Finite element methods or Elp oblems Variational formulation The poisson problem March19&31.2003
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1 Motivation The Poisson problem has a strong formulation a minimization formulation and a weak formulation T weak formulations are more general than the strong formulation in terms of regularity and admissible data SLIDE 2 The minimization/weak formulations are defined by: a space X; a bilinear The minimization/weak formulations identify ESSENTIAL boundary conditions NATURAL boundary conditions ed in a The points of departure for the finite element method are the weak formulation(more generally) the minimization statement (if a is SPD) 2 The dirichlet problem 2.1 Strong Formulation Find u such that in Q on t The boundary condition u=0 is denoted"homogeneous Dirichlet. We consider Dirichlet boundary conditions in Section 4 and Q is a domain in R with boundary r
❙ ❚❯✰❱❳❲☎❨✰❩✵❱❬❲❭❯❫❪ ❴❉❵❜❛❞❝❃❡✂❢ ❣✿❤❥✐❧❦✰♠❬♥♣♦rqsqs♥♣t✆✉❧✈s♥①✇❧②③❦❭④⑤✐⑦⑥①q⑧⑥⑨q❶⑩✞✈s♥①t❧❷✓❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t❃❻ ⑥❫④⑨♦③t❧♦③④❼♦③❽❋⑥❾⑩s♦③♥♣t✆❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✮❻❧⑥❾t➁➀✛⑥❼➂✏❦❋⑥♣➃⑨❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✮➄ ❣✿❤❥✐❧❦➅④⑨♦❿t❧♦③④⑨♦❿❽✷⑥❍⑩✞♦❿♥①t❉➆➇➂✏❦❋⑥♣➃➈❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t⑦q✢⑥❾✈✞❦➉④⑨♥♣✈✞❦➅❷♣❦❭t⑦❦❭✈➊⑥❾②✏⑩s✐⑦⑥♣t✺⑩s✐❧❦➋q❶⑩✞✈s♥①t❧❷ ❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✻♦③t✆⑩✞❦❭✈✞④✢q❥♥❾❸❬✈s❦❋❷♣❺❧②r⑥❾✈✞♦➌⑩❶➍✩⑥♣t⑦➀✛⑥♣➀➎④⑨♦rqsqs♦❿✇⑦②❿❦✰➀❧⑥❾⑩✞⑥❧➄ ❴❉❵❜❛❞❝❃❡➐➏ ❣✿❤❥✐❧❦❥④❼♦③t❧♦③④⑨♦❿❽✷⑥❍⑩s♦③♥♣tP➆➇➂✏❦❋⑥❾➃➑❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t➁q✠⑥♣✈s❦❥➀➎❦❭➒⑦t❧❦❋➀❫✇▲➍P➓✙⑥➑q➔✉➁⑥♣→✍❦❥➣↔❻①⑥↕✇⑦♦❿②③♦❿t⑦❦❋⑥❾✈ ❸❹♥♣✈✞④✧➙❉❻❧⑥⑨②❿♦③t❧❦❋⑥♣✈✏❸❹♥♣✈✞④⑤➛❾➄ ❣✿❤❥✐❧❦➑④❼♦③t❧♦③④⑨♦❿❽✷⑥❍⑩s♦③♥♣tP➆➇➂✏❦❋⑥❾➃⑨❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t➁q❥♦③➀➎❦❋t❜⑩s♦❿❸❹➍ ➜✒➝❧➝➎➜✒➞❤❥➟❶➠↕➡➋✇❉♥♣❺⑦t⑦➀❧⑥❾✈✞➍✻→❭♥♣t⑦➀➎♦❿⑩s♦③♥♣t➁q❭➢ ➤♦❿✈✞♦③→➊✐⑦②❿❦❭⑩❥➥➦✈s❦❭➧⑦❦❋→✍⑩s❦✷➀✻♦③t➉➣↔❻ ➞➠❳❤⑧➨➫➩✸➠↕➡➈✇➁♥①❺❧t⑦➀❧⑥♣✈s➍✆→✍♥♣t➁➀➎♦➌⑩✞♦❿♥①t⑦q❋➢ ➞❦❋❺❧④✢⑥❾t❧t✆➥➭✈✞❦✍➧⑦❦✷→☎⑩s❦✷➀✛♦❿t➅➙❉➯❶➛❾➄ ❴❉❵❜❛❞❝❃❡➐➲ ❣✿❤❥✐❧❦↕➳❧➵❾➸❞➺⑦➻➽➼✦➵➔➾✓➚♣➪➶➳⑦➹❍➘☎➻➷➴➎➘s➪➑❸❹♥①✈❥⑩s✐❧❦✸➬➮➺⑦➸❞➻➶➪✓➪❭➱❿➪✍✃✢➪❭➺⑦➻➑✃⑨➪❭➻❹❐❧➵✷➚✆⑥♣✈s❦①➓ ⑩✞✐❧❦✰➂✒❦✷⑥❾➃✢❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t❮❒❹④⑨♥①✈s❦➑❷♣❦❋t❧❦❭✈➊⑥❾②③②③➍➎❰✍❻ ♥♣✈ ⑩s✐❧❦✰④⑨♦③t❧♦❿④⑨♦③❽❋⑥❾⑩s♦③♥♣t➅q❶⑩➊⑥❍⑩✞❦❭④⑨❦❭t❜⑩✦❒❹♦❿❸✗➙✢♦③q ➝♠➤❰☎➄ Ï ÐÒÑ➉Ó❂ÔÕ❲✍Ö✗❲❭×➮Ñ➅Ø✍Ó❥❱ÚÙ❏Ö❳❯❫Û➅Ø✍Ó↕Ü Ý❥Þ➔ß à✒á➎â❧ã✒ä✸å❁æ⑧ã✗â⑦çéè➫ê❶ë✠á❧ì❶ã✒ä ❴❉❵❜❛❞❝❃❡îí ï♦❿t➁➀✆ð✹qs❺⑦→➊✐✛⑩s✐➁⑥❍⑩ ñ➫ò❼ó ð ô õ ♦❿t✹ö ð ô ÷ ♥♣t➉ø ù❐❧➪✸ú➊➵❾➴➎➺❉➚♣➹❾➘☎û✦ü✞➵❾➺❉➚❾➸❞➻➽➸ý➵❾➺✰ð✛ôþ÷❼➸r➼❥➚①➪✍➺P➵❍➻➶➪➊➚➐ÿ❞❐⑦➵❍✃✢➵✁❜➪✍➺P➪➊➵❍➴▲➼✄✂✓➸❞➘☎➸ýü➊❐▲➱❿➪❭➻✆☎✞✝✠✟➉➪⑧ü➊➵❍➺⑦➼☎➸ý➚♣➪❭➘ ✡➪❭➴➎✃✢➹❍➺⑦➺↔ú➊➵❍➴➎➺P➚♣➹❾➘☎û➉ü➊➵❍➺P➚❍➸❞➻➷➸ý➵❍➺⑦➼☞☛✍✌✏✎ ✌✏✑ ➸❞✃✸➳⑦➵➇➼❭➪➊➚✓✒➉➸❞➺✕✔P➪➊ü❭➻➽➸ý➵❾➺✗✖✙✘➫➹❍➺P➚✛➸❞➺▲❐❧➵❾✃⑨➵✚❜➪✍➺P➪✞➵❾➴▲➼ ✂✓➸❞➘☎➸ýü✞❐➎➱❿➪✍➻✸ú➊➵❍➴➎➺P➚♣➹❾➘☎û✻ü✞➵❾➺❉➚❾➸❞➻➽➸ý➵❾➺❧➼✦➸❞➺✛✔✶➪➊ü✍➻➷➸ý➵❍➺☞✜✢☎ ➂⑧✐❧❦❋✈s❦ òó✤✣✦✥ó ✥✢✧ó✩★ ✥ ó ✥✢✪ ó ⑥❾t➁➀✛ö✭♦③q➫⑥❼➀❧♥♣④✢⑥❾♦③t✆♦③t➉➟➩ó ➂⑧♦➌⑩✞✐✛✇❉♥♣❺⑦t⑦➀❧⑥❾✈✞➍✻ø➮➄ ✫
equire that a be“ lipschitzian.” We recall functie K such that Jw(a)-w(y)I< Kle-yI for all r, y of interest. a domain Q is Lipschitzian if the boundary f at any point admits a locally Lipschitzian repre- sentation - it can t be too wiggly or singular. Note also that, unless otherw. indicated, we will be speaking of open domains Q (e.g, Q=(0, 1),which de not include 0 and 1); the closure of such a domain will be denoted Q(e.g 2.2 Minimization Principle The finite element method is not based on the strong form, but rather a min ization statement or, more generally, a weak formulation. We must thus develop and understand these formulations before proceeding with the finite ele 2.2.1 Statement SLIDE 5 Find he X=u sufficiently smooth X here is a linear space, the precis e definition of which will be given shortly; we shall also make "sufficiently smooth"precise during the course of this lecture. los Note 1 Notation We explain here some of the notation that we will be using. First arg min The lol e)of";C subset or subspace of”; means“ for all;彐 means" there exists”;|(ands.t)
✬✮✭✰✯✲✱✳✭✴✱✳✵✁✶✸✷✺✹✼✻✽✱✾✵✁✱✓✿✳❀❂❁❃✵✁✱✾❄❆❅❇✶✸❄❉❈❋❊✓✱❍●❆■✄❁❏▲❑✍▼✚❅❂❁❃❄❃◆✍❁❖✶✙✭✴P✞◗❙❘❚✱✾✵✁✱✓▼✓✶✙✷❃✷❯❄❃❅❱✶✙❄❲✶❨❳✓❀❂✭▲▼✍❄❩❁❖❬✸✭✠❬❭❳ ❪❑✍✶✙❫✓❴❵❬✙✭✴✱❜❛✙✶✙✵✮❁❖✶❝❊✍✷❞✱✳✹❢❡✼✹✤❑✍✶✙❄❣❁❤❑❃✐❯✱✮❑❥✶❦■❧❁❏✢❑✍▼✚❅❂❁❃❄❃◆♠▼✓❬✸✭▲♥✙❁❃❄❣❁❖❬✙✭♦❁❳♣❄❃❅❱✱✳✵✚✱❦✱✚q❝❁❤❑✮❄❩❑❜✶❵▼✚❬✸✭❇❑✮❄r✶✙✭❱❄ st❑✮❀❱▼✚❅✉❄❆❅❇✶✸❄✇✈ ❡②①❆③✴④❯⑤✗❡②①❖⑥❂④⑦✈⑨⑧⑩s❶✈ ③☞⑤✗⑥⑨✈❂❳✳❬✸✵❜✶✸✷❃✷⑨③❸❷✁⑥❶❬❭❳✩❁❃✭✢❄✆✱✍✵✁✱✳❑✮❄✆P❥❹❺♥❻❬✙❼❜✶✙❁❃✭✉❈❽❁❤❑ ■✄❁❏▲❑✍▼✚❅❂❁❃❄❃◆✍❁❖✶✙✭✕❁❳✼❄❆❅❇✱❨❊✓❬✙❀❂✭✴♥❝✶✙✵✮❫✩❾❍✶✙❄❿✶✙✭✢❫✇❏❇❬✸❁❃✭❱❄➀✶❝♥✸❼✩❁❃❄❩❑②✶❥✷❞❬✏▼✓✶✙✷❃✷➁❫♣■✄❁❏✢❑✍▼✓❅➂❁❃❄❆◆✍❁❖✶✙✭✕✵✁✱r❏▲✵✁✱✍➃ ❑✍✱✳✭✢❄✆✶✸❄❩❁❖❬✸✭➅➄➆❁❃❄❿▼✓✶✙✭➈➇❄❿❊✚✱✩❄r❬⑦❬❦✻❧❁➉✯✏✯✸✷➁❫♠❬✙✵➊❑✮❁❃✭✲✯❝❀❂✷❞✶✸✵✍P❉➋②❬✸❄✆✱❨✶✙✷➁❑✳❬✾❄❆❅❇✶✸❄❖✹✽❀❂✭✢✷❞✱✳❑✚❑②❬✸❄❃❅❱✱✳✵✮✻❧❁❤❑✍✱ ❁❃✭▲♥✸❁❖▼✓✶✙❄r✱✚♥✙✹➀✻✽✱♣✻❧❁❃✷❃✷✽❊✓✱✩❑❣❏❇✱✓✶⑦➌✸❁❃✭✲✯✉❬✞❳✤➍❝➎▲➏✍➐➑♥❝❬✸❼♣✶✸❁❃✭❇❑❲❈ ❪✱✏P✺✯❻P❞✹❢❈➓➒➔①❩→❇❷⑦➣✏④✓✹↔✻↕❅➂❁❖▼✓❅✗♥❻❬⑦✱✳❑ ✭▲❬✸❄②❁❃✭✴▼✳✷➁❀❱♥❻✱✕➙➛✶✙✭✴♥➛➜✮❴✏➝✾❄❃❅❱✱✉▼✍✷❞❬➞❑✮❀❂✵✁✱➅❬✞❳☞❑✮❀❱▼✓❅➑✶♦♥❝❬✙❼❜✶✙❁❃✭➓✻❧❁❃✷❃✷➊❊✚✱✉♥❝✱✍✭▲❬✸❄✆✱✓♥ ❈ ❪✱✏P✺✯✲P❞✹ ❈✠➒⑩➟→❇❷⑦➣✳➠➡❴✸P ➢↔➤❩➢ ➥➧➦✁➨❉➦❭➩➫➦❭➭➈➯↕➲❇➦❭➳✽➨➸➵❵➺✢➦❭➨❉➻➈➦❭➼✤➽❭➾ ➚✴❅❇✱✇✐➪✭✢❁❃❄✆✱✾✱✳✷➡✱✳❼❜✱✳✭✢❄✤❼❜✱✍❄❃❅❱❬⑦♥❵❁❤❑❜✭▲❬✙❄✼❊✓✶➞❑✍✱✓♥❵❬✸✭♦❄❆❅❇✱❥❑✮❄❩✵✁❬✸✭✲✯②❳✳❬✙✵✮❼✩✹✼❊✍❀❂❄❉✵✁✶✸❄❃❅❱✱✳✵❦✶❵❼❨❁❃✭❱➃ ❁❃❼❨❁❞◆⑦✶✸❄❩❁❖❬✸✭➶❑✓❄r✶✙❄r✱✍❼♣✱✍✭❱❄❜❬✙✵✓✹❥❼❜❬✙✵✁✱♠✯✲✱✳✭✴✱✳✵✚✶✙✷❃✷➁❫➞✹☞✶➛✻✽✱✓✶⑦➌✾❳✳❬✙✵✮❼❨❀❂✷❞✶✸❄❩❁❖❬✸✭▲P➹❘♠✱✉❼❨❀➂❑✮❄✩❄❆❅➂❀➂❑ ♥❝✱✍❛✙✱✳✷❞❬✓❏✗✶✸✭▲♥♣❀❂✭✴♥❝✱✍✵✓❑✮❄✆✶✸✭▲♥♣❄❆❅❇✱✳❑✍✱➪❳✳❬✙✵✮❼❨❀❂✷❞✶✸❄❩❁❖❬✸✭❇❑②❊✓✱❖❳✳❬✸✵✁✱↔❏✴✵✁❬✏▼✚✱✓✱✓♥✙❁❃✭➂✯❥✻❧❁❃❄❆❅☞❄❆❅❇✱❢✐➪✭✢❁❃❄✆✱❲✱✍✷❞✱✍➃ ❼❜✱✳✭✢❄✽❼❜✱✳❄❆❅❇❬✏♥❝P ➘⑨➴❤➘⑨➴❖➷ ➬⑨➮⑦➱❇➮⑦✃➂❐✗✃➂❒✴➮ ❮▲❰✲Ï❃Ð❸Ñ➅Ò ÓÕÔ❞➐✢Ö × ➒➑Ø✸Ù✚Ú➛Û♣Ô➡➐ ÜÕÝ❝Þàß ①❖❡✤④ á❿â➏⑦Ù✁➏ ã②➣ ä❋➒➓å⑦æ❺ç❭è❇é❜ê✍Ô❞➏⑦➐✲ë✁ì➡í✾ç✁Û❨➍➂➍❝ëâ ✈➂æ✴✈ î❥➒➑→❇ï✽❷ äð❅❱✱✳✵✚✱➊❁❤❑➊✶✩✷➁❁❃✭✴✱✚✶✸✵❉❑❣❏❱✶❝▼✓✱✳✹✄❄❆❅❇✱✽❏✴✵✁✱✓▼✳❁❤❑✍✱❲♥❻✱❖✐✽✭❱❁❃❄❣❁❖❬✙✭❵❬❭❳❉✻↕❅❂❁❖▼✚❅☞✻❧❁❃✷❃✷⑨❊✓✱➀✯❝❁❃❛➞✱✍✭♠❑✁❅❇❬✸✵✮❄❩✷➁❫➞➝❧✻✽✱ ❑✁❅❇✶✸✷❃✷Õ✶✙✷➁❑✳❬❦❼❜✶⑦➌❻✱✗●❩❑✮❀✳ñ❶▼✍❁❖✱✳✭✢❄❩✷➉❫♣❑✓❼❜❬✏❬✙❄❆❅❸◗↔❏▲✵✁✱✓▼✍❁❤❑✳✱♣♥✸❀❂✵✮❁❃✭✲✯❦❄❃❅❱✱✩▼✓❬✙❀❂✵✓❑✍✱✩❬✞❳✼❄❆❅➂❁❤❑✼✷❞✱✓▼✍❄❩❀❂✵✁✱✏P Ø✸➐✢Ö ß ①❖❡✤④❢➒ ➣ ò ó➂ôöõ❡ö÷ õ ❡ ø ù✳ú û Ü➈üý✍þÜ➈üÿ✁✂ ⑤ ó➂ô☎✄ ❡ ✂✝✆ ãò ✞✠✟☛✡✌☞✎✍ ✞✠✟✏✡✒✑☛✡✔✓✕✟✗✖ ✘➅➏❜➏✚✙❂➎❇ì❤Ø✸Ô➡➐ â➏⑦Ù✁➏❜ç✁➍❝Û♣➏❨➍✜✛✽ëâ➏❜➐❇➍❝ë✚Ø✙ë✚Ô❞➍❻➐✛ëâØ✙ë á➏ áÔ➡ì➡ì✣✢✢➏❜è❱ç✁Ô➡➐❇Ú✥✤❨Ó↕Ô➡Ù✓ç✞ë❲Ø✸Ù✚Ú➪Û♣Ô➡➐ Û♣➏⑦Ø❝➐❱ç✧✦✞ëâ➏❥Ø✸Ù✚Ú❝è❱Û❨➏⑦➐✲ë✇ëâØ✙ë❲Û♣Ô➡➐❇Ô➡Û❨Ô✩★✍➏✏ç✫✪ ✬♠ëâØ✙ë②Ô❤ç✭✪⑨ëâ➏❜Û❨Ô➡➐❇Ô➡Û♣Ô✮★➂✱✳✵✕①❖Ø❻ç✼➍❝➎❇➎▲➍❻ç✁➏⑦Ö❵ë✁➍ ëâ➏❥Û♣Ô❞➐❇Ô➡Û♠❀❂❼❥④✌✤✰✯â➏❦ç❭í➂Û✱✢✢➍❻ì✳✲❶Û♣➏✏Ø✸➐❱ç✧✦✁Ô❞➐➅ëâ➏❦ç✁➏✳ë❦①❆➍❻Ù②ç✁➎❱Ø❝ê✍➏✏④✤➍✜✛✴✬✥✵✳✶ Û♣➏⑦Ø❝➐❱ç✧✦✚Ø ç✁è✷✢❱ç✁➏✳ë➊➍❝Ù➊ç❭è✷✢✢ç❭➎❱Ø❻ê✳➏❨➍✜✛✴✬✥✵✏✸➅Û♣➏✏Ø✸➐❱ç✹✦✕✛❆➍❻Ù✇Ø❝ì❞ì✺✬✷✵✼✻❵Û♣➏⑦Ø❝➐❱ç✹✦✞ëâ➏✍Ù✚➏②➏✚✙❂Ô❤ç✞ë✓ç✴✬✥✵✽✈✄①❖Ø❝➐❱Ö❵ç✭✤ ë✽✤ ④ ò
