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 ➴