ESFBNFS 'PS B ESFBNFS JT POF XIP DBO POMZ ėOE IJT XBZ CZ NPPOMJHIU BOE IJT QVOJTINFOU JT UIBU IF TFFT UIF EBXO CFGPSF UIF SFTU PG UIF XPSMEu 0TDBS 8JMEF. 2 $PODFQUPT 1SFMJNJOBSFT. . &ňńĵķĽŃň ĸĹ ĺŊłķĽŃłĹň. &O FTUB TFDDJÓO WBNPT B JOUSPEVDJS OPDJPOFT EF BMHVOPT FTQBDJPT RVF TF VUJMJ[BSÃO FO FTUF EPDV NFOUP &ňńĵķĽŃň ĸĹ #ĵłĵķļ %FėOJDJÓO &TQBDJPT EF #BOBDI 4FB (X, k · k) VO FTQBDJP OPSNBEP DPNQMFUP TF EJDF FTQBDJP EF #BOBDI %FėOJDJÓO 'VODJPOBM BDPUBEP 4FBO f : X → R VO GVODJPOBM Z X FT VO FTQBDJP EF #BOBDI f FT BDPUBEP TJ FYJTUF c VO OÙNFSP SFBM UBM RVF x ∈ X |f (x)| ≤ ckxk.. . f FT VO GVODJPOBM DPOUJOVP TJ FT BDPUBEP %FėOJDJÓO 'VODJPOBM $PFSDJWP 6O GVODJPOBM f : X → R Z x ∈ X TF EJDF DPFSDJWP TJ MJN. kxk→+∞. f (x) = +∞. . .
(14) %FėOJDJÓO 'VODJPOBM $POWFYP 4FB(X, || · ||) VO FTQBDJP EF #BOBDI Z TFB f : X → R VO GVODJPOBM 4J QBSB UPEP λ ∈ (0, 1) TF DVNQMF RVF f (λu + (1 − λ)v) ≤ λf (u) + (1 − λ)f (v),. . FM GVODJPOBM f TF EFOPNJOB DPOWFYP %FėOJDJÓO 'VODJPOBM 2VBTJDPOWFYP 4FB (X, || · ||) VO FTQBDJP EF #BOBDI S VO DPOKVOUP DPO WFYP OP WBDÎP Z f : S → R VO GVODJPOBM 4J QBSB UPEP α Sα = {x ∈ S / f (x) ≤ α} FT DPOWFYP f TF EJDF RVBTJDPOWFYP 5FPSFNB 4FB (X, || · ||) VO FTQBDJP EF #BOBDI S VO DPOKVOUP DPOWFYP OP WBDÎP Z f : S → R VO GVODJPOBM 4J f FT DPOWFYP FOUPODFT QBSB UPEP α ∈ R FM DPOKVOUP Sα = {x ∈ S / f (x) ≤ α} FT DPOWFYP %FNPTUSBDJÓO -B EFNPTUSBDJÓO FTUÃ EFUBMMBEB FO <-JPOT QÃH >. . 5FPSFNB 4FB (X, || · ||) VO FTQBDJP EF #BOBDI S VO DPOKVOUP DPOWFYP OP WBDÎP Z BDPUBEP EF VO FTQBDJP SFĚFYJWP EF #BOBDI Z f : S → R VO GVODJPOBM DPOUJOVP RVBTJDPOWFYP &OUPODFT f UJFOF VO QVOUP NJOJNBM FO 4 %FNPTUSBDJÓO -B EFNPTUSBDJÓO FTUÃ EFUBMMBEB FO <-JPOT QÃH >. . 5FPSFNB $POTJEFSFNPT FM GVODJPOBM . J(y) = J1 (y) + J2 (y). %POEF MBT GVODJPOFT Ji , i = 1, 2 TPO DPOUJOVBT DPOWFYBT Z TFNJDPOUJOVB JOGFSJPSNFOUF FO MB UPQPMPHÎB EÊCJM Z UBM RVF J(y) → +∞, cuando ||y|| → +∞, QBSB UPEP y ∈ W01,n+1. . 4J y → J1 (y) FT EJGFSFODJBCMF QFSP J2 OP FT OFDFTBSJBNFOUF EJGFSFODJBCMF Z J FT FTUSJDUBNFOUF DPOWFYP &OUPODFT u ∈ W01,n+1 FT FM ÙOJDP FMFNFOUP UBM RVF J(u) = JOGy∈W 1,n+1 J(y) Z FTUÃ DBSBDUFSJ[BEB QPS 0. J1′ (u) · (y − u) + J2 (y) − J2 (u) ≥ 0. .
(15) %FNPTUSBDJÓO WFS <-JPOT QÃH >. . %FėOJDJÓO &TQBDJP %VBM %BEP X VO FTQBDJP OPSNBEP FM DPOKVOUP EF UPEBT MBT GVODJPOFT MJOFBMFT BDPUBEBT FO X TF EFOPNJOB FTQBDJP EVBM X ′ Z FTUB EPUBEP EF MB OPSNB kf k = TVQ. x∈X x6=0. kf (x)k kxk. . %FėOJDJÓO &TQBDJP SFĚFYJWP 4FB X VO FTQBDJP EF #BOBDI Z J VOB JOZFDDJÓO DBOÓOJDB EF X FO X ′′ 4F EJDF RVF X FT SFĚFYJWP TJ J(X) = X ′′ %FėOJDJÓO &TQBDJPT EF )JMCFSU 4FB V VO FTQBDJP WFDUPSJBM Z (·, ·) : V × V → R RVF TBUJTGBDF (x + y, z) = (x, z) + (y, z) (αx, y) = α(x, y) (x, y) = (y, x) (x, x) ≥ 0 (x, x) = 0 ⇔ x = 0 (·, ·) FT VO QSPEVDUP JOUFSOP Z (V, (·, ·)) FT VO FTQBDJP DPO QSPEVDUP JOUFSOP Z TF EFOPNJOB FTQBDJP EF )JMCFSU TJ FT DPNQMFUP DPO MB NÊUSJDB RVF EFėOF FM QSPEVDUP FTDBMBS &ňńĵķĽŃň .ĹĸĽĶŀĹň %FėOJDJÓO &TQBDJP .FEJCMF 6O DPOKVOUP X EPUBEP EF VOB σÃMHFCSB A FT MMBNBEP DPOKVOUP NF EJCMF Z FT OPUBEP QPS (X, A ) -PT FMFNFOUPT EF MB σÃMHFCSB A TPO EFOPNJOBEPT DPOKVOUPT A NFEJCMFT %FėOJDJÓO &TQBDJP NFEJEP 4FB (X, A ) VO FTQBDJP NFEJCMF 6OB NFEJEB TPCSF (X, A ) FT VOB GVODJÓO µ : A −→ R+ RVF WFSJėDB MBT QSPQJFEBEFT TJHVJFOUFT µ(0) = 0; Z QBSB UPEB TVDFTJÓO EF FMFNFOUPT EJTKVOUPT (An )n∈N EF A . µ. [. n∈N. . An =. X. µ(An ).. n∈N. -B USJQMFUB (X, A , µ) TF EFOPNJOB FTQBDJP NFEJEP " UPEP FMFNFOUP " EF A EFOPNJOBSFNPT MB DBOUJEBE µ(A) µNFEJEB EF ".
(16) %FėOJDJÓO 'VODJÓO NFEJCMF 4FBO (X, A ) Z (Y, B) EPT FTQBDJPT NFEJCMFT %FDJNPT RVF VOB GVODJÓO f EF 9 FO : FT (A , B)NFEJCMF TJ QBSB UPEP B ∈ B UFOFNPT f −1 (B) ∈ A %FėOJDJÓO 'VODJÓO JOUFHSBCMF 4FBO (X, A , µ) VO FTQBDJP NFEJEP Z f : X −→ [−∞, +∞] VOB GVODJÓO NFEJCMF %JSFNPT RVF f FT µJOUFHSBCMF TJ MBT DBOUJEBEFT Z. +. f (x) dµ(x) y X. Z. f − (x) dµ(x). . X. TPO ėOJUBT -B JOUFHSBM EF FTUÃ GVODJÓO FTUB EFėOJEB QPS Z. f (x) dµ(x) = X. Z. f + (x) dµ(x) + intX f − (x) dµ(x). . X. %FėOJDJÓO 1SPQJFEBEFT WÃMJEBT µDUQ 4FB (X, A , µ) VO FTQBDJP NFEJEP %FDJNPT RVF VOB QSPQJFEBE P (x) RVF EFQFOEF EF VO QVOUP x ∈ X FT WÃMJEB FO DBTJ UPEBT QBSUFT P µDUQ TJ FM DPOKVOUPT EF MPT x ∈ X FO EPOEF FTUÃ QSPQJFEBE OP TF WFSJėDB FT VO DPOKVOUP EF µ−NFEJEB OVMB P TJ FT VO DPOKVOUP µ−EFTQSFDJBCMF 5FPSFNB 5FPSFNB EF MB DPOWFSHFODJB EPNJOBEB EF -FCFTHVF 4FBO (X, A , µ) VO FTQBDJP NFEJEP Z f VOB GVODJÓO Z (fn )n∈R VOB TVDFTJÓO EF GVODJPOFT BNCBT A NFEJCMFT EFėOJEBT TPCSF X RVF UPNB WBMPSFT FO R )BDFNPT MBT TJHVJFOUFT IJQÓUFTJT QBSB µDBTJ UPEP x ∈ X, TF UJFOF MJNn→+∞ fn (x) = f (x) Z FYJTUF VOB GVODJÓO JOUFHSBCMF g : X −→ R UBM RVF QBSB UPEP O TF WFSJėDB MB EFTJHVBMEBT |fn (x) ≤ g(x)| µDBTJ UPEBT QBSUFT FO X &OUPODFT f FT VOB GVODJÓO JOUFHSBCMF Z MJN. n→+∞. Z. fn (x) dµ(x) = X. Z. f (x) dµ(x). X. %FNPTUSBDJÓO &M EFUBMMF EF MB EFNPTUSBDJÓO EFM UFPSFNB FTUÃ EFTDSJUP FO <$IBNPSSP QÃH > . &ňńĵķĽŃň Lp %FėOJDJÓO &TQBDJPT -p 4FB Ω ⊂ RN Z p ∈ R EPOEF 0 ≤ p < ∞ TF EFėOF -p BM FTQBDJP EF MBT GVODJPOFT NFEJCMFT f : Ω → R RVF TBUJTGBDFO Z. Ω. |f (x)|p dx < ∞..
(17) &M FTQBDJP -p FTUÃ EPUBEP EF MB OPSNB kf kLp =. Z. p. |f (x)| dx. Ω. 1/p. WFS <#SÊ[JT QÃH > %FėOJDJÓO &TQBDJP -∞ &M FTQBDJP -∞ FT FM FTQBDJP EF MBT GVODJPOFT NFEJCMFT f UBM RVF |f (x)| ≤ C DUQ FO Ω DPO $ DPOTUBOUF &M FTQBDJP -∞ QPTFF MB OPSNB kf kL∞ = JOG{C; |f (x)| ≤ C DUQ FO Ω} WFS <#SÊ[JT QÃH > 5FPSFNB 3FQSFTFOUBDJÓO EF 3JFT[ 4FB 1 < p < ∞ Z ϕ ∈ (Lp )′ &OUPODFT FYJTUF VO ÙOJDP ′ u ∈ Lp UBM RVF hϕ, f i =. Z. uf QBSB UPEP f ∈ Lp . %FNPTUSBDJÓO 7FS <#SÊ[JT 5FP *7>. . &ňńĵķĽŃň ĸĹ 4ŃĶŃŀĹŋ %FėOJDJÓO &TQBDJPT W 1,p 4FB VO DPOKVOUP BCJFSUP Ω ⊂ RN Z 1 ≤ p ≤ ∞ FM FTQBDJP EF 4PCPMFW W 1,p TF EFėOF DPNP. W 1,p (Ω) =. . u ∈ Lp Z. ∃ g1 , g2 , g3 , . . . , gN ∈ Lp (Ω) UBM RVF ∂ϕ =− u Ω ∂xi. Z. gi ϕ Ω. ∀ϕ ∈. Cc∞ (Ω) ∀i. . . = 1, 2, . . . , N . &M FTQBDJP W 1,p (Ω) DPO 1 ≤ p < ∞ FTUÃ EPUBEP EF MB TJHVJFOUF OPSNB . kukW 1,p = kukLp. n X ∂u + ∂xi i=1. Lp. 1/p . .. .