means“ such that.”Alo,Uand∩ indicate" union”and“ intersection,"and means"set minus"(i.e, a\B is A with B removed) Functionals A functional takes as input a member of a set or space(here X), and returns a scalar. We summarize this in the case above as J: X-R. which means J takes as input a member of X, and yields as output a real number. More generally, the notation W: X-Y means that W is a function(or application from X, the input(domain) space, to Y, the output (range) space; if y is IR, W is a funct Over all functions w in X that satisfies V-u We give a geometric picture in the next lecture--J(w) paraboloid, the bottom of which occurs at w =u and takes on the value J(a) Note 3 Physical interpretatio e many cases in which this minimization principle(also known the Dirichlet principle) has a meaningful and intuitive significance - often an energy statement. For example, if u is a velocity potential for uncompress- ible flow, then(say for f =0 and inhomogeneous Dirichlet conditions --see Section 4)J(a) is the kinetic energy, and minimizing J thus corresponds to nimizing energy. However, there are also cases(e.g, if u is tem perature) hich a physical interpretation is rather strained, more of an a posteriori jus- tification than any particular ly useful perspective. For our purposes here we eed only the mathematical properties of the minization principle; the physical interpretation is not central 2.2.2 Proof SLIDE 7
✾❀✿✭❁❃❂✥❄❆❅✴❄✴❇✥❈❊❉✹❋✴❉●❁❍❋✭■ ❏▲❑◆▼✺❄✔❖●P✏◗❘❁✜❂✥❙✎❚❱❯✮❂●❙❲❯✩❈✭❁❍❋✒✿❳❅✔❇✷❂✥❯✮❖❨❂✥❏❆❁✜❂✥❙❩❅✴❯✮❂❬❋✒✿✫❭❊❄✔✿✽❈✌❋✴❯✩❖❃❂✼P ❏❀❁✜❂●❙✎❪ ✾❀✿✭❁❃❂✥❄❀❅✴❄✴✿✚❋✳✾❀❯✮❂✗❇✥❄✒❏❴❫❵❯❛■ ✿❨■✮P●❜❝❪❡❞✝❯✩❄✳❜❣❢✳❯✮❋✴❉❳❞❤❭✒✿✫✾❀❖❍✐❃✿✽❙✥❥✌■ ❦✠❧☛♠✌♥✹♦ ♣rq●s✉t❬♠✔✈✕❧✗s①✇❲②④③ ❑❩⑤❵❇✥❂✥❈✌❋✒❯✮❖❨❂✥❁✜▼❲❋❊❁✜⑥❨✿✭❄⑦❁❨❄①❯✮❂✷⑧✥❇❲❋r❁⑨✾❀✿✫✾✱⑩●✿✭❭⑦❖✜⑤❶❁❷❄✴✿✚❋⑦❖❃❭⑦❄✴⑧✥❁❨❈✚✿❸❫❵❉✥✿✫❭✒✿✳❹❺❥✌P❬❁✜❂✥❙❻❭✒✿✚❋✴❇✥❭✴❂✥❄ ❁✎❄✒❈✫❁❃▼✩❁❃❭✭■✎❼❺✿✹❄✔❇✷✾❀✾❆❁✜❭✒❯✩❽✫✿❾❋✒❉✷❯✺❄❿❯✩❂➀❋✒❉✷✿✰❈✭❁❃❄✴✿✰❁✜⑩➁❖❍✐❃✿❾❁❨❄❾➂✣➃✱❹➅➄➇➆➈❷P✉❢✳❉✷❯✩❈❊❉➉✾❻✿✽❁✜❂✥❄ ➂▲❋✒❁✜⑥❨✿✭❄✱❁❃❄❸❯✩❂✷⑧✷❇✷❋❻❁✧✾❀✿✫✾✱⑩●✿✭❭❿❖❃⑤➊❹➉P①❁❃❂✥❙➌➋✗❯✮✿✭▼✩❙✥❄❿❁❃❄❿❖❃❇❲❋✒⑧✷❇❲❋❀❁✧❭✴✿✽❁✜▼⑦❂✗❇✷✾✱⑩●✿✭❭✭■❳➍❳❖❃❭✒✿ ➎✿✭❂✷✿✫❭❊❁✜▼✩▼✩➋❃P❬❋✴❉✷✿❷❂✥❖✜❋✒❁✜❋✴❯✩❖❃❂✧➏❤➃✔❹➐➄➒➑❤✾❀✿✭❁✜❂●❄❡❋✴❉✥❁✜❋◆➏➇❯✺❄✳❁❿⑤❵❇✥❂✥❈✌❋✒❯✮❖❨❂➌❫➓❖❃❭◆❁❃⑧✷⑧✷▼✩❯✩❈✭❁❍❋✴❯✩❖❃❂➁❥ ⑤❵❭✒❖❃✾➔❹➀P●❋✴❉✷✿❻❯✮❂✥⑧✷❇❲❋❆❫✴→❃➣✜↔❀↕✜➙➜➛✏❥➝❄✴⑧✥❁❃❈✫✿❃P●❋✴❖✹➑❾P➁❋✴❉✷✿✱❖❃❇❲❋✒⑧✷❇❲❋❆❫✕➞✴↕✜➛❬➟❬➠➡❥➢❄✴⑧✥❁❃❈✫✿❃➤✏❯✮⑤r➑❱❯✺❄➢➆➈❷P ➏➇❯✩❄✳❁✱⑤❵❇✷❂✥❈✚❋✴❯✩❖❃❂✥❁❃▼④■ ➥➁➦❬➧➜➨✼➩✠➫ ➆➭❂❴❢r❖❨❭✒❙✥❄✫➃ ➯✐❨✿✫❭✳❁✜▼✩▼☛⑤❵❇✷❂✥❈✚❋✴❯✩❖❃❂✥❄➊➲➳❯✩❂❴❹➀P ➵❴❋✒❉✥❁❍❋➢❄✒❁❍❋✒❯✩❄✔➸✥✿✭❄ ➺➢➻❻➼ ➵ ➽ ➾ ❯✮❂❳➚ ➵ ➽ ➪ ❖❃❂❴➶ ✾❆❁✜⑥❨✿✭❄➝➂⑦❫❵➲➝❥✳❁❃❄➊❄✴✾❀❁❃▼✮▼➹❁❃❄➊⑧➁❖❨❄✒❄✔❯✩⑩✷▼✩✿❃■ ➘➷➴ ➬➠➮➟❃➙➜➱❍➠◆↕r➟❬➠❊➣❍↔❆➠✚✃❛➞✌➙➓❐➹❒✏➙➓❐✚✃❛❮❲➞✴➠r➙➜➛✱✃❵❰✷➠❡➛✏➠✴Ï❨✃➁Ð✮➠❊❐✚✃❛❮❲➞✴➠➊ÑÒ➂⑦❫❵➲➝❥✉➙✺Ó➊↕❍➛❻➙➜➛✭Ô⑦➛●➙➜✃Õ➠➊→✜➙➜↔❀➠✫➛✷Ó✌➙➓➣❍➛✏↕❍Ð ❒✥↕❍➞✴↕❨Ö✒➣✜Ð✮➣❍➙➓→❍×❡✃❵❰✷➠❻Ö❊➣❍✃❛✃Õ➣✜↔Ø➣✔Ù❷Ú➹❰✗➙➓❐❊❰✎➣✽❐❊❐✚❮❲➞❊Ó❿↕✜✃➹➲Û➽Ü➵▲↕❍➛✏→❆✃➭↕✭Ý❨➠✌Ó✱➣✜➛❳✃❵❰✷➠❿➱➡↕✜ÐÞ❮✥➠❿➂✣❫➓➵✏❥✭ß ❦✠❧☛♠✌♥✧à á❷â●ã➹③✽✈➭t❍✇❲②➊✈➓s✣♠✌♥✜ä✫å⑦ä❃♥✷♠✒✇☛♠✔✈✕❧✗s æ❉✷✿✭❭✴✿❾❁❃❭✴✿❀✾❆❁✜❂✗➋✎❈✭❁❃❄✴✿✭❄⑨❯✩❂➌❢✳❉✷❯✩❈❊❉✠❋✒❉✷❯✩❄❸✾❻❯✩❂✷❯✩✾❀❯✮❽✽❁❍❋✴❯✩❖❃❂➌⑧✷❭✒❯✩❂✥❈✚❯✩⑧✷▼✩✿❳❫➓❁✜▼✺❄✴❖✹⑥✗❂✷❖❍❢✳❂➌❁❃❄ ❋✴❉✥✿✱ç➷❯✮❭✒❯✩❈❊❉✥▼✮✿✫❋➷⑧✷❭✒❯✮❂●❈✚❯✩⑧✷▼✮✿➡❥✳❉✥❁❨❄➢❁❆✾❀✿✽❁✜❂✷❯✩❂➎⑤❵❇✥▼➮❁❃❂✥❙❴❯✮❂❬❋✒❇✷❯è❋✒❯✮✐❨✿✱❄✴❯➎❂✷❯✮➸●❈✫❁❃❂✥❈✚✿⑨éê❖✜⑤➜❋✒✿✫❂✎❁❃❂ ❅✔✿✭❂✷✿✫❭➎➋✠❄✕❋❊❁❍❋✒✿✫✾❀✿✫❂❬❋✭■ ❏Øë✷❖❃❭❿✿✚ì✷❁✜✾❀⑧✷▼✩✿❃P①❯✮⑤➊➵❝❯✩❄✱❁✧✐❃✿✭▼✮❖❲❈✫❯è❋✕➋❺⑧●❖❃❋✴✿✫❂❬❋✒❯✩❁❃▼⑦⑤❵❖❃❭✱❯✮❂●❈✚❖❃✾❀⑧✷❭✒✿✭❄✒❄✔í ❯✩⑩✷▼✮✿❾î✥❖❍❢❷P✼❋✒❉✷✿✫❂ï❫➓❄✒❁➡➋❳⑤❵❖❃❭❀➾▲➽ð➪✧❁❃❂✥❙➌❯✮❂✥❉✷❖❃✾❀❖➎✿✫❂✷✿✭❖❃❇✥❄❿ç➢❯✩❭✴❯✺❈❊❉✷▼✩✿✚❋✱❈✫❖❃❂✥❙✷❯è❋✒❯✮❖❨❂✥❄⑨éñ❄✔✿✭✿ ò ✿✽❈✌❋✴❯✩❖❃❂❝ó❬❥❾➂⑦❫❵➲➷❥✱❯✺❄❿❋✒❉✷✿✹⑥❬❯✩❂✷✿✫❋✴❯✺❈ô✿✭❂✷✿✫❭➎➋❨P⑦❁✜❂✥❙➉✾❻❯✩❂✷❯✩✾❀❯✮❽✭❯✮❂➎ ➂☎❋✴❉✗❇✥❄❀❈✫❖❃❭✒❭✴✿✽❄✔⑧➁❖❃❂✥❙✥❄❷❋✴❖ ✾❀❯✮❂✥❯✮✾❀❯✩❽✫❯✩❂➎ ✿✭❂✷✿✫❭➎➋❨■➝õ➢❖❍❢r✿✭✐❃✿✭❭✭P✥❋✴❉✷✿✭❭✴✿❀❁✜❭✒✿❿❁✜▼✺❄✴❖✰❈✫❁❨❄✔✿✽❄❿❫➓✿❃■ ➎ ■✩P☛❯è⑤✉➵✠❯✺❄➢❋✒✿✫✾❀⑧➁✿✫❭❊❁❍❋✴❇✥❭✴✿➡❥✳❯✩❂ ❢✳❉✷❯✺❈❊❉✎❁❾⑧✥❉❬➋❲❄✴❯✩❈✭❁✜▼➮❯✮❂❬❋✒✿✫❭✒⑧✷❭✴✿✫❋✒❁✜❋✴❯✩❖❃❂✧❯✺❄➢❭❊❁❍❋✒❉✷✿✫❭⑨❄✕❋✒❭✒❁❃❯✮❂✥✿✭❙☛P➁✾❀❖❃❭✒✿❿❖✜⑤✉❁❃❂❝↕❿❒✥➣➡Ó✌✃Õ➠✫➞✌➙➓➣❍➞✌➙①ö✕❇●❄✕í ❋✴❯✮➸●❈✭❁❍❋✴❯✩❖❃❂➀❋✴❉✥❁❃❂❝❁✜❂✗➋✠⑧✥❁✜❭✴❋✴❯✺❈✚❇✥▼✩❁❃❭✴▼✩➋❺❇●❄✔✿✫⑤❵❇✷▼➊⑧●✿✭❭✒❄✴⑧●✿✽❈✌❋✒❯✮✐❨✿❃■❺ë✥❖❃❭❻❖❃❇✷❭✱⑧✷❇✷❭✒⑧●❖❬❄✔✿✽❄❿❉✥✿✫❭✒✿ô❢❡✿ ❂✷✿✭✿✭❙❾❖❨❂✷▼✩➋❀❋✴❉✷✿⑨✾❆❁❍❋✒❉✷✿✫✾❆❁❍❋✒❯✩❈✭❁✜▼✏⑧✷❭✒❖❃⑧➁✿✫❭✴❋✴❯✩✿✭❄r❖✜⑤❶❋✒❉✷✿⑨✾❀❯✮❂✷❯✩❽✭❁✜❋✴❯✩❖❃❂✰⑧✷❭✴❯✩❂✥❈✫❯✮⑧✷▼✩✿❃➤✗❋✒❉✷✿➝⑧✷❉✗➋✗❄✴❯✺❈✫❁✜▼ ❯✩❂❨❋✒✿✫❭✒⑧✷❭✒✿✚❋✒❁✜❋✴❯✩❖❃❂✹❯✩❄➊❂✥❖✜❋➢❈✚✿✭❂❬❋✴❭❊❁✜▼❛■ ÷❶ø✺÷❶ø✺÷ ù⑨ú❍û❶û➁ü ➥➁➦❬➧➜➨✼➩þý ➴
f(u +u)dA Note u]r =ur=0, which ensures that wr=0, and hence is a member of X ted on t 2 Vu. VudA-lofud4 J(u) Jo Vu Vu- fud4 6J(u) first variation We can think of(u+y)as a "Taylor "series about u. Since j is only guadratic, it is not surprising that J(u+u) contains a constant term, a linear (in u) term(a "gradient), and a guadratic (in u)term(a"Hessian")-and then terminates 0)=/w,d- f udA u[-v2u-f)dA We know the gradient of a function vanishes at its minimizer; it is thus not sur prising that the first variation of a functional the gradient times a test function) anishes at its minimizer. Here n is the unit normal on t Note 4 G TTh Much of our analysis here is based on humble integration by parts, which in r space dimensions is essentially one of greens Theorems. The necessary
