result is demonstrated most easily in indicial notation. In particular, we note auau dA dA ax ni ds uVu·ndS一 sIn ce we ar J(u+y=J(0)+2/ u VudA, VeX J(u)>J(u),V∈X,u≠u a is the minimizer of j(w) What PDEs admit such a inimization statement? PDEs associated with oper ators that are SPD (symmetric positive definite). We define this more precisely, and indicate how the FEM (inite element method) proceeds in the absence of this property, in a future lecture. For now, we focus on the simplest case almost all of which turns out to be directly relevant to the more general case We could also derive the result above by applying the general machinery of the calculus of variations. In this sense, we may view -Vu=f Euler o Euler- Lagrange equations as sociated with minimization of the functional J b Exercise 1 Consider the problem -xz=1,0<a<l, u(0)=u(1) 0, with solution (1-a). Show by explicit calculation that 8J(u) Jo urUz -U dr=0 for all (smooth) v such that v(0)=v(1)=0.B 2.3 Weak Formulation 2.3.1 Statement SLIDE 11 Find a∈ X such thatã❬ä✕å♦æ✓çéè➙ê❛å❭ë❩ä✄ì✷í➂î➑å❡è❬ã❬ï❆è♦ä✕ëðì✷í✿å➊è➙ä✧ï➂å♦ê➏ç×ñ➡ê➏î❚ê×î❲ë❩ê❛ò✥ê❛ï❆ç❋î✓í➂è❬ï❫è❬ê➏í✿î✁ó❄ôqî❚õ❲ï➂ã➊è❬ê×ò✄æ✓ç×ï➂ã✕ö➔÷tä❴î❲í❆è♦ä è♦ø➑ï❫è ù úüû✎ý û➑þ✓ÿ û✁￾ û✎þ✓ÿ ✂☎✄ ✆ ù ú ✝ û û✎þ❩ÿ ✞ ý û✁￾ û✎þ✓ÿ ✟✡✠ ý û☞☛✌￾ û✎þ❩ÿ✕û✎þ❩ÿ ✍ ✂☎✄ ✆ ù ✎ ý û✁￾ û➑þ✓ÿ✑✏✒ ÿ ✂✔✓ ✠ ù ú ý û☛ ￾ û✎þ❩ÿ✧û✎þ✓ÿ ✂☎✄ ✆ ù ✎ ý✖✕✗￾✙✘ ✏✒ ✂✔✓ ✠ ù ú ý✖✕☛ ￾ ✂☎✄✛✚ ÷❺ø✓ä✕ã♦ä❋÷tä✘ø❲ï✢✜✿ä✛æ➑å➊ä✧ë✤✣➙ï❆æ❲å❬å✦✥★✧✒ø✓ä✄í✿ã♦ä✕ì è♦í❉ò✥í➂î✔✜✿ä✄ã♦è✎è❬ø✓ä✩✜✿í➂ç×æ✓ì✷ä❋ê➏î✻è❬ä✦✪➂ã❙ï❆ç❫ê×î✻è♦í❣ï❺å♦æ✓ã✬✫✶ï➂ò✄ä è♦ä✕ã♦ì➡ó✮✭❣í❆è❬ä❄÷tä❄ï➂ë❩í✿õ❩è❣è❬ø✓ä❴ò✄í➂î✔✜➂ä✕î✻è♦ê×í➂î➡í✯✫◆å♦æ✓ì✷ì✲ï❫è♦ê×í➂î➈í✰✜✿ä✄ã✆ã❬ä✄õ✎ä✕ï❆è♦ä✧ë➈ê➏î➑ë❩ê×ò✄ä✕å✕ö➑ø❲ä✄ã❬ä ✫✱ã❬í➂ì✲✱↔è♦í✙✳☛å♦ê➏î➑ò✥ä❭÷tä➙ï➂ã♦ä↔ê×î■ô✴ ☛ ó ✵✁✶☎✷✹✸✻✺✽✼✯✾ ✿❁❀￾✡❂✌ý ❃ ❄❆❅ ❇ ❈❊❉ ✆ ✿❁❀￾ ❉ ❂ ✱ ✳ ù ú ✕❄ý✗✘★✕❴ý ✂☎✄ ❃ ❄❆❅ ❇ ❋❍●❏■✦❑★▲ ▼❖◆P◆❘◗❚❙❯● ✚❲❱ ý✙❳❩❨ ❬ ✿❁❀P❭❉❫❪ ✿❴❀￾❉ ✚ ❱ ❭ ❳❵❨ ✚ ❭❜❛✆ ❝ ￾ ￾ ê×å❡❞❣❢✐❤sì✷ê➏î✓ê×ì✷ê❦❥✕ä✄ã✆í❧✫ ✿❁❀❣❭❉ ♠ ✱ ♥❢✐♦✰❞q♣❁r❚s✉t❡♦❧✈✯✇✗①✹❞②t❆③⑤④⑥❢❵♦⑦✇✡①✹⑧⑤①✹✇✡①❦⑨⑩♦✰❞❶①P❷✰⑧❵t❆❞❖♦✰❞❖❤✌✇❸❤✌⑧❹❞✦❺✗♣❴r❻s✉t❡♦✢t⑥t✦❷★④✦①P♦✰❞❖❤⑥✈✡❼❴①✹❞✹❢❵❷⑥❽✐❤✦❾❆❿ ♦✰❞❖❷✰❾⑥t❏❞❣❢✐♦✯❞②♦✯❾✬❤✮➀✖♣❴r➂➁❶t❆➃✰✇✗✇❸❤✌❞❶❾❆①P④❁❽✐❷✰t❆①✹❞➄①✹➅✰❤❡✈❧❤➄➆❁⑧❹①✹❞➇❤➉➈❧➊ ♥❤➋✈➌❤P➆✩⑧✁❤➋❞❣❢✔①➍t❏✇❸❷✰❾✬❤❴❽✁❾✬❤⑥④✦①➍t✦❤✌➎➏➃✢➐ ♦✰⑧☞✈✑①✹⑧☞✈✰①P④⑥♦✰❞❖❤❸❢✐❷✯❼➑❞✹❢⑤❤✡➒②s✉➓➔➁→➆✩⑧⑤①✹❞❖❤➣❤✦➎❦❤✦✇↔❤✦⑧⑤❞❚✇↔❤✦❞❣❢✐❷⑩✈⑥➈↔❽✁❾↕❷★④⑥❤⑥❤⑥✈✢t❸①✹⑧➙❞❣❢✐❤❵♦❧➛✌t✦❤✌⑧☞④⑥❤➣❷➝➜ ❞❣❢✔①➍t❡❽☞❾✬❷↕❽⑤❤✌❾❆❞❶➃✢➐✤①✹⑧➞♦✡➜⑥③✖❞➄③✖❾✬❤✙➎❦❤⑥④✦❞➄③✖❾✬❤⑩➊➟➒✻❷✯❾✙⑧☞❷✰❼②➐❡❼✩❤❻➜✌❷★④✦③✔t❩❷✯⑧➠❞❣❢✐❤➣t❆①✹✇✮❽✁➎❦❤✌t❆❞⑦④⑥♦✢t✦❤❍➡ ♦✰➎➢✇↔❷✰t⑥❞❫♦✰➎✹➎②❷➉➜❡❼q❢✖①P④↕❢✑❞❶③✖❾❆⑧✐t❻❷✯③✖❞✉❞❖❷➤➛⑥❤✡✈✰①✹❾✬❤⑥④✦❞➄➎➏➃✙❾✬❤✌➎➥❤✌➅✰♦✰⑧❹❞✉❞❖❷❸❞❣❢✐❤✤✇↔❷✯❾✬❤➋➦☎❤✦⑧✁❤✦❾✬♦✰➎②④↕♦✰t✦❤★➊ ♥❤✡④⑥❷✯③✖➎❦✈❵♦✰➎→t✦❷❍✈❧❤✦❾❆①✹➅✰❤✡❞✹❢⑤❤✡❾✬❤❆t❆③✖➎➏❞❏♦➌➛↕❷✯➅✢❤❸➛✦➃❩♦⑥❽❧❽☞➎➏➃✰①✹⑧☎➦❵❞❣❢✐❤❡➦☎❤✦⑧✁❤✦❾✬♦✰➎❘✇↔♦➌④↕❢✖①✹⑧✁❤✦❾❆➃❵❷➝➜✤❞❣❢✐❤ ④⑥♦✰➎❦④✦③✖➎➏③✔t➣❷➝➜❸➅✢♦✯❾❆①P♦✰❞❶①P❷✰⑧⑤t✌➊➨➧❆⑧➠❞❣❢✔①➍t❸t✦❤✌⑧⑤t✦❤❆➐❚❼✩❤✙✇❸♦✰➃➩➅★①P❤✦❼ ✠ ✕✗￾ ✆➭➫ ♦✢t➤❞❣❢✐❤❸s❲③✖➎➥❤✌❾➣❷✯❾ s❲③✖➎❦❤✦❾❆❿➄➯q♦✦➦❧❾✬♦✰⑧✔➦➌❤✡❤⑥➲✌③⑤♦✯❞➄①P❷✯⑧✐t⑦♦✢t⑥t✦❷★④✦①P♦✰❞❖❤⑥✈➣❼❴①✹❞✹❢❍✇✗①✹⑧❹①✹✇✗①❦⑨⑩♦✰❞❶①P❷✰⑧➟❷➉➜✤❞✹❢⑤❤✩➜⑥③✖⑧☞④✌❞❶①P❷✰⑧☞♦✰➎ ✿ ➊ ➳➸➵➋➺②➻✰➼➌➽✢➾P➚❧➻➙➪❯➶í➂î❲å♦ê×ë✓ä✄ã↔è♦ø✓ä✷õ✓ã❬í❧➹❲ç➏ä✕ì ✠ ￾✁➘⑩➘ ✆ ✱✿ö✻➴✑➷ þ ➷✛✱➂ö ￾ ❀ ➴ ❉ ✆ ￾ ❀ ✱ ❉ ✆ ➴✓ö❋÷❺ê➏è♦ø⑧å♦í➂ç×æ❩è♦ê×í➂î ￾ ✆➮➬ ☛ þ ❀ ✱ ✠ þ❉ ó➙➱✯ø✓í❫÷✛➹✯ñ✳ä✌✃❩õ✓ç×ê×ò✄êéè✲ò✄ï➂ç×ò✄æ✓ç❛ï❫è♦ê×í➂î➌è❬ø❲ï❫è❸❐ ✿ ◗ ❀￾❉ ✆ ❒ ➬ ● ￾☞➘❆ý✯➘ ✠ ý ✂þ ✆ ➴✗✫✱í✿ã❣ï❆ç×ç ❀ å♦ì✷í✻í➂è♦ø❉ ý å➊æ❲ò❙ø①è♦ø➑ï❫è ý ❀ ➴ ❉ ✆ ý ❀ ✱ ❉ ✆ ➴✓ó ❮Ï❰➄Ð ÑÓÒ✻Ô❘Õ➭Ö❫×❁Ø⑤ÙÛÚ✮Ü➉Ô❘Ý✐Þ➉×✉ß àâá➥ã✻áPä åâæ★ç✐æ★è✔éêè✔ë☞æ ✵✁✶☎✷✹✸✻✺✽✼✐✼ ìê➏î➑ë ￾❯❳➣❨ å➊æ➑ò❙ø❍è❬ø❲ï❫è í