(18) 4J p = ∞ MB OPSNB FT kukW 1,∞ =. N X i=1. JOG. (. ∂u ≤ C DUQ FO Ω ∂xi. ). . -PT FTQBDJPT W m,p (Ω) QBSB m ≥ 2 TF EFėOFO QPS W. m,p. =. . u∈W. m−1,p. ∂u ∈ W m−1,p , ∀i = 1, 2, . . . , n (Ω) : ∂xi. . . &TUPT FTQBDJPT FTUÃO EPUBEPT EF MB OPSNB. kukW m,p. 1/p n Z X k∇α ukp dx = i=1. . Ω. &M FTQBDJP W 1,2 (Ω) FT DPOPDJEP DPNP H 1 (Ω) RVF FT VO FTQBDJP EF )JMCFSU DPO FM QSPEVDUP FTDBMBS N X ∂u ∂v , (u, v)H 1 = (u, v)L2 + ∂xi ∂xi i=1 Z FTUÃ BTPDJBEP B MB OPSNB EF W 1,2 (Ω) EBEB QPS . 1SPQPTJDJÓO 4J 1 ≤ p ≤ ∞ W 1,p FT VO FTQBDJP EF #BOBDI .ÃT BÙO TJ 1 < p < ∞ W 1,p FT VO FTQBDJP SFĚFYJWP &M FTQBDJP H 1 FT VO FTQBDJP EF )JMCFSU TFQBSBCMF %FNPTUSBDJÓO 7FS <#SÊ[JT QÃH 5FP *9>. . %FėOJDJÓO &TQBDJPT W01,p &M FTQBDJP W01,p FT MB BEIFSFODJB EF C01 (Ω) FO W 1,p &M FTQBDJP W01,p QPTFF MB OPSNB JOEVDJEB QPS FM FTQBDJP EF #BOBDI W 1,p H01 = W01,2 FT VO FTQBDJP EF )JMCFSU DPO FM QSPEVDUP FTDBMBS EFėOJEP FO 5FPSFNB %FTJHVBMEBE EF 1PJODBSÊ %BEP Ω BCJFSUP Z BDPUBEP &OUPODFT QBSB 1 ≤ p ≤ ∞ FYJTUF VOB DPOTUBOUF c UBM RVF kukLp ≤ ck∇ukLp QBSB UPEP u ∈ W01,p . %FNPTUSBDJÓO 7FS <#SÊ[JT QÃH 5FP *9>. .
(19) . . %ĽĺĹŇĹłķĽĵĶĽŀĽĸĵĸ. %FėOJDJÓO -JQTDIJU[ DPOUJOVJEBE %BEPT XF Y EPT FTQBDJPT OPSNBEPT Z F : X → Y TF EJDF -JQTDIJU[ DPOUJOVB TJ FYJTUF VOB DPOTUBOUF QPTJUJWB L UBM RVF kF (x1 ) − F (x2 )kY ≤ Lkx1 − x2 kX. QBSB UPEP x1 , x2 ∈ X.. . %FėOJDJÓO %FSJWBEB EF (ÄUFBVY 4FBO X VO FTQBDJP OPSNBEP U VO BCJFSUP OP WBDÎP EF X Z F : U ⊆ X → R VOB GVODJÓO 'JKBEP VO WFDUPS OP OVMP h ∈ X Z u0 ∈ U TJ F (u0 + th) − F (u0 ) t→0 t. DF (u0 ) = MJN. . FYJTUF Z MJOFBM TF EFOPNJOB EFSJWBEB EF (ÄUFBVY EF F FO MB EJSFDDJÓO h ∈ X FO FM QVOUP u0 ∈ U %FėOJDJÓO %FSJWBEB EF 'SÊDIFU 4F EJDF RVF F FT EJGFSFODJBCMF FO u0 FO FM TFOUJEP EF 'SÊDIFU TJ FYJTUF VOB BQMJDBDJÓO MJOFBM Z DPOUJOVB L : X → R UBM RVF F (u0 + h) + F (u0 ) − L(h) = 0. h→0 khk. F ′ (u0 ) MJN. . 1SPQPTJDJÓO 4FB F : U ⊆ X → R VOB GVODJÓO EJGFSFODJBCMF FO FM TFOUJEP EF 'SÊDIFU FO u0 FOUPODFT F FT EJGFSFODJBCMF FO FM TFOUJEP EF (ÄUFBVY FO FTF QVOUP Z BEFNÃT DF (u0 ) = F ′ (u0 ). . .͐ʼnŃĸŃň ĸĹ ĵńŇŃŎĽŁĵķĽł łŊŁ͐ŇĽķĵ. &O FTUB TFDDJÓO SFWJTBSFNPT NÊUPEPT EF BQSPYJNBDJÓO RVF OPT BZVEBSÃO B DBMDVMBS OVNÊSJDBNFO UF MB TPMVDJÓO EF OVFTUSP QSPCMFNB 3ĹĻŀĵň ĸĹ ĽłʼnĹĻŇĵķĽł łŊŁ͐ŇĽķĵ " DPOUJOVBDJÓO QSFTFOUBSFNPT EPT NÊUPEPT RVF VUJMJ[BSFNPT QBSB DBMDVMBS JOUFHSBMFT FO FTUF EP DVNFOUP .͐ʼnŃĸŃ ĸĹŀ ńŊłʼnŃ ŁĹĸĽŃ &M NÊUPEP EFM QVOUP NFEJP DPOTJTUF FO SFFNQMB[BS MB GVODJÓO f TPCSF FM JOUFSWBMP [a, b] QPS MB GVODJÓO DPOTUBOUF JHVBM BM WBMPS BMDBO[BEP QPS f FO FM QVOUP NFEJP EFM JOUFSWBMP [a, b] <2VBSUFSPOJ FU BM > a+b I0 (f ) = (b − a)f . 2.
(20) 1BSB BQSPYJNBS EF NBOFSB NÃT QSFDJTB MB JOUFHSBM I(f ) QPEFNPT EJWJEJS FM JOUFSWBMP [a, b] FO m TVCJOUFSWBMPT EF BODIP H = (b − a)/m QBSB m ≥ 1 : FO DBEB TVCJOUFSWBMP UPNBNPT MPT OPEPT EF BQSPYJNBDJÓO xk = a + (2k + 1)H/2 QBSB k = 0, . . . , m − 1 TF PCUJFOF MB TJHVJFOUF GÓSNVMB DPNQVFTUB <2VBSUFSPOJ FU BM > I0,m (f ) = H. m−1 X. f (xk ).. k=0. .͐ʼnŃĸŃ ĸĹŀ ʼnŇĵńĹķĽŃ -B GÓSNVMB TF PCUJFOF SFFNQMB[BOEP f QPS FM QPMJOPNJP EF -BHSBOHF EF HSBEP VOP DPO MPT OPEPT x0 = a Z x1 = b <2VBSUFSPOJ FU BM > I1 (f ) =. b−a [f (a) + f (b)]. 2. 5BNCJÊO QPEFNPT BQSPYJNBS MB JOUFHSBM I(f ) EJWJEJFOEP FM JOUFSWBMP FO m TVCJOUFSWBMPT Z SFFNQMB [BEP f DPO MPT SFTQFDUJWPT QPMJOPNJPT EF -BHSBOHF FO MPT OPEPT xk = a + kH QBSB k = 0, . . . , m Z H = (b − a)/m <2VBSUFSPOJ FU BM > TF PCUJFOF MB GÓSNVMB EFM USBQFDJP DPNQVFTUB m−1 HX I1,m (f ) = (f (xk ) + f (xk+1 )), 2 k=0. . m ≥ 1.. .͐ʼnŃĸŃň ĸĹ 0ńʼnĽŁĽŐĵķĽł ňĽł ŇĹňʼnŇĽķķĽŃłĹň. %BEB VOB GVODJÓO PCKFUJWP f : Rn → R UPNBNPT FM QSPCMFNB EF NJOJNJ[BDJÓO NJO f (x). . x∈Rn. EFOPNJOBEP QSPCMFNB EF PQUJNJ[BDJÓO TJO SFTUSJDDJPOFT 4J UPNBNPT x∗ TPMVDJÓO EF P NÎOJ NP HMPCBM Z NÎOJNP MPDBM TJ FYJTUF R > 0 UBM RVF f (x∗ ) ≤ f (x). ∀x ∈ B(x∗ , R),. . "TVNJSFNPT RVF f ∈ C 1 (Rn ) FM HSBEJBOUF EF f FO FM QVOUP x TF EFėOF DPNP ∇f (x) = ∂f EPOEF ∂x FT MB EFSJWBEB QBSDJBM i. . ∂f ∂f (x), . . . , ∂x1 ∂xn. ⊤. .
(21) . : MB EFSJWBEB EJSFDDJPOBM FO MB EJSFDDJÓO w TF EFėOF ∂f f (x + αw) − f ((x) (x) = MJN . α→0 ∂w α. . ∂f 4BCFNPT RVF ∂w (x) = ∇f (x)⊤ w <2VBSUFSPOJ FU BM >. 1SPQPTJDJÓO %BEPT f ∈ C 1 (Rn ) w ∈ R α ∈ R Z ξ ∈ (x, x + αw) MB FYQBOTJÓO EF 5BZMPS EF f FT f (x + αw) − f (x) = α∇f (ξ)⊤ w %FėOJDJÓO 1VOUPT DSÎUJDPT %BEB VOB GVODJÓO f : Rn → R x∗ TF EJDF QVOUP DSÎUJDP TJ ∇f (x∗ ) = 0.. . 1SPQPTJDJÓO x∗ ∈ Rn FT VO NÎOJNP MPDBM EF f : Rn → R EPOEF f ∈ C 1 (B(x∗ , R)) QBSB R > 0 DPOWFOJFOUF TJ x∗ FT VO QVOUP DSÎUJDP 7FS <2VBSUFSPOJ FU BM QÃH > .͐ʼnŃĸŃň ĸĹ #ΌňŅŊĹĸĵ -ĽłĹĵŀ -B JEFB EF FTUPT NÊUPEPT FT FODPOUSBS VOB TPMVDJÓO BQSPYJNBEB EFM QSPCMFNB B QBSUJS EF VO QVOUP JOJDJBM x0 ∈ Rn TF HFOFSB VOB TVDFTJÓO EF FMFNFOUPT x1 , x2 , . . . , xn </PDFEBM BOE 8SJHIU QÃH > UBM RVF f (xk ) < f (xk+m ). QBSB k ∈ {1, 2, . . . , n} Z m ∈ {1, 2, . . . , n − 1} -PT FMFNFOUPT xk DPO k ∈ {1, 2, . . . , n} TF DBMDVMBO BTÎ xk = xk−1 + αk wk. . EPOEF wk FT MB EJSFDDJÓO RVF OPT HBSBOUJ[B FM EFTDFOTP EF f Z αk FT MB EJTUBODJB RVF QPEFNPT NPWFSOPT FO MB EJSFDDJÓO wk FTUB EJTUBODJB TF BQSPYJNB B QBSUJS EFM QSPCMFNB NJO f (xk + αwk ) α>0. . EPOEF α TF EFOPNJOB MBSHP EFM QBTP -B TPMVDJÓO FYBDUB EF EB VOB NFKPS BQSPYJNBDJÓO DPO MB EJSFDDJÓO wk QFSP FODPOUSBSMB FT DPTUPTB F JOOFDFTBSJB QPS MP RVF TF VUJMJ[BO NÊUPEPT QBSB BQSPYJNBS FM MBSHP EFM QBTP.
(22) . x0 x1. w. k. 'JHVSB %FTDFOTP EFM BMHPSJUNP FO DBEB JUFSBDJÓO. θk −∇fk 'JHVSB %JSFDDJÓO EF EFTDFOTP. %ĽŇĹķķĽł ĸĹ ĸĹňķĹłňŃ -B EJSFDDJÓO EF EFTDFOTP wk QBSB MPT NÊUPEPT JUFSBUJWPT TF UPNB VTVBMNFOUF DPNP wk = −Bk−1 ∇fk .. . %POEF B FT VOB NBUSJ[ EFėOJEB QPTJUJWB Z TJNÊUSJDB </PDFEBM BOE 8SJHIU > &O HFOFSBM QBSB π RVF MB GVODJÓO f EFTDJFOEB FM ÃOHVMP FOUSF MB EJSFDDJÓO wk Z −∇fk EFCF TFS NFOPS RVF 2. -B DPOEJDJÓO QBSB TFS VOB EJSFDDJÓO EF EFTDFOTP FT dk ∇fk = kdk k · k∇fk k DPT θk < 0.. .
(23) )BZ VOB WBSJFEBE EF NÊUPEPT TFHÙO MB FMFDDJÓO EF MB EJSFDDJÓO EF EFTDFOTP /PTPUSPT OPT FOGP DBSFNPT FO MPT NÊUPEPT EF UJQP EFTDFOTP NÃT QSPGVOEP .͐ʼnŃĸŃ ĸĹŀ ĸĹňķĹłňŃ Ł̾ň ńŇŃĺŊłĸŃ &M NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP FT VO NÊUPEP EF EFTDFOTP RVF CVTDB MB TPMVDJÓO EFM QSP CMFNB DPO VOB EJSFDDJÓO EF EFTDFOTP wk = −∇f (uk ) &TUF NÊUPEP TFSÃ VUJMJ[BEP QBSB FODPOUSBS MB TPMVDJÓO EF MB FDVBDJÓO EF NPWJNJFOUP QBSB MPT ĚVJEPT EF )FSTDIFM#VMLMFZ RVF FT FM UFNB QSJODJQBM EF FTUF EPDVNFOUP 5PNBSFNPT MB GVODJÓO PCKFUJWP f : Rn → R UPNBNPT FM QSPCMFNB EF NJOJNJ[BDJÓO EBEP FO NJOn f (u) u∈R. EFOPNJOBEP QSPCMFNB EF PQUJNJ[BDJÓO TJO SFTUSJDDJPOFT 4J UPNBNPT u∗ TPMVDJÓO EF 1PS TFS VO NÊUPEP EF CÙTRVFEB MJOFBM EFėOJNPT BM QBTP uk+1 DPNP uk+1 = uk + αk · wk. . EPOEF αk UBNBÒP EFM QBTP FO MB LFTÎNB JUFSBDJÓO Z wk EJSFDDJÓO EF EFTDFOTP FO MB LFTÎNB JUFSBDJÓO &M PCKFUJWP QSJODJQBM EFM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP FT RVF MB GVODJÓO f EFDSFDF MP NÃT SÃQJEP QPTJCMF FO DBEB JUFSBDJÓO 1BSB RVF MB GVODJÓO f EFDJFOEF SÃQJEBNFOUF FO MB EJSFDDJÓO −∇f (un ) QPS MP RVF TF UPNB MB EJSFDDJÓO EF EFTDFOTP DPNP wn = −∇f (un ). . -B EFėOJDJÓO EF MB EJSFDDJÓO FO DVNQMF DPO Z FM NÊUPEP EF CÙTRVFEB MJOFBM TF EFOP NJOB EFTDFOTP NÃT QSPGVOEP </PDFEBM BOE 8SJHIU 2VBSUFSPOJ FU BM > %FTQVÊT EF RVF MB EJSFDDJÓO EF EFTDFOTP FTUÃ EFUFSNJOBEB UFOFNPT RVF EFėOJS DVBOUP QPEFNPT NPWFSOPT FO EJDIB EJSFDDJÓO QBSB QPEFS FODPOUSBS MB TPMVDJÓO EFM QSPCMFNB %F <%F -PT 3FZFT > TBCFNPT RVF MB MPOHJUVE EFM QBTP FTUÃ EFėOJEB QPS αk = BSH NJO{f (yk + αwk )}. α>0. 1BSB SFTPMWFS FTUF QSPCMFNB EF NJOJNJ[BDJÓO VUJMJ[BSFNPT MB DPOEJDJÓO EF "SNJKP. .
(24) $ŃłĸĽķĽł ĸĹ "ŇŁĽľŃ %BEP RVF SFTPMWFS OVNÊSJDBNFOUF FT DPNQMJDBEP WBNPT B BQSPYJNBS TV TPMVDJÓO f (un + αn wn ) − f (un ) −→ 0. DVBOEP n → ∞. . 6OB GPSNB EF FODPOUSBS αn TF EFOPNJOB DPOEJDJÓO P SFHMB EF "SNJKP RVF DPOTJTUF FO UPNBS FTDBMBSFT s, ρ > 0 DPO ρ ∈ (0, 1) Z σ ∈ (0, 1/2) &TDPHFNPT αn ∈ {s, sρ, sρ2 , · · · } f (un ) − f (un − αwn ) ≥ +σα∇f (un )⊤ · wn. . f (uk ). α∇f (uk ) · wk αk 'JHVSB $POEJDJÓO EF "SNJKP. .
(25) i*U JT JNQPSUBOU UP SFNFNCFS UIBU XF BMM IBWF NBHJD JOTJEF VTu +, 3PXMJOH. 3 1SPCMFNBT 2VBTJMJOFBMFT &O FTUF DBQÎUVMP BCPSEBSFNPT FM QSPCMFNB EFM Q-BQMBDJBOP RVF FT QBSUF JNQPSUBOUF FO MB FDVB DJÓO EF NPWJNJFOUP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ $PNFO[BSFNPT SFWJTBOEP MBT FDVBDJPOFT EF 1PJTPO BTPDJBEBT BM PQFSBEPS Q-BQMBDJBOP -VFHP NPTUSBSFNPT RVF FTUF UJFOF TPMVDJÓO ÙOJDB FO FM FTQBDJP W01,p DPO 1 < p < ∞Z NPTUSBSFTNPT VOB BQSPYJNBDJÓO OVNÊSJDB CBTBEB FO FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP. . &ķŊĵķĽŃłĹň 2ŊĵňĽŀĽłĹĵŀĹň ĸĹ 4ĹĻŊłĸŃ ŃŇĸĹł. 6OB FDVBDJÓO EJGFSFODJBM RVBTJMJOFBM EF TFHVOEP PSEFO FTUà EFėOJEB QPS <.JFSTFNBON > n X. i,j=1. aij (x, u, ∇u)uxi xj + b(x, u, ∇u) = 0.. . FO Ω ⊂ Rn EPOEF u : Ω → R Z BTVNJFOEP RVF aij = aji 1PEFNPT DBSBDUFSJ[BS DPNP n X. i,j=1. aij (x, u, ∇u)χxi χxj = 0. . -BT FDVBDJPOFT RVBTJMJOFBMFT BTPDJBEBT B MB DBSBDUFSJ[BDJÓO EF MB FDVBDJÓO OP UJFOFO TPMVDJÓO χ DVBOEP ∇χ 6= 0 <.JFSTFNBON QÃH > .
(26) %FėOJDJÓO &VBDJÓO RVBTJMJOFBM FMÎQUJDB -B FDVBDJÓO RVBTJMJOFBM TF EJDFO FMÎQUJDBT TJ MB NB USJ[ aij (x, z, p) FT EFėOJEB QPTJUJWB QBSB DBEB (x, z, p) ∈ U {(x, z, p) : x ∈ Ω, z ∈ R, p ∈ Rn }. . 1ŇŃĶŀĹŁĵ ĸĹŀ ń-ĵńŀĵķĽĵłŃ. &O FTUB TFDDJÓO UPNBSFNPT FM PQFSBEPS Q-BQMBDJBOP EBEP QPS △p y ≡ ∇ · |∇y|p−2 ∇y. DPO 1 < p < ∞. . RVF FT VO FKFNQMP NVZ JOUFSFTBOUF EF PQFSBEPS RVBTJMJOFBM Z BEFNÃT FT QBSUF EF MB FDVBDJÓO EF NP WJNJFOUP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ RVF FT FM UFNB QSJODJQBM FO FTUF EPDVNFOUP &O <#BSSFĨ BOE -JV (MPXJOTLJ BOE .BSSPDDP )VBOH FU BM > Z <1FSBM > TF BOBMJ[B FM TJHVJFOUF QSPCMFNB %BEP VO BCJFSUP Ω ⊂ Rn -JQTDIJU[ DPOUJOVB DPO 1 < p < ∞ IBMMBS y UBM RVF p−2 −∇ · |∇y| ∇y = f DPO f ∈ W −1,p (Ω) Z p′ = ′. . y =0. FO Ω FO ∂Ω. p ′ EPOEF W −1,p (Ω) FT FM FTQBDJP EVBM EF W 1,p (Ω) p−1. 1BSB QSPCBS RVF UJFOF TPMVDJÓO QSFTFOUBSFNPT FM TJHVJFOUF UFPSFNB 5FPSFNB %BEP Ω BDPUBEP Z f ∈ W −1,p FM QSPCMFNB UJFOF VOB TPMVDJÓO y ∈ W01,p FO FM TFOUJEP TJHVJFOUF ′. Z. Ω. |∇y|. p−2. (∇y, ∇w) dx =. Z. f w dx Ω. QBSB UPEP w ∈ W01,p. %FNPTUSBDJÓO 7FS <1FSBM QÃH > "TÎ FM QSPCMFNB FRVJWBMF B FODPOUSBS y ∈ W01,p RVF DVNQMB Z. Ω. |∇y|. p−2. (∇y, ∇v) dx =. Z. Ω. f v dx QBSB UPEP v ∈ W01,p. . .
(27) 1ŇŃĶŀĹŁĵ ĸĹ ŁĽłĽŁĽŐĵķĽł &M QSPCMFNB QVFEF TFS DBSBDUFSJ[BEP DPNP VO QSPCMFNB EF NJOJNJ[BDJÓO EFėOJNPT 1 J(y) = p. Z. p. Ω. |∇y| dx −. Z. . f y dx Ω. DPO EFSJWBEB EF (BUFBVY FO MB EJSFDDJÓO w EBEB QPS &OUPODFT FM QSPCMFNB TF QVFEF SFTPMWFS WBSJBDJPOBMNFOUF NJOJNJ[BOEP FM GVODJPOBM DPSSFTQPOEJFOUF BM QSPCMFNB EF NJOJNJ[BDJÓO <$BTBT BOE 'FSOÃOEF[ (MPXJOTLJ BOE .BSSPDDP > . NJO J(y). y∈W01,p. EPOEF. 1 J(y) = p. Z. p. Ω. |∇y| dx −. Z. . f y dx Ω. %ĽňķŇĹʼnĽŐĵķĽł &O FTUB TFDDJÓO EJTDSFUJ[BSFNPT BQMJDBOEP FM NÊUPEP EF FMFNFOUPT ėOJUPT <#BSSFĨ BOE -JV )VBOH FU BM > EFėOJFOEP Ω ⊂ R2 5PNBOEP Ωh DPNP MB BQSPYJNBDJÓO QPMJHPOBM EF Ω [ h τ EPOEF T h FT VOB QBSUJDJÓO EF Ωh FO FM TFOUJEP EF $JBSFU FO VO OÙNFSP ėOJUP EF ZΩ = τ ∈T h. USJÃOHVMPT τ EJTKVOUPT DPO MBEP h "EFNÃT TJ UPNBNPT EPT USJÃOHVMPT DVBMFTRVJFSB EF MB QBSUJDJÓO TVT DMBVTVSBT TPO EJTKVOUBT P B MP NÃT DPNQBSUFO VO MBEP P WÊSUJDF &O MPT TJHVJFOUFT HSÃėDPT UFOFNPT MB QBSUJDJÓO T h QBSB FM EPNJOJP [0, 1] × [0, 1] Z [−1, 1] × [−1, 1] SFTQFDUJWBNFOUF <"MCFSUZ FU BM #BSSFĨ BOE -JV > Solution of the Problem 1. 1. 0.9. 0.8. 0.8. 0.6. 0.7. 0.4. 0.6. 0.2. 0.5. 0. 0.4. −0.2. 0.3. −0.4. 0.2. −0.6. 0.1. −0.8. 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 'JHVSB .BMMB SFDUBOHVMBS FO [0, 1] × [0, 1] DPO h = 1/11. 1. −1 −1. −0.8. −0.6. −0.4. −0.2. 0. 0.2. 0.4. 0.6. 0.8. 'JHVSB .BMMB DJSDVMBS FO [−1, 1] × [−1, 1] DPO h = 1/11. 1.
(28) 4FB τ ∈ T h VO USJBOHVMP EF MB QBSUJDJÓO TVT WÊSUJDFT UJFOF DPPSEFOBEBT (xj , yj ) (x3 , y3 ). τ (x1 , y1 ). (x2 , y2 ). " MB QBSUJDJÓO T h BTPDJBNPT FM FTQBDJP ėOJUP EJNFOTJPOBM <#BSSFĨ BOE -JV QÃH > TJHVJFOUF n o h S h = χ ∈ C(Ω ) : χ|τ MJOFBM QBSB UPEP τ ∈ T h ⊂ W01,p (Ωh ).. . NJO J(y). . "TÎ FM QSPCMFNB EJTDSFUJ[BEP FT. y h ∈S h. EPOEF. 1 J(y) = p. %FėOJNPT MBT GVODJPOFT ϕ DPNP . Z. p. Ωh. |∇y| dx −. . Z. 1 EFU 1 1 ϕj (x, y) = 1 EFU 1 1 1 ∇ϕj (x, y) = 2|τ |. x. Z. f y h dx Ωh. y. . xj+1 yj+1 xj+2 yj+2 xj yj xj+1 yj+1 xj+2 yj+2 yj+1 − yj+2 xj+2 − xj+1. . !. . .. . EPOEF MPT ÎOEJDFT TF UPNBO FO NÓEVMP Z |τ | SFQSFTFOUB FM ÃSFB EFM USJÃOHVMP <"MCFSUZ FU BM > 5PNBNPT BSh = {ϕ1 , ϕ2 , . . . , ϕn } CBTF S h QBSB DBEB USJÃOHVMP τ DPO WÊSUJDFT (xi , yi ) DPO i = 1, 2, 3. Z ϕ1 , ϕ2 , ϕ3 MBT DPSSFTQPOEJFOUFT GVODJPOFT EF MB CBTF ϕj (xk , yk ) = δjk ,. j, k = 1, 2, 3. .
(29) 6O FMFNFOUP DVBMFTRVJFSB y ∈ S h TF FTDSJCF DPNP MB DPNCJOBDJÓO EF MBT GVODJPOFT EF MB CBTF h. y =. n X. yi ϕi. i=0. h. Z∇ y=. n X i=0. yi ∇ϕi .. . "TÎ FM GVODJPOBM EBEP FO FTUÃ EBEP QPS 1 → y J(y) = C h − f h⊤ − p. . DPO C h = (Ci ) EPOEF. Ci =. 3 XZ X. τ ∈T h. %POEF |∇y | = h. 3 X k=1. τ. p. yk ϕk. QBSB i = 1, . . . , n. dx. . k=1. yk ∇ϕk FO DBEB USJÃOHVMP τ UPNBOEP MB GPSNB EJTDSFUB ∂1h ∂2h. ∇h =. !. .. . 1BSB DBEB USJÃOHVMP τ DPO WÊSUJDFT (x1 , y1 ), (x2 , y2 ) Z (x3 , y3 ) TF UJFOF . ∂ϕi ∂x 1 ∂ϕi ∂1h = ∂x2 ∂ϕ i ∂x 3 . "TÎ FM HSBEJBOUF EJTDSFUP FTUÃ EBEP QPS Z . ∂ϕi ∂y1 ∂ϕi ∂2h = ∂y2 QBSB i = 1, 2, 3. ∂ϕ i ∂y 3 . . ∂ϕi yi ∂Y. . − → h ∇y = ∂ϕi yi ∂Y. QBSB i = 1, 2, 3 . <(PO[ÃMF[ "OESBEF > EPOEF Y = (x1 , x2 , x3 ) Z Y = (y1 , y2 , y3 ). . .
(30) 1BSB FYQSFTBS MB |∇y h | EFėOJNPT MB TJHVJFOUF GVODJÓO ξ :R6 → R3. ξ(u)k : = |(pk , pk+3 )⊤ | QBSB k = 1, 2, 3. "TÎ MB JOUFHSBM. R. Ω. . |∇y|p TF BQSPYJNB BTÎ XZ. τ ∈T h. p. τ. |∇y| dx =. XZ. τ ∈T h. ξ(∇y)p dx.. . τ. "QMJDBOEP FM NÊUPEP EFM QVOUP NFEJP BQSPYJNBSFNPT FM WBMPS EF MB JOUFHSBM BTÎ Ci =. XZ. τ ∈T h. τ. ξ(∇y)p dx ≈. X. τ ∈T h. |τ |. ξ(∇y)p 2. QBSB i = 1, . . . , n.. . 1PS PUSP MBEP f h = (fi ) DPO fi =. XZ. τ ∈T h. f ϕi (x) dx τ. QBSB i = 1, · · · , n. . DVZB BQSPYJNBDJÓO FO DBEB τ TF DBMDVMB FO (xc , yc ) RVF FT FM DFOUSP EF HSBWFEBE EFM USJÃOHVMP <"M CFSUZ FU BM QÃH >. (x3 , y3 ). (xc , yc ) (x1 , y1 ). BTÎ Z. 1 f ϕi dx ≈ EFU 6 τ. (x2 , y2 ). x 2 − x 1 x3 − x1 y2 − y1 y3 − y1. !. f (xc , yc ). .
(31) Z fi =. X1. τ ∈T h. . 6. EFU. x 2 − x1 x 3 − x 1 y2 − y1 y3 − y1. !. . f (xc , yc ). .͐ʼnŃĸŃ ĸĹŀ ĸĹňķĹłňŃ Ł̾ň ńŇŃĺŊłĸŃ ńĵŇĵ Ĺŀ ńŇŃĶŀĹŁĵ ĸĹŀ ń-ĵńŀĵķĽĵłŃ. &O FTUB TFDDJÓO UPNBSFNPT FM QSPCMFNB EF NJOJNJ[BDJÓO EJTDSFUJ[BEP RVF DBSBDUFSJ[B BM QSPCMFNB EFM Q-BQMBDJBOP Z VUJMJ[BSFNPT FM NÊUPEP EF CÙTRVFEB MJOFBM EFM EFTDFOTP NÃT QSPGVOEP WJTUP FO FM DBQÎUVMP QBSB FODPOUSBS TV TPMVDJÓO BQSPYJNBEB 1BSB BQMJDBS FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP EFėOJSFNPT EPT UJQPT EF EJSFDDJPOFT EF EFTDFOTP FNQMFBEPT FO <)VBOH FU BM ;IPV FU BM > -VFHP EFUFSNJOBSFNPT MB MPOHJUVE EFM QBTP EF EFTDFOTP VTBOEP MB DPOEJDJÓO EF "SNJKP Z MVFHP FODPOUSBSFNPT MB TPMVDJÓO EF 1BSB DPNFO[BS DPO FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP UPNBSFNPT FM QSPCMFNB EJTDSFUJ[BEP NJO J(y h ). . y h ∈S h. DPO h. J(y ) =. Z. h p. Ωh. |∇y | −. Z. f yh.. . Ωh. %FėOJSFNPT MB BQSPYJNBDJÓO EF MB TPMVDJÓO EF FO FM OÊTJNP QBTP DPNP yn+1 = yn + αn wn. n ∈ R+. . EPOEF αn FT MB MPOHJUVE EFM OÊTJNP QBTP Z wn FT MB EJSFDDJÓO EFM OÊTJNP QBTP 1BSB FTDPHFS MB EJSFDDJÓO EF EFTDFOTP QBSB FM QSPCMFNB OPT CBTBSFNPT FO <)VBOH FU BM ;IPV FU BM > FO FTUPT BSUÎDVMPT TF EFėOF MB EJSFDDJÓO EF EFTDFOTP BQMJDBEB BM QSPCMFNB EFM Q-BQMBDJBOP EF MBT GPSNBT RVF EFTDSJCJSFNPT B DPOUJOVBDJÓO .͐ʼnŃĸŃ ĸĹŀ ĸĹňķĹłňŃ Ł̾ň ńŇŃĺŊłĸŃ ńŇĹķŃłĸĽķĽŃłĵĸŃ %F BDVFSEP DPO FM UFYUP EF <)VBOH FU BM > UPNBSFNPT MB EJSFDDJÓO EF EFTDFOTP wn DPNP Z. ′. Ωh. ∇wn ∇v = −J (yn )v = −. Z. Ωh. |∇yn |. p−2. (∇yn , ∇v) +. Z. fv Ωh. . QBSB UPEP v ∈ S h Z QPS MB GPSNB RVF TF TFMFDDJPOB wn FT FM SFQSFTFOUBOUF EF 3JFT[ QBSB FM GVODJPOBM −J ′ (yn ) FO FM FTQBDJP S h <)VBOH FU BM QÃH > $PO MB EJSFDDJÓO FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP TF DPOPDF DPNP QSFDPOEJDJPOBEP <)VBOH FU BM ;IPV FU BM > &O <)VBOH FU BM QÃH > TF QSVFCB RVF FT VOB EJSFDDJÓO EF EFTDFOTP.
(32) . 0USB GPSNB EF QSFTFOUBS FT B h w n = Ah yn − f h. . DPO B h = (bij ) EPOEF bij =. XZ. τ ∈T h. τ. QBSB i, j = 1, . . . , n. (∇ϕi , ∇ϕj ),. . Ah = (aij ) Z FYQSFTBOEP aij DPO MB EFėOJDJÓO EF |∇h y| EBEB FO Z MB GVODJÓO ξ EFėOJEB FO UFOFNPT aij =. XZ. τ ∈T h. τ. . h. ξ ∇ y. Z f h = (fj ) EPOEF fj =. p−2. XZ. τ ∈T h. ∇ϕi , ∇ϕj dx QBSB i, j = 1, . . . , n f ϕj. . QBSB j = 1, . . . , n.. . τ. Z FWBMVBOEP f FO FM DFOUSP EF NBTB τ UFOFNPT RVF Z. 1 f ϕj dx ≈ EFU 6 τ. !. x2 − x1 x3 − x1 y2 − y1 y3 − y1. !. . f (xc , yc ). .͐ʼnŃĸŃ ĸĹŀ ĸĹňķĹłňŃ Ł̾ň ńŇŃĺŊłĸŃ ńŃłĸĹŇĵĸŃ $POTJEFSBOEP FM FTQBDJP S h ֒→ H01 (Ωh ) EFėOJEP FO DPO MB OPSNB QPOEFSBEB |·. |2yn. =. Z. Ω. (ξ + |∇yn |p−2 )|∇ · |2. . UPNBSFNPT MB EJSFDDJÓO EF EFTDFOTP wn EBEB FO <)VBOH FU BM ;IPV FU BM > DPNP Z. Ωh. (ǫ + |∇yn |. p−2. ′. )∇wn ∇v = −J (yn )v = −. Z. Ωh. |∇yn |. p−2. (∇yn , ∇v) +. Z. fv Ωh. . EPOEF FM QBSÃNFUSP ǫ FT VOB DPOTUBOUF QPTJUJWB QFRVFÒB MB RVF QFSNJUF BM BMHPSJUNP HFOFSBS VO SF TVMUBEP BÙO TJ ∇y = 0 <)VBOH FU BM QÃH > Z QPS MB GPSNB EF TFMFDDJÓO wn FT FM SFQSFTFOUBOUF EF 3JFT[ EFM GVODJPOBM J ′ (yn ) FO FM FTQBDJP S h DPO MB OPSNB &M NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP DPO MB EJSFDDJÓO wn EFėOJEB FO TF EFOPNJOB QPO.
(33) EFSBEP <)VBOH FU BM ;IPV FU BM > -B FYQSFTJÓO TF QVFEF SFQSFTFOUBS DPNP B h w n = Ah yn − f h. . DPO B h = (bij ) EPOEF bij =. XZ . τ ∈T h. τ. ǫ + ξ(∇h y)p−2 (∇ϕi , ∇ϕj ),. QBSB i, j = 1, . . . , n. . DPO Ah Z f h EBEBT FO Z SFTQFDUJWBNFOUF -ŃłĻĽʼnŊĸ ĸĹŀ ńĵňŃ &M NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP DPOTJTUFO FO FODPOUSBS B QBSUJS EFM QBTP xn VOB EJSFDDJÓO wn RVF OPT QFSNJUB DBMDVMBS FM TJHVJFOUF QBTP xn+1 Z DPNP WJNPT FO FM DBQÎUVMP MB MPOHJUVE RVF QPEFNPT NPWFSOPT FO MB EJSFDDJÓO wn TF BQSPYJNB SFTPMWJFOEP FM QSPCMFNB EF NJOJNJ[BDJÓO NJO J(yn + αwn ) α>0. . EPOEF α TF EFOPNJOB MBSHP EFM QBTP Z TV TPMVDJÓO OP TF DBMDVMB EF NBOFSB FYBDUB QPS FM DPTUP DPNQV UBDJPOBM RVF SFRVJFSF 1BSB SFTPMWFS OVFTUSP QSPCMFNB WBNPT B VUJMJ[BS MB DPOEJDJÓO EF "SNJKP </PDF EBM BOE 8SJHIU > QSFTFOUBEB FO MB EFėOJDJÓO RVF BQMJDBEB BM QSPCMFNB EFM Q-BQMBDJBOP EJTDSFUJ[BEP FTUÃ EBEP QPS MB TJHVJFOUF EFėOJDJÓO %FėOJDJÓO $POEJDJÓO EF "SNJKP QBSB FM Q-BQMBDJBOP %BEPT MPT FTDBMBSFT s, ρ > 0 DPO ρ ∈ (0, 1) Z σ ∈ (0, 1/2) 5PNBNPT α ∈ {s, sρ, sρ2 , · · · } J(yn ) − J(yn − αwn ) ≥ −σαJ ′ (yn )v · wn 1BSB BQMJDBS MB DPOEJDJÓO EF "SNJKP DBEB FO EFCFNPT TFHVJS MPT QBTPT TJHVJFOUFT. .
(34) "MHPSJUNP "SNJKP 'JKBS s, ρ, σ y0 J1 = J(y0 ) Z n = 0 SFQFUJS 5PNBS y = y0 %FėOJS arm FM DSJUFSJP EF QBSBEB EBEP FO "DUVBMJ[BS α = sρ1/n y = y + αwn $BMDVMBS arm = J1 − J(y) IBTUB RVF MB DPOEJDJÓO EF QBSBEB TF DVNQMB %FTQVÊT EF EFėOJS MB MPOHJUVE EFM QBTP MB EJSFDDJÓO EF EFTDFOTP Z Z EBEB MB DPOWFSHFODJB QSFTFOUBEB FO <)VBOH FU BM TFDDJÓO > QPEFNPT BQMJDBS FM NÊUPEP EFM EFTDFO TP NÃT QSPGVOEP BM QSPCMFNB EF NJOJNJ[BDJÓO RVF DBSBDUFSJ[B BM QSPCMFNB EFM Q-BQMBDJBOP NFEJBOUF MPT TJHVJFOUFT QBTPT <%F -PT 3FZFT > "MHPSJUNP 'JKBS y0 Z n = 0 SFQFUJS &TDPHFS DSJUFSJP EF QBSBEB &ODPOUSBS wn DPO %FUFSNJOBS αn VUJMJ[BOEP DPO FM BMHPSJUNP EF "SNJKP "DUVBMJ[BS y = y + αn wn IBTUB RVF MB DPOEJDJÓO EF QBSBEB TF DVNQMB. . &ŎńĹŇĽŁĹłʼnŃň /ŊŁ͐ŇĽķŃň. &O FTUB TFDDJÓO QSFTFOUBSFNPT EPT FYQFSJNFOUPT OVNÊSJDPT MPT DVBMFT NPTUSBSBO FM SFOEJNJFOUP TF MPT NÊUPEPT EFM EFTDFOTP NÃT QSPGVOEP QSFDPOEJDJPOBEP Z QPOEFSBEP SFTQFDUJWBNFOUF &ŎńĹŇĽŁĹłʼnŃ 1BSB FTUF QSPCMFNB WBNPT B DPOTJEFSBS FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QPOEFSBEP FO FM DVBESBEP VOJEBE Z UPNBSFNPT FM UBNBÒP B MB NBMMB EF 202 (h = 1/20) Z p = 1, 5 QBSB SFTPMWFS NJO J(y h ). y h ∈S h. .
(35) DPO. Z. Z. f yh.. . f (x1 , x2 ) = TJO(πx1 ) · TJO(πx2 ). . h. J(y ) =. h p. Ωh. |∇y | −. 1BSB FTUF FYQFSJNFOUP UPNBSFNPT f DPNP MB GVODJÓO. Ωh. : ėKBSFNPT y0 DPNP MB TPMVDJÓO EFM QSPCMFNB − △ y0 = TJO(πx1 ) · TJO(πx2 ) . y0. FO [0, 1] × [0, 1]. . FO ∂([0, 1] × [0, 1]). =0. 1BSB BQMJDBS FM NÊUPEP EFM EFTDFOTP QPOEFSBEP JOJDJBMJ[BSFNPT FM QSPCMFNB EBOEP y = y0 error = 1 ξ = 1 × 10−4 Z w0 DPNP Z. Ωh. (ǫ + |∇y0 |. −0.5. )∇w0 ∇v = −. Z. Ωh. |∇y0 |. −0.5. (∇y0 , ∇v) +. Z. fv Ωh. . Z wn DPNP Z. Ωh. (ǫ + |∇yn |. −0.5. )∇wn ∇v = −. Z. Ωh. |∇yn |. −0.5. (∇yn , ∇v) +. Z. fv Ωh. . %FėOJNPT FM FSSPS EF BQSPYJNBDJÓO DPNP ′. error = |J (y)w| = −. Z. Ωh. |∇y|. −0.5. (∇y, ∇w) +. Z. fw Ωh. . Z FM DSJUFSJP EF QBSBEB DPNP error ≤ 1 × 10−4 1PS PUSP MBEP EFėOJSFNPT α DPNP MB MPOHJUVE EFM QBTP NFEJBOUF MB DPOEJDJÓO EF "SNJKP UPNBOEP J1 = J(y) Z MBT DPOTUBOUFT ρ = 0.8 σ = 1 × 10−4 arm = 1 × 106 , s = 0.7 Z FM DSJUFSJP EF QBSBEB EBEB FO "M BQMJDBS FM BMHPSJUNP EFM EFTDFOTP NÃT QSPGVOEP QPOEFSBEP BM QSPCMFNB PCUVWJNPT MPT TJHVJFOUFT SFTVMUBEPT |J ′ (y)v| ×10−4. /ÙNFSP EF JUFSBDJPOFT 5JFNQP . T. "EFNÃT yh PCUFOJEB QPS FM BMHPSJUNP UJFOF MB TJHVJFOUF SFQSFTFOUBDJÓO HSÃėDB.
(36) . 'JHVSB 4PMVDJÓO EFM QSPCMFNB . &O FM HSÃėDP QPEFNPT WFS VOB TVQFSėDJF SFHVMBS RVF SFQSFTFOUB B MB TPMVDJÓO y h EF . 'JHVSB &SSPS EF BQSPYJNBDJÓO. &M HSÃėDP NVFTUSB DPNP FM FSSPS EF BQSPYJNBDJÓO EFTDJFOEF FO DBEB JUFSBDJÓO EFM NÊUPEP &O FTUF FYQFSJNFOUP QPEFNPT WFS RVF FM FSSPS EFTDJFOEF EF GPSNB DBTJ MJOFBM DPO DBEB JUFSBDJÓO.
(37) &ŎńĹŇĽŁĹłʼnŃ 1BSB FTUF QSPCMFNB WBNPT B DPOTJEFSBS FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QPOEFSBEP FO FM DJSDVMP Z UPNBSFNPT FM UBNBÒP B MB NBMMB EF 302 (h = 1/30) Z p = 9 QBSB SFTPMWFS NJO J(y h ). . y h ∈S h. DPO h. J(y ) =. Z. h 9. Ωh. |∇y | −. 1BSB FTUF FYQFSJNFOUP UPNBSFNPT f DPNP MB GVODJÓO. Z. f yh. . Ωh. . f (x1 , x2 ) = 10 : ėKBSFNPT y0 DPNP MB TPMVDJÓO EFM QSPCMFNB − △ y0 = 10 . y0 =. FO Ωh FO ∂(Ωh ). 0. 1BSB BQMJDBS FM NÊUPEP EFM EFTDFOTP QPOEFSBEP JOJDJBMJ[BSFNPT FM QSPCMFNB EBOEP y = y0 error = 1 ǫ = 1 × 10−4 Z w0 DPNP Z Z wn. Z. Ωh. Ωh. ∇w0 ∇v = −. Z. ∇wn ∇v = −. Z. %FėOJNPT FM FSSPS EF BQSPYJNBDJÓO DPNP ′. 7. Ωh. |∇y0 | (∇y0 , ∇v) +. 7. Ωh. error = |J (y)w| = −. |∇yn | (∇yn , ∇v) +. Z. 7. Ωh. Z. Z. |∇y| (∇y, ∇w) +. fv. . fv. . Ωh. Ωh. Z. fw Ωh. . Z FM DSJUFSJP EF QBSBEB DPNP error ≤ 1 × 10−4 1PS PUSP MBEP EFėOJSFNPT B MB MPOHJUVE EF QBTP α NFEJBOUF MB DPOEJDJÓO EF "SNJKP 5PNBOEP J1 = J(y) Z MBT DPOTUBOUFT ρ = 0.5 Z σ = 1 × 10−4 arm = 1 × 105 , s = 0.5 %FTQVÊT EF BQMJDBS FM BMHPSJUNP EFM EFTDFOTP NÃT QSPGVOEP QPOEFSBEP BM QSPCMFNB PCUVWJNPT MPT SFTVMUBEPT EF MB TJHVJFOUF UBCMB.
(38) |J ′ (y)v| ×10−4. /ÙNFSP EF JUFSBDJPOFT 5JFNQP . T. "EFNÃT yh PCUFOJEB QPS FM BMHPSJUNP UJFOF MB TJHVJFOUF SFQSFTFOUBDJÓO HSÃėDB. 'JHVSB 4PMVDJÓO EF MB FDVBDJÓO . 'JHVSB error EF BQSPYJNBDJÓO. &M HSÃėDP SFQSFTFOUB FM FSSPS EF BQSPYJNBDJÓO FO DBEB JUFSBDJÓO EFM NÊUPEP &O MB DVSWB TF QVFEF WFS DPNP PTDJMB FM FSSPS FO BMHVOBT JUFSBDJPOFT FTUF DPNQPSUBNJFOUP EFM FSSPS FT FM FTQFSBEP FO MB BQMJDBDJÓO EFM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP.
(39) i*WF MFBSOFE UIBU QFPQMF XJMM GPSHFU XIBU ZPV TBJE QFPQMF XJMM GPSHFU XIBU ZPV EJE CVU QFPQMF XJMM OFWFS GPSHFU IPX ZPV NBEF UIFN GFFMu .BZB "OHFMPV. 4 &TUVEJP EFM ĚVJEP EF )FSTDIFM#VMLMFZ &O FTUF DBQÎUVMP BOBMJ[BSFNPT FM DPODFQUP EF ĚVJEP WJTDPQMÃTUJDP 1BSB MP RVF SFWJTBSFNPT MFZFT Z DPODFQUPT RVF EFTDSJCFO FM DPNQPSUBNJFOUP EF FTUPT NBUFSJBMFT -VFHP QSFTFOUBSFNPT MPT NPEFMPT DMÃTJDPT EF ĚVJEPT WJTDPQMÃTUJDPT Z OPT DPODFOUSBSFNPT FO FM FTUVEJP EFM ĚVJEP EF )FSTDIFM#VMLMFZ. . $ĵŇĵķʼnĹŇ͠ňʼnĽķĵň ĺ͠ňĽķĵň ĸĹ ŀŃň ĺŀŊĽĸŃň. &O FTUB TFDDJÓO QSFTFOUBSFNPT DPODFQUPT GÎTJDPT RVF OPT BZVEBSÃO B DPNQSFOEFS FM NPWJNJFOUP EF VO ĚVJEP FO VO NFEJP DPOUJOVP 4FB Z ∈ Rn VOB SFHJÓO EF VO ĚVJEP x ∈ Z QBSUÎDVMB P QVOUP EFM ĚVJEP RVF TF NVFWF FO VO UJFNQP t y(x, t) MB WFMPDJEBE EF MB QBSUÎDVMB x FO Z Z EFOTJEBE EF NBTB ρ(x, t) %FėOJDJÓO 1SJODJQJP EF DPOTFSWBDJÓO EF NBTB &M QSJODJQJP EF DPOTFSWBDJÓO EF NBTB FO GPSNB EJGFSFODJBM FTUBCMFDF RVF ∂ρ + ∇ · (ρy) = 0 ∂t FTUB FDVBDJÓO FT DPOPDJEB DPNP FDVBDJÓO EF DPOUJOVJEBE <$IPSJO BOE .BSTEFO > %FėOJDJÓO #BMBODF EF NPNFOUP &O VO ĚVJEP FO NPWJNJFOUP TF QSPEVDF VO HSBEJBOUF EF WFMPDJ EBE Z VOB USBOTGFSFODJB EF NPNFOUP QPS WFMPDJEBE EF ÃSFB Z TV GPSNB EJGFSFODJBM FT ρ. ∂y = −∇p + ρf ∂t . .
(40) %FėOJDJÓO 'MVJEPT JODPNQSFTJCMFT &O MPT ĚVJEPT JODPNQSFTJCMFT MB EFOTJEBE EVSBOUF FM ĚVKP QFS NBOFDF DPOTUBOUF ρ = c EPOEF c FT VOB DPOTUBOUF <-BOEBV BOE -JGTIJU[ > 1BSB MPT ĚVJEPT JODPN QSFTJCMFT FM QSJODJQJP EF DPOTFSWBDJÓO EF NBTB EBEP FO TF SFEVDF B ∇·y =0. . 'ŀŊĽĸŃň ŋĽňķŃňŃň 1BSB PCUFOFS MBT FDVBDJPOFT RVF EFTDSJCFO VO ĚVJEP WJTDPTP JODMVJSFNPT DPODFQUPT BEJDJPOBMFT FO MBT FDVBDJPOFT BOUFSJPSNFOUF WJTUBT QBSB VO ĚVJEP JEFBM %FėOJDJÓO 7JTDPTJEBE -B WJTDPTJEBE P SP[BNJFOUP JOUFSOP BQBSFDF QPS VOB USBOTGFSFODJB EF JN QVMTP FOUSF QVOUPT EFM ĚVJEP DPO VOB WFMPDJEBE B PUSPT QVOUPT EFM ĚVJEP RVF UJFOFO WFMPDJEBE NFOPS <$IP SJO BOE .BSTEFO > : QBSB MPT ĚVJEPT WJTDPTPT TF UJFOF MB GVFS[B FO S QPS VOJEBE EF ÃSFB= −∇p + ∇ · σ EPOEF MB NBUSJ[ σ TF EFOPNJOB UFOTPS EF FTUSÊT <$IPSJO BOE .BSTEFO > %FėOJDJÓO 5FOTPS EF FTUSÊT σ &M UFOTPS EF FTUSÊT EFTDSJCF MBT GVFS[BT JOUFSOBT FO VO NBUFSJBM FMÃTUJDP EFCJEP B TV EFGPSNBDJÓO %FėOJDJÓO &DVBDJÓO EF $BVDIZ -B FDVBDJÓO EF $BVDIZ SFMBDJPOB MB EFėOJDJÓO EF CBMBODF EF NPNFOUP FO VO WPMVNFO SFHVMBS DPO FM DPODFQUP EF UFOTPS EF FTUSÊT ρ. ∂y = ρf + ∇ · σ ∂t. . %FėOJDJÓO &TUSÊT DPSUBOUF τ &T MB GVFS[B RVF TF SFRVJFSF QBSB RVF VOB VOJEBE EF ÃSFB EF VO ĚVJEP P NBUFSJBM TF NVFWB TPCSF PUSP Z TF NJEF FO N/m2 &O MPT ĚVJEPT DPNP FM BHVB BMDPIPM τ FT EJSFDUBNFOUF QSPQPSDJPOBM BM DBNCJP EF WFMPDJEBEFT FOUSF QPTJDJPOFT EJGFSFOUFT EFM ĚVJEP %FėOJDJÓO 5BTB $PSUBOUF E -B UBTB DPSUBOUF P HSBEJBOUF EF MB WFMPDJEBE FT MB NFEJEB EFM DBNCJP ∆y EF MB WFMPDJEBE EBEB QPS -B UBTB DPSUBOUF P UBTB EF EFGPSNBDJÓO FT ∆x E= EPOEF ∆ SFQSFTFOUB MB WBSJBDJÓO. ∆y ∆x. .