ÿ✁✄✂✆☎✞✝✠✟☛✡✌☞✎✍ ✏✒✑✓✕✔ ✖✘✗ ✙✛✚ ✜ ✢ ✣✥✤ ✦ ✟ ✤✧✦★✢✩✣ ✚ ✜ ✡ ☞ ✤✩✦★✢✧✣ ✚ ✜ ✪ ✝ ✫ ✬ ✭✯✮✠✰ ✗✱✟✲✡✳☞ ✪✵✴ ✰ ✗✶✟✷✡✌☞ ✪✹✸✻✺ ✼ ✭✌✮✾✽ ✗✱✟☛✡✌☞ ✪✹✸✿✺❁❀ ❂❄❃❆❅❈❇❉✟❋❊ ●❍✝✞☞✹❊ ●■✝❑❏✓▲◆▼P❖✯◗✶❘❙❖❚❇✥❯❲❱❙❳❩❨❬❇★❱❭❅✱❖✓❪❫❅✵☎❴❊ ●■✝❑❏❩▲✒❪❫❯✎❵✷❖✓❇✄❯✎❘❙❇❴◗❛❱❴❪❍❜✷❇✄❜✷❝❙❇✄❨❴❃❡❞❣❢✐❤ ❥❣❇❙❘❙❪❫❦❧❦♠❅❧❖❲❪❫❅✛☎❴❊ ●✳❜✲❇❙❪❫❯❲❱✒☎♥❨♦❇★❱★❅♣❨★◗✶❘✥❅q❇❙❵✲❅q❃sr✘▲✘❅❧❖❲❪❫❅t◗❛❱❬▲✉❇✄✈❫❪❫❦✇❳❲❪❆❅❈❇❙❵■❃❫❯①rt❤ ②✎③✻④❧⑤♠⑥❚⑦ ✖✘✗✶✟☛✡✳☞ ✪ ✝ ✫ ✬ ✭❩✮⑧✰✟ ✴ ✰ ✟ ✸✿✺ ✼ ✭❩✮ ✽ ✟ ✸✿✺ ⑨s⑩❡❶✒❷ ✡ ✭❩✮ ✰ ✟ ✴ ✰ ☞ ✸✻✺ ✼ ✭✯✮✠✽ ☞ ✸✿✺ ❸❩⑨♠❹♠⑩q❶❺❷ ❻❨❙❱★❅◆✈❫❪❫❨★◗✶❪❆❅❼◗✶❃❆❯ ✡ ✫ ✬ ✭✮ ✰ ☞ ✴ ✰ ☞ ✸✿✺ ❽♥❾①❿✱➀➂➁❺➃➅➄➆ ❾ ➇➈❇❣❘❙❪❫❯✲❅✱❖✯◗❧❯❩➉❄❃➊❞◆✖✘✗✶✟❋✡❴☞ ✪ ❪✩❱✆❪➌➋✥➍❲❪❆➎❫❦➏❃❫❨✿➐❺❱✄❇✥❨★◗✶❇✥❱✆❪➂❝❙❃❫❳❩❅➑✟✵❤t➒✎◗❧❯✎❘❙❇✒✖❚◗❛❱❺❃❆❯❲❦✇➎❄➓✄❳❲❪➂❵❆❨♦❪❫❅♣◗✶❘✥▲ ◗❧❅✁◗❛❱t❯✎❃❫❅➔❱★❳❩❨♣→➔❨★◗❛❱❙◗❧❯✯➣❭❅✱❖✓❪❆❅➔✖❋✗✶✟✛✡s☞ ✪ ❘❙❃❫❯➑❅❈❪❆◗❧❯✓❱❺❪❭❘❬❃❆❯✓❱★❅q❪❫❯❲❅✹❅q❇✄❨★❜❄▲♠❪↔❦✇◗❧❯➔❇❬❪❆❨➙↕♣◗❧❯❭☞✿➛❉❅q❇✄❨★❜➜↕q❪ ➋❧➣❆❨♦❪✿❵❫◗✶❇✄❯❲❅✥➐❈➛✩▲✘❪❫❯➔❵☛❪✷➓✄❳❲❪➂❵❆❨♦❪❫❅♣◗✶❘✷↕❈◗❧❯✷☞➂➛s❅q❇✥❨★❜➝↕❡❪✐➋❧➞❄❇✥❱❬❱★◗✶❪❆❯✓➐q➛❴➟➠❪❆❯✎❵❴❅❧❖❲❇✥❯➡❅q❇✥❨★❜☛◗❧❯✎❪❆❅❈❇✥❱✄❤➢②✎③✻④❧⑤♠⑥❚➤ ➥ ✖✻➦✻✗✶✟✪ ✝ ✭✯✮✠✰✟ ✴ ✰ ☞ ✸✻✺ ✼ ✭✯✮ ✽ ☞ ✸✿✺ ✝ ✭✯✮s✰ ✴ ✗✱☞✰ ✟✪✁✸✿✺ ✼ ✭✯✮ ☞✰❴➧ ✟ ✸✻✺ ✼ ✭✯✮ ✽ ☞ ✸✻✺ ✝ ✭ ●➨☞ ❏ ✰ ✟ ✴➑➩➫ ✸✯➭ ✡ ✭✮ ☞➑➯ ✼ ✰➧ ✟ ✼ ✽ ✤ ✦★✢ ✣ ➲ ➳ ✸✻✺ ✝ ❏❺➵ ➸❄☞✾➺①❢ ➻❣➼ ➇➈❇✒➉❆❯✎❃❆▼✳❅❧❖❲❇✒➣❆❨♦❪✿❵❫◗✶❇✄❯❲❅✘❃❡❞↔❪✘❞❙❳❩❯➔❘✥❅♣◗✶❃❫❯■✈✩❪❆❯❲◗❛❱♦❖✓❇✥❱↔❪❆❅P◗❧❅❼❱❺❜❴◗❧❯➑◗❧❜❴◗➏➽✧❇✄❨❙➾✵◗❧❅✛◗❛❱✆❅✱❖✯❳✯❱❺❯✎❃❆❅♠❱★❳❩❨★➚ →✎❨★◗❛❱★◗❧❯✯➣❭❅✱❖✓❪❫❅P❅❧❖❲❇ ❻❨❙❱★❅✹✈❫❪❫❨★◗✶❪❆❅❼◗✶❃❆❯■❃❡❞❺❪❋❞❙❳❩❯➔❘✥❅♣◗✶❃❫❯➔❪❫❦✹↕❈❅✱❖✓❇✘➣➂❨♦❪➂❵❆◗✶❇✥❯➑❅♠❅♣◗❧❜✷❇✥❱✒❪❭❅❈❇✥❱★❅❆❞❙❳❩❯✎❘✄❅❼◗✶❃❆❯❆➛ ✈❫❪❫❯❲◗❛❱♦❖❲❇★❱❴❪❆❅✉◗❧❅❼❱↔❜☛◗❧❯❲◗❧❜☛◗➏➽✧❇✥❨✄❤❺➞❴❇✥❨❬❇ ➩➫ ◗❛❱❭❅❧❖❲❇s❳❩❯➑◗❧❅◆❯➔❃❫❨★❜✲❪❫❦P❃❆❯①r✉❤ ➪❚➶✹➹★➘■➴ ➷✲➬✁➮❲➱✧➱✲➬❩✃◆❐➅➷✷❒✿➘✻➘❆✃❭❮➱➈❰❺Ï◆➘✯➶✯❒✿➘❆Ð✌➱ Ñ➈Ò➑Ó❙✑ ➀❆❿✛➀Ò➁✒Ô✔Ô❆Õ×Ö✯Ø♦Ù❛Ø ✑✓➁ Ù❛Ø✉Ú➑Ô➂Ø ✕Û ➀✔❍✑✯Ò✓ÜÚ❲Õ Ù✔✻✂❬✄Ý➁❙Ô✂ Ù➏➀✔ Ú✯Ö✲Þ❲Ô❆➁✂ Ø✕ß✯à✑Ù Ó❙✑ Ù✔ ✑ÙÝ➂✑✓➁❣Ø♦Þ❲ÔÓ✄➙ÛÙÜ✷✄✔Ø➊Ù×➀✔Ø✆Ù×Ø Ø♦Ø ✄✔✻✂ Ù×Ô➂Õ➏Õ×Ö✾➀✔❲ ➀❆❿✘á↔➁✕✄✔✁â Ø ✏✒✑✓➀➂➁✄ÜØ ✍✉✏✒✑✓❭✔✓✕Ó✄Ø❬Ø♦Ô➂➁♦Ö ➼