bove slide is(1+C2)1/2. This can also be shown by explicit construction of u (see Lecture 2 from earlier in the course) The fact that u is regular when f is regular(and in IR2, the domain is suitably regular) has very important implications as regards the convergence rate of the of a priori and a poste st es 1.2.1 Strong form SLIDE 7 2=(0,1) Find a such tha u(0) (1) for given∫, 1.2.2 Minimization statement SLIDE 8 {U∈H1(9)|v(0)=0} arg f wdr-g w(1 This follows from the previous lecture, noting that rn g v da is here ju 9v(1). We can also show this explicitly by integrating by parts to find dJ,(u) ∫0v{-x-∫}dr+v(1){uz(1)-g}=0,V∈X 1.2.3 Weak Statement Find u∈ X such that yu∈X +gv(1),VU∈Xö✿÷✟ø❿ùPú❛û✳ü❍ý♣þ✵ú➄ý❍û➄ÿ✁￾✄✂✆☎✞✝✠✟☛✡☛☞✌✝✎✍✄✏✒✑✓ý♣û✒✓➐ö✎✔♥ö✿ü♣û✳ø❏÷✟ú➄û☛✑✓ø✖✕✗✔➽÷✙✘➧ú✛✚✢