(41) . σ=µ EPOEF µ FT MB WJTDPTJEBE EFM ĚVJEP. . ∆y ∆x. . . 'ŀŊĽĸŃň /ĹŌʼnŃłĽĵłŃň ŏ /Ń /ĹŌʼnŃłĽĵłŃň -PT ĚVJEPT RVF UJFOFO WJTDPTJEBE µ 6= 0 TF QVFEFO EJWJEJS FO EPT DMBTFT TFHÙO TV SFMBDJÓO FOUSF FM FTUSÊT EF DPSUF Z MB UBTB DPSUBOUF %FėOJDJÓO 'MVJEPT /FXUPOJBOPT &O MPT ĚVJEPT /FXUPOJBOPT MB SFMBDJÓO FOUSF FM FTUSÊT EF DPSUF Z MB UBTB DPSUBOUF FT MJOFBM FTUF UJQP EF NBUFSJBMFT DVNQMFO DPO 6O DPNQPSUBNJFOUP /FXUPOJBOP TF PCTFSWB FO UPEPT MPT HBTFT FO MÎRVJEPT Z TPMVDJPOFT DPO QPDP QFTP NPMFDVMBS DPNP QPS FKFNQMP BDFJUF BMDPIPM BHVB CFODFOP HMJDFSJOB FM BJSF FOUSF PUSPT %FėOJDJÓO 'MVJEPT /P /FXUPOJBOPT -PT ĚVJEPT OP /FXUPOJBOPT UJFOFO VOB SFMBDJÓO OP MJOFBM FOUSF MB UBTB DPSUBOUF Z FM FTUSÊT EF DPSUF <2VPD )VOH BOE /HPD%JFQ > &TUF HSVQP EF NBUFSJBMFT TF EJWJEF TFHÙO TV SFMBDJÓO FOUSF MB UBTB DPSUF Z FM FTUSÊT EF DPSUF FO 'MVJEPT 1TFVEPQMÃTUJDPT &TUPT ĚVJEPT TF DBSBDUFSJ[BO QPS MB EJTNJOVDJÓO EF TV WJTDPTJEBE Z TV FTUSÊT DPSUBOUF DPO FM BVNFOUP EF MB UBTB EF DPSUF "MHVOPT FKFNQMPT EF FTUF UJQP EF ĚVJEPT TPO MB NPTUB[B BMHVOBT DMBTFT EF QJOUVSBT TVTQFOTJPOFT BDVPTBT EF BSDJMMB FOUSF PUSPT 'MVJEPT %JMBUBOUFT -PT ĚVJEPT EJMBUBOUFT TF DSBDUFSJ[BO QPS FM VO BVNFOUP EFM FTUSÊT DPSUBOUF DPO FM BVNFOUP EF EF MB UBTB EF DPSUBOUF FT EFDJS RVF IBZ VO BVNFOUP EF MB WJTDPTJEBE DPO FM BVNFOUP EF MB UBTB DPSUBOUF &TUF UJQP EF ĚVJEPT TPO NÃT DPNVOFT RVF MPT ĚVJEPT QTFV EPQMÃTUJDPT WBSJPT FKFNQMPT EF ĚVJEPT RVF QPTFFO FTUF DPNQPSUBNJFOUP TPO MB NBOUFDB MBT TVTQFOTJPOFT EF BMNJEÓO MB IBSJOB EF NBÎ[ MB BSFOB NPKBEB EJÓYJEP EF UJUBOJP FUD 'MVJEPT 7JTDPQMÃTUJDPT -PT ĚVJEPT WJTDPQMÃTUJDPT TF DBSBDUFSJ[B QPS TFS DPNQPSUBSTF DPNP VO TÓMJEP IBTUB TPCSFQBTBS VO FTUSÊT DPSUBOUF NÎOJNP Z B QBSUJS EF EJDIP WBMPS DPNJFO[B B ĚVJS DPNP VO MÎRVJEP.
(42) . PT ÃTUJD DPQM T J 7 PT. E. &TUSÊT EF DPSUF. 'MVJ. σy. PT UJD T Ã QM. P. E FV 1T. EP VJ 'M. P BO. J PO XU F T/. T. UFT BO MJ BU %. 5BTB EF DPSUF 'JHVSB (SÃėDP EF EJGFSFOUFT ĚVJEPT TFHÙO TV SFMBDJÓO FOUSF MB UBTB EF DPSUF Z FM FTUSÊT EF DPSUF. 'ŀŊĽĸŃň 7ĽňķŃńŀ̾ňʼnĽķŃň &O FTUÃ TFDDJÓO FTUVEJBSFNPT B MPT ĚVJEPT WJTDPQMÃTUJDPT VOP EF DVZPT NPEFMPT NÃT DPOPDJEPT FT FM NPEFMP EF RVF TPO JNQPSUBOUFT QBSB FTUVEJBS MPT ĚVJEPT EF )FSTDIFM#VMLMFZ %FėOJDJÓO -ÎNJUF FMÃTUJDP σy &M MÎNJUF FMÃTUJDP FT FM FTGVFS[P DPSUBOUF NÎOJNP P FTGVFS[P VNCSBM RVF OFDFTJUB VO ĚVJEP WJTDPQMÃTUJDP QBSB QBTBS EF VO DPNQPSUBNJFOUP TÓMJEP B VOP MÎRVJEP &M MÎNJUF FMÃTUJDP WBSJB EFQFOEJFOEP EFM ĚVJEP -PT ĚVJEPT WJTDPQMÃTUJDPT QPS TFS OP /FXUPOJBOPT UJFOFO VOB SFMBDJÓO OP MJOFBM FOUSF FM FTUSÊT EF DPSUF Z MB UBTB DPSUBOUF OP DVNQMF DPO &TUPT ĚVJEPT TF DPNQPSUBO DPNP VO TÓMJEP SÎHJEP DVBOEP FM FTUÊT DPSUBOUF FT NFOPS BM MÎNJUF FMÃTUJDP Z DPNP VO ĚVJEP FO DBTP DPOUSBSJP 6O DPN QPSUBNJFOUP WJTDPQMÃTUJDP TF FWJEFODJB FO BMHVOPT ĚVJEPT DPNP FM DIPDPMBUF MB BSDJMMB MB QBTUB EF EJFOUFT FNVMTJPOFT NBZPOFTB NBOUFRVJMMB EFSJWBEPT EFM QFUSÓMFP FUD Ex #. F ". Ey. 'JHVSB 'MVJEP WJTDPQMÃTUJDP FOUSF EPT QMBDBT.
(43) &O MB ėHVSB SFQSFTFOUBNPT VO ĚVJEP WJTDPQMÃTUJDP FOUSF EPT QMBDBT <#PHFS BOE )BMNPT 2VPD )VOH BOE /HPD%JFQ > MB GVFS[B QBSB EFTQMB[BS VOB QBSUÎDVMB VOB EJTUBODJB dx FT F dx QBSB VO ĚVJEP /FXUPOJBOP MB QBSUÎDVMB SFDPSSF EFTEF FM QVOUP JOJDJBM IBTUB A NJFOUSBT RVF FO MPT ĚVJEPT WJTDPQMÃTUJDPT TF NPWFSà IBTUB FM QVOUP B EFCJEP B RVF MB GVFS[B BQMJDBEB BM ĚVJEP EFCF TVQFSBS FM MÎNJUF FMÃTUJDP QBSB DPNFO[BS FM NPWJNJFOUP $ŀĵňĽĺĽķĵķĽł ĸĹ ŀŃň ĺŀŊĽĸŃň 7ĽňķŃńŀ̾ňʼnĽķŃň " DPOUJOVBDJÓO QSFTFOUBSFNPT MPT NPEFMPT DMÃTJDPT EF ĚVJEPT WJTDPQMÃTUJDPT RVF TPO #JOHIBN $BTTPO Z )FSTDIFM#VMLMFZ $BTTPO. &TUSÊT EF DPSUF. )FSTDIFM#VMLMFZ n>1 #JOHIBN. )FSTDIFM#VMLMFZ n<1. σy. 5BTB EF DPSUF 'JHVSB .PEFMPT WJTDPQMÃTUJDPT DMÃTJDPT Z TV SFMBDJÓO DPO MB UBTB DPSUBOUF Z FM FTUSÊT EF DPSUF. "OUFT EF SFWJTBS MPT NPEFMPT EF #JOHIBN $BTTPO Z )FSTDIFM#VMLMFZ SFWJTBSFNPT FM TJHVJFOUF DPODFQUP %FėOJDJÓO /ÙNFSP EF 0MESPZE &M OÙNFSP EF 0MESPZE FT VO OÙNFSP BEJNFOTJPOBM RVF SFMBDJPOB QSPQJFEBEFT GÎTJDBT EFM ĚVJEP TV WFMPDJEBE Z MBT DBSBDUFSÎTUJDBT EFM NFEJP Z FTUà EBEP QPS σy cL EPOEF L FT MB MPOHJUVE EFM EVDUP σy FT FM MÎNJUF FMÃTUJDP Z c FT FM HSBEJBOUF EF MB QSFTJÓO Z FTUB EBEP QPS dp c=− dy " DPOUJOVBDJÓO QSFTFOUBSFNPT MPT NPEFMPT EF FTUPT ĚVJEPT FO VOB UVCFSÎB DJSDVMBS EF MPOHJUVE L dp Z c = − DPOTUBOUF QPTJUJWP FT HSBEJBOUF EF MB QSFTJÓO dy Od =.
(44) .PEFMP EF #JOHIBN "TVNJNPT RVF σy FM MÎNJUF FMÃTUJDP EFM ĚVJEP σ FT FM FTUSÊT EF DPSUF Z TV SFMBDJÓO FTUÃ EBEB QPS . σ = σy + µE. EPOEF µ FT FM QBSÃNFUSP EF WJTDPTJEBE Z E FT MB UBTB EF DPSUF : TV WFMPDJEBE DBSBDUFSÎTUJDB 2 V = cLµ Z TV OÙNFSP EF 0MESPZE EBEP QPS Od =. σy cL. . EPOEF 0 ≤ Od ≤ 0.5 <)VJMHPM BOE :PV > &M NPEFMP DMÃTJDP EF #JOHIBN FO <(PO[ÃMF[ "OESBEF )VJMHPM BOE :PV .JUTPVMJT > Od σij = E Eij + Eij si |σij | > Od Eij = 0 si |σij | ≤ Od. EPOEF E FT FM UFOTPS EF EFGPSNBDJÓO Z σij FT FM UFOTPS EF FTUSÊT .PEFMP EF $BTTPO. 1BSB FTUF UJQP EF ĚVJEPT MB SFMBDJÓO FOUSF FM MÎNJUF FMÃTUJDP Z FM FTUSÊT EF DPSUF FTUÃ EBEB QPS √. σ=. √. σy +. p. . Kc E. EPOEF Kc FT VO QBSÃNFUSP EF WJTDPTJEBE Z QPS DPNPEJEBE TF FTDSJCF h. σ = σ y + Kc +. p. σy Kc |E|. − 12. i. E. . cL2 Z TV OÙNFSP EF 0MESPZE EFėOJEP FO Kc &M NPEFMP EF $BTTPO <)VJMHPM BOE :PV .JUTPVMJT > FT 4V WFMPDJEBE DBSBDUFSÎTUJDB FT V =. " 1/2 # Od Od Eij si |σij | > Od σij = E Eij + 1 + 2 E Eij = 0 si |σij | ≤ Od. .
(45) DPO σij FT FM UFOTPS EF FTUSÊT Z Eij FT FT UFOTPS EF EFGPSNBDJÓO 1PS ÙMUJNP FYQMJDBSFNPT DPO NBZPS EFUBMMF FO MB TJHVJFOUF TFDDJÓO FM NPEFMP EF )FSTDIFM#VMLMFZ. . .ŃĸĹŀŃ ĸĹ )ĹŇňķļĹŀ #ŊŀĿŀĹŏ. -PT ĚVJEPT EF )FSTDIFM #VMLMFZ TPO ĚVJEPT OP /FXUPOJBOPT OP DVNQMFO DPO Z WJTDPQMÃT UJDPT DPNÙONFOUF MMBNBEPT NBUFSJBMFT TFNJTÓMJEPT Z TPO HFOFSBMNFOUF JSSFHVMBSFT F JNQSFEFDJCMFT FO TV NPWJNJFOUP QPS TV PSEFOBNJFOUP NPMFDVMBS "MHVOPT FKFNQMPT EF ĚVJEPT EF )FSTDIFM#VMLMFZ TPO QJOUVSBT QSPEVDUPT BMJNFOUJDJPT QPMÎNFSPT QMÃTUJDPT DSVEPT QFTBEPT QSPEVDUPT GBSNBDÊVUJDPT FOUSF PUSPT <.JUTPVMJT > &M NPEFMP NBUFNÃUJDP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ FO VOB UVCFSÎB DJSDVMBS EF MPOHJUVE L σ Z TV FTUSÊT EF DPSUF FT DPO TFDDJÓO L/2 FM OÙNFSP EF 0MESPZE Od = c·L σ = σy + KH |E|n−1 E. . EPOEF σy FT FM MÎNJUF FMÃTUJDP E FT MB UBTB EF DPSUF Z KH FT FM QBSÃNFUSP EF WJTDPTJEBE dp Z MB WFMPDJEBE EFM ĚVJEP DPNP 5PNBOEP VOB QSFTJÓO DPOTUBOUF QPTJUJWB c = − dy. V =L. . 1 cL n KH. . &M NPEFMP EF )FSTDIFM#VMLMFZ FO R3 <#PHFS BOE )BMNPT )VJMHPM BOE :PV .FT TFMNJ FU BM 4BSBNJUP 4BZFE"INFE FU BM > FTUB EBEP QPS Od n−1 σij = E Eij + Et Eij si |σij | > Od Eij = 0 si |σij | ≤ Od. . EPOEF σij FT FM UFOTPS EF FTUSÊT Eij FT VO UFOTPS EF EFGPSNBDJÓO Od FT FM OÙNFSP EF 0MESPZE E FT MB UBTB EF DPSUF Z n OÙNFSP NBZPS RVF DFSP EFOPNJOBEP DPFėDJFOUF EF QPEFS $VBOEP n = 1 FM NPEFMP EF )FSTDIFM#VMLMFZ TF SFEVDF BM NPEFMP EF #JOHIBN : EBEP RVF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ TPO JODPNQSFTJCMFT MBT FDVBDJPOFT RVF EFTDSJCFO TV NP.
(46) WJNJFOUP FTUÃO EBEBT FO Z ∂y =∇ · σ + f ∂t ∇ · y =0. . EPOEF σ FTUÃ EFTDSJUB FO TJ SFFTDSJCJNPT Z UFOFNPT RVF . ∇·σ+f =0. . 'ŀŊĽĸŃ ĸĹ )ĹŇňķļĹŀ#ŊŀĿŀĹŏ Ĺł Ŋłĵ ʼnŊĶĹŇ͠ĵ. &O FTUB TFDDJÓO FTUVEJBSFNPT FM ĚVJEP EF )FSTDIFM#VMLMFZ FTUBDJPOBSJP Z MBNJOBS FO VOB TFDDJÓO DJSDVMBS QBSB MP DVBM UPNBNPT VO TJTUFNB PSUPHPOBM x1 x2 x3 Z BOBMJ[BSFNPT FM NPWJNJFOUP EFM ĚVJEP FOUSF EPT TFDDJPOFT x3 = 0 Z x3 = L EPOEF L FT MB MPOHJUVE EF MB UVCFSÎB x1. . x3 L. x2. $POTJEFSBNPT VO NPWJNJFOUP MBNJOBS FO VOB UVCFSÎB QPS DBNCJP EF QSFTJÓO Z EFTQSFDJBOEP MBT GVFS[BT EF DPOUBDUP FOUSF FM ĚVJEP Z MB UVCFSÎB 5PNBSFNPT VOB QSFTJÓO QPS VOJEBE EF MPOHJUVE DPOTUBOUF F JHVBM B c QPS MP RVF. p(x3 ) = 0 TJ x3 = 0 p(x3 ) = −cL TJ x3 = L $PNP FTUBNPT DPOTJEFSBOEP VO ĚVKP MBNJOBS <(PO[ÃMF[ "OESBEF > MB WFMPDJEBE EFM ĚVJEP TF SFEVDF B ∂y y = (0, 0, y) Z =0 ∂x3.
(47) QPS MP RVF y = y(x1 , x2 ) -B UBTB EF EFGPSNBDJÓO EFTDSJUB FO DPO MB WFMPDJEBE y = y(x1 , x2 ) FTUB EBEB QPS MB NBUSJ[ . 0 1 E= 0 2 ∂y ∂x1. 0 0 ∂y ∂x2. ∂y ∂x1 ∂y ∂x2 0. . . 1PS PUSP MBEP TVTUJUVZFOEP FO MB FDVBDJÓO EF NPWJNJFOUP QBSB MPT ĚVJEPT EF )FSTDIFM#VMLMFZ EBEB FO MB GVFS[B f EBEB QPS MB QSFTJÓO EFTDSJUB UFOFNPT FM TJHVJFOUF TJTUFNB EF FDVBDJPOFT ∂p =0 ∂x1 ∂p =0 ∂x2 ∂σ31 ∂σ32 ∂p ∂x = ∂x + ∂x 3 1 2. Z. p = −cx3. . . "EFNÃT TJ SFFNQMB[BNPT MPT SFTVMUBEPT EF FO FM NPEFMP EF )FSTDIFM#VMLMFZ UFOFNPT RVF ∂σ31 ∂σ32 + , ∂x1 ∂x2 E3,i σ3,i = Od + |E3,1 + E3,2 |n−1 E3,i QBSB i = 1, 2 FO Ω, |E3,1 + E3,2 | y = 0 FO Γ. c =. . &M TJTUFNB TPO MBT FDVBDJPOFT RVF EFTDSJCFO FM NPWJNJFOUP MBNJOBS EF VO ĚVJEP EF )FSTDIFM#VMLMFZ FO VOB UVCFSÎB.
(48) . . 'ŃŇŁŊŀĵķĽł ŋĵŇĽĵķĽŃłĵŀ. &O MB TFDDJÓO BOUFSJPS PCUVWJNPT MBT FDVBDJPOFT RVF EFTDSJCFO FM NPWJNJFOUP EF MPT ĚVJEPT EF )FSTDIFM#VMLMFZ FO GPSNB EJGFSFODJBM FO FTUB TFDDJÓO VUJMJ[BSFNPT MBT FDVBDJPOFT QBSB SFQSFTFOUBS FM QSPCMFNB FO GPSNB WBSJBDJPOBM -B EFTJHVBMEBE WBSJBDJPOBM RVF EFTDSJCF FM NPWJ NJFOUP EF )FSDIFM#VMLMFZ TF QSFTFOUB FO MB TJHVJFOUF QSPQPTJDJÓO. 1SPQPTJDJÓO %BEP y TPMVDJÓO EF &OUPODFT y ∈ W01,n+1 Z TBUJTGBDF MB EFTJHVBMEBE WBSJBDJPOBM EF TFHVOEP UJQP a(y, v − y) + Od j(v) − Od j(y) ≥ (c, v − y) QBSB UPEP v ∈ W01,n+1. . %POEF a(y, v) = j(y) =. Z. ZΩ. (c, v − y) = c. Ω Z. | ▽ y|n−1 (∇y · ∇v) dx |∇y(x)|dx Ω. (v − y)dx. %FNPTUSBDJÓO &O NVMUJQMJDBNPT QPS (v − y) ∈ W01,n+1 F JOUFHSBNPT Z. T. Ω. ∇ · (σ3,1 , σ3,2 ) (v − y)(x) dx = −c. Z. Ω. (v − y)(x)dx QBSB UPEP v ∈ W01,n+1. Ω. (v − y)(x)dx QBSB UPEP v ∈ W01,n+1. *OUFHSBOEP QPS QBSUFT MB FDVBDJÓO BOUFSJPS Z. T. Ω. (σ3,1 , σ3,2 ) ∇(v − y)(x) dx = −c. Z. . 1PS PUSP MBEP EF PCUFOFNPT σ3,1 = |E3,1 + E3,2 |n−1 E3,1 (y) + σ3,2 = |E3,1 + E3,2 |n−1 E3,2 (y) +. Od E3,1 (y) |E3,1 (y) + E3,2 (y)|. . Od E3,2 (y) |E3,1 (y) + E3,2 (y)|. .
(49) SFFNQMB[BNPT Z FO FM MBEP J[RVJFSEP EF MB FDVBDJÓO UFOFNPT Z. . Od E3,1 (y) |E3,1 (y) + E3,2 (y)|n−1 + |E3,1 (y) + E3,2 (y)| Ω ·. E3,2 (y)|n−1 · +E3,2 (y) +. Od E3,2 (y) |E3,1 (y) + E3,2 (y)|. !. , |E3,1 (y) + E3,2 (y)|T. ! · (∇(v − y)(x))dx. . "HSVQBOEP UÊSNJOPT FO Z. Ω. ". |E3,1 (y) + E3,2 (y)|n−1 +. . E3,1 (y), E3,2 (y) ·. QBSB UPEP v ∈. Od |E3,1 (y) + E3,2 (y)|. !T. · (▽(v − y))(x)dx . W01,n+1 . 1PS MB EFėOJDJÓO EFM UFOTPS EF FTUSÊT EBEB FO UFOFNPT RVF ∇y = E3,1 (y) + E3,2 (y). . BTÎ MB FYQSFTJÓO FT Z Ω. |∇y|. n−1. ∇y (∇y) + Od |∇y|. . · (∇(v − y))dx. . Z SFFNQMB[BOEP FO MB FDVBDJÓO PCUFOFNPT Z. Ω. ". |∇y|. n−1. . (∇y) ·(∇(v−y)(x))dx+. Z Ω. Z ∇y ·(∇(v−y)(x))dx = c (v−y)(x)dx Od |∇y| Ω . QBSB UPEP v ∈ W01,n+1 1PS PUSP MBEP UPNBOEP EF FM UÊSNJOP Z Ω. ∇y Od |∇y|. . · ∇(v − y))(x) dx.
(50) UFNFNPT Z Ω. ∇y Od |∇y|. . · ∇(v − y)) = Od. ≤ Od. Z Ω. Z. Z ∇y ∇y , ∇v dx − Od , ∇y dx |∇y| |∇y| Ω. |∇y||∇v| dx − Od |∇y|. Ω. Z. Ω. |∇y| dx. Z BQMJDBNPT MB EFTJHVBMEBE EF $BVDIZ4DIXBS[ Z Ω. ∇y Od |∇y|. . · (∇(v − y)) ≤ Od. Z. Ω. |∇v|dx − Od. Z. Ω. |∇y|dx. . EF Z MB EFTJHVBMEBE WBSJBDJPOBM FT Z. Ω. |∇y|. n−1. . (∇y) · (∇(v − y)(x))dx + Od. Z. Ω. |∇v|dx − Od. Z. Ω. |∇y|dx ≤ c. Z. Ω. (v − y)(x)dx . TJ UPNBNPT a(v, y) =. j(y) =. Z Z. (c, v − y) = c. Ω. |∇y|n−1 (∇y · ∇v) dx. Ω. |∇y(x)|dx. Z. Ω. (v − y)dx. FO MB EFTJHVBMEBE WBSJBDJPOBM FT a(y, v − y) + Od j(v) − Od j(y) ≥ (c, v − y) QBSB UPEP v ∈ W01,n+1 .. .
(51) . . 'ŃŇŁŊŀĵķĽł ķŃŁŃ Ŋł ńŇŃĶŀĹŁĵ ĸĹ ŃńʼnĽŁĽŐĵķĽł. &O FTUB TFDDJÓO QSFTFOUBSFNPT FM NPEFMP EF )FSTDIFM#VMLMFZ FO VOB UVCFSÎB DPNP VO QSPCMFNB EF NJOJNJ[BDJÓO &TUB GPSNVMBDJÓO OPT QFSNJUJSà BQMJDBS FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP QBSB IBMMBS VOB TPMVDJÓO BM QSPCMFNB 5PNBSFNPT FM QSPCMFNB EF NJOJNJ[BDJÓO NJO. y∈W01,n+1. EPOEF J(y) =. . 1 n+1. Z. Ω. |∇y|. n+1. . J(y). dx + Od. Z. Ω. |∇y|dx − c. Z. y dx. . Ω. %POEF MB TPMVDJÓO EFM QSPCMFNB FT TPMVDJÓO EF MB EFTJHVBMEBE WBSJBDJPOBM &ŎĽňʼnĹłķĽĵ ŏ ŊłĽķĽĸĵĸ ĸĹ ŀĵ ňŃŀŊķĽł ńĵŇĵ Ĺŀ .ŃĸĹŀŃň ĸĹ )ĹŇķļĹŀ#ŊŀĿŀĹŏ 1BSB BQMJDBS DVBMRVJFS BMHPSJUNP EF SFTPMVDJÓO OVNÊSJDB B QSJNFSP QSPCBSFNPT RVF QPTFF VOB TPMVDJÓO ÙOJDB 5FPSFNB 4FB J : W01,n+1 −→ R EFėOJEP QPS 1 J(y) = n+1 QPTFF VO QVOUP NJOJNBM. Z. Ω. |∇y|. n+1. dx + Od. Z. Ω. |∇y|dx − c. Z. y dx. . Ω. %FNPTUSBDJÓO %BEP RVF W01,n+1 FT VO FTQBDJP EF #BOBDI SFĚFYJWP BDPUBEP Z OP WBDÎP QPEF NPT VUJMJ[BS FM 5FPSFNB QBSB NPTUSBS RVF J UJFOF VO QVOUP NJOJNBM 1BSB BQMJDBS FM 5FPSFNB OFDFTJUBNPT QSPCBS RVF J FT VO GVODJPOBM RVBTJDPOWFYP 1BSB QSPCBS RVF J : W01,n+1 −→ R FT VO GVODJPOBM RVBTJDPOWFYP QSPCBSFNPT RVF FT DPOWFYP 5PNFNPT u, y ∈ W01,n+1 DPO n ≥ 1 Z λ ∈ [0, 1] Z Z 1 n+1 |∇(λu + (1 − λ)y)| dx + Od |∇(λu + (1 − λ)y)| dx J(λy + (1 − λ)y) = n+1 Ω Ω Z − c (λu + (1 − λ)y) dx Ω.
(52) %BEP RVF |∇(λu + (1 − λ)y)| ≥ 0 TBCFNPT RVF |∇(λu + (1 − λ)y)|n+1 FT VOB GVODJÓO DSFDJFOUF FO R+ QBSB n > 0 QPS MP RVF J(λu + (1 − λ)y) ≤. . 1 n+1. + Od. Z. Z. Ω. |∇(λu)| Z. |∇λu| +. Ω. Ω. n+1. dx +. Z. Ω. |∇((1 − λ)y)|. n+1. . dx +. Z Z (λu) dx + (1 − λ)y dx |(1 + λ)∇y| du − c Ω. Ω. "IPSB λ ≤ 1 (1 − λ) ≤ 1 UFOFNPT RVF λ ≥ λn+1 Z (1 − λ) ≥ (1 − λ)n+1 MP RVF JNQMJDB J(λu + (1 − λ)y) ≤. . 1 n+1. Z Z n+1 n+1 λ |∇u| dx + (1 − λ) |∇y| dx + Ω. Ω. Z Z Z Z + Od λ |∇u| + (1 + λ) |∇y| dx − c λ (u) dx + (1 − λ) y) dx Ω. ≤λ. ". 1 n+1. + (1 − λ). ". Ω. Z. Ω. |∇u|. 1 n+1. Z. n+1. Ω. dx + Od. Ω. Z. Ω. |∇u|dx − c. |∇y|n+1 dx + Od. Z. Ω. Ω. Z. #. u dx + Ω. |∇y|dx − c. Z. y dx Ω. #. ≤ λJ(u) + (1 − λ)J(y) J FT VO PQFSBEPS DPOWFYP Z BQMJDBOEP FM 5FPSFNB J FT RVBTJDPOWFYP 1PS FM 5FPSFNB J QP TFF VO QVOUP NJOJNBM FO W01,n+1 1PS FM 5FPSFNB BOUFSJPS TBCFNPT RVF FM QSPCMFNB QPTFF VOB TPMVDJÓO "IPSB WBNPT BOBMJ[BS MB VOJDJEBE EF MB TPMVDJÓO FO W01,n+1 QBSB FM QSPCMFNB QBSB MP DVBM QSFTFOUBSFNPT FM TJHVJFOUF UFPSFNB <-JPOT QÃH > 1BSB BQMJDBS FM UFPSFNB B OVFTUSP GVODJPOBM UFOFNPT RVF QSPCBS RVF J FT FTUSJDUBNFOUF DPO WFYP 1SPQPTJDJÓO &M GVODJPOBM J : W01,n+1 → R FT FTUSJDUBNFOUF DPOWFYP.
(53) %FNPTUSBDJÓO 5PNFNPT u, y ∈ W01,n+1 DPO n ≥ 1 Z λ ∈ (0, 1) Z Z 1 n+1 J(λy + (1 − λ)y) = |∇(λu + (1 − λ)y)| dx + Od |∇(λu + (1 − λ)y)| dx n+1 Ω Ω Z − c (λu + (1 − λ)y) dx Ω. %BEP RVF |∇(λu + (1 − λ)y)| ≥ 0 TBCFNPT RVF |∇(λu + (1 − λ)y)|n+1 FT VOB GVODJÓO DSFDJFOUF FO R+ QBSB n > 0 QPS MP RVF J(λu + (1 − λ)y) ≤. . 1 n+1. + Od. Z. Ω. Z. Ω. |∇(λu)|. |∇λu| +. Z. Ω. n+1. dx +. Z. Ω. |∇((1 − λ)y)|. n+1. . dx +. Z Z (λu) dx + (1 − λ)y dx |(1 + λ)∇y| du − c Ω. Ω. "IPSB λ < 1 (1 − λ) < 1 UFOFNPT RVF λ > λn+1 Z (1 − λ) > (1 − λ)n+1 MP RVF JNQMJDB J(λu + (1 − λ)y) <. . 1 n+1. Z Z n+1 n+1 λ |∇u| dx + (1 − λ) |∇y| dx + Ω. Ω. Z Z Z Z + Od λ |∇u| + (1 + λ) |∇y| dx − c λ (u) dx + (1 − λ) y) dx Ω. Ω. Ω. Ω. # Z Z Z 1 |∇u|n+1 dx + Od |∇u|dx − c u dx + <λ n+1 Ω Ω Ω " # Z Z Z 1 n+1 + (1 − λ) |∇y| dx + Od |∇y|dx − c y dx n+1 Ω Ω Ω ". < λJ(u) + (1 − λ)J(y) J FT VO FTUSJDUBNFOUF DPOWFYP Z QPEFNPT BQMJDBS FM 5FPSFNB . .
(54) 4J UPNBNPT FM GVODJPOBM UFOFNPT J(y) =. . 1 n+1. Z. . 1 n+1. Z. n+1. Ω. |∇y|. Ω. |∇y|n+1 dx. dx + Od. Z. Ω. |∇y|dx − c. Z. y dx Ω. Z J1 (y) =. J2 (y) = Od. Z. Ω. |∇y|dx − c. Z. y dx Ω. %POEF J FT VO PQFSBEPS FTUSJDUBNFOUF DPOWFYP J1 FT VO PQFSBEPS EJGFSFODJBCMF Z J2 FT DPOWFYP QPS MB DPOWFYJEBE EF J QPEFNPT BQMJDBS FM UFPSFNB B : QPEFNPT BTFHVSBS RVF FYJTUF VO ÙOJDP u ∈ W01,n+1 UBM RVF J1′ (u) · (y − u) + J2 (y) − J2 (u) ≥ 0. . 1BSB PCUFOFS MB FYQSFTJÓO FO GPSNB FYQMJDJUB QSJNFSP EFSJWBSFNPT J1. 1 n+1. Z. Ω. | ▽ y(x)|. n−1. ′. (u) dx =. Z. Ω. |∇y|n−1 (∇y, ∇u) dx. . : SFFNQMB[BOEP J2 Z FO TF UJFOF RVF u ∈ W01,n+1 TF DBSBDUFSJ[B BTÎ Z. Ω. |∇y|. n−1. (∇y, ∇v) dx + Od. Z. Ω. |∇y|dx − Od. Z. Ω. |∇u|dx ≥ c. Z. Ω. y dx − c. Z. u dx Ω.
(55) i6OMFTT TPNFPOF MJLF ZPV DBSFT B XIPMF BXGVM MPU /PUIJOH JT HPJOH UP HFU CFĪFS *UT OPUu %S 4FVTT ĉF -PSBY. 5 4PMVDJÓO /VNÊSJDB EFM 1SPCMFNB EF )FSTDIFM#VMLMFZ &O FTUF DBQÎUVMP SFTPMWFSFNPT OVNÊSJDBNFOUF MB FDVBDJÓO EF NPWJNJFOUP EF MPT ĚVJEPT EF )FSTDIFM #VMLMFZ 1BSB MP DVBM USBOTGPSNBSFNPT FM PQFSBEPS J EFėOJEP FO FO VO PQFSBEPS EJGFSFODJBCMF NFEJBOUF VOB SFHVMBSJ[BDJÓO EF )VCFS %FTQVÊT EJTDSFUJ[BSFNPT FM QSPCMFNB SFHVMBSJ[BEP BQMJDBOEP FM NÊUPEP EF FMFNFOUPT ėOJUPT -VF HP VUJMJ[BSFNPT FM NÊUPEP JUFSBUJWP EFM EFTDFOTP NÃT QSPGVOEP QBSB IBMMBS TV TPMVDJÓO. . 3ĹĻŊŀĵŇĽŐĵķĽł ĸĹŀ ńŇŃĶŀĹŁĵ. 3FUPNBOEP FM QSPCMFNB EF NJOJNJ[BDJÓO NJO. y∈W01,n+1. DPO. 1 J(y) = n+1. Z. Ω. |∇y|. n+1. . J(y). dx + Od. Z. Ω. |∇y| dx − c. Z. y dx. . Ω. 4BCFNPT RVF UJFOF TPMVDJÓO ÙOJDB Z J FT GVODJPOBM OP EJGFSFODJBCMF EFCJEP B TV TFHVOEP UÊS NJOP FTUF IFDIP JNQJEF BQMJDBS EJSFDUBNFOUF FM NÊUPEP EFM EFTDFOTP NÃT QSPGVOEP B OVFTUSP QSP CMFNB "EFNÃT TF QVFEF UFOFS QSPCMFNBT EF JOOFTUBCJMJEBE QPS FM UÊSNJOP OP EJGFSFODJBCMF .
(56) 3ĹĻŊŀĵŇĽŐĵķĽł ĸĹ )ŊĶĹŇ 1SPQPOFNPT VTBS VOB SFHVMBSJ[BDJÓO EF UJQP )VCFS EFM UÊSNJOP OP EJGFSFODJBCMF OP EJGFSFODJBCMF QPSRVF FTUF QSPDFEJNJFOUP IB TJEP VUJMJ[BEP FO NPEFMJ[BDJPOFT TJNJMBSFT DPNP #JOHIBN <%F MPT 3FZFT (PO[ÃMF[ "OESBEF > -B SFHVMBSJ[BDJÓO BQMJDBEB BM UÊSNJOP OP EJGFSFODJBCMF EFM GVODJPOBM FTUÃ EBEB QPS ψγ : Rn −→ R. ψγ : ∇y −→ ψγ (∇y) =. Od Od2 si |∇y| > Od|∇y| − 2γ γ γ |∇y|2 2. . . Od si |∇y| ≤ γ. EPOEF Od FT FM OÙNFSP EF 0MESPZE Z γ FT VO OÙNFSP QPTJUJWP NBZPS RVF DFSP. 1.5 1.4 1.2 1 1 0.8 0.6 0.5. 0.4 0.2 0 2. 0. 0. 2 0. −2. −1. 0. 'JHVSB 7BMPS BCTPMVUP. 1. −2. 2. −2. −1. 0. 1. 'JHVSB 3FHVMBSJ[BDJÓO EF )VCFS . (SÃėDBNFOUF QPEFNPT WFS RVF MB GVODJÓO ψ FO MB ėHVSB TVBWJ[B MB DVSWB EF | · | EF MB ėHVSB Z TJ UPNBNPT γ MP TVėDJFOUFNFOUF HSBOEF ψ TF BQSPYJNB B | · | -FNB ψγ FT VOB GVODJÓO EJGFSFODJBCMF QBSB UPEP w ∈ Rn %FNPTUSBDJÓO %BEP RVF ψγ FT VOB GVODJÓO QPS USBNPT QSPCFNPT QSJNFSP RVF MJN ψ FYJTUF QPS MP RVF WBNPT B FODPOUSBS MPT MÎNJUFT MBUFSBMFT Z QSPCBS RVF FTUPT TPO JHVBMFT. |w|→ Od γ. 2.
Documento similar
Es el tipo de valoración —y los cambios fisiológicos como tales; ni tampoco las Es el tipo de valoración —y los cambios fisiológicos como tales; ni tampoco las creencias o la
T ) Or ccmifsion y mandato de los feñores del Confejo de fu A K cezavhe viftocidiícurío que ha hecho Don f rancifco deMiranr da y p3z,Capdlan de fu Mageíbd^en ia Real Capilla de
In the PACS spectrum we can clearly see that the continuum emission and much of the line emission is focused near the position of the submillimeter source, 2MASS 20581767+4353310,
c io , promete fu Mageílad Catholica aplicar de fu parce codo ei cuidado pofiiblc para abolirías.,, y hazerlas evitar por todos medios en lo venidero ; y igualmence fu
ANÁLISIS TRANSITORIO DEL COMPORTAMIENTO DEL FLUJO DE FANGOS PARA LA OPTIMIZACIÓN DEL CONSUMO DE ENERGÍA MEDIANTE LA DETERMINACIÓN DE UN RÉGIMEN DE FUNCIONAMIENTO INTERMITENTE
Durante la exploración, puede borrar temporalmente canales específicos de la secuencia de exploración pulsando la tecla programada con la función borrar/añadir a exploración
Spatial Distribution of Continuum and Lines Although we detect substantial line emission from three sources (V1057 Cyg, V1735 Cyg, and HBC 722), we must consider whether the lines
químicamente de otra forma, con exclusión de los de la partida 15.16; mezclas o preparaciones no alimenticias de grasas o de aceites animales o vegetales, o de fracciones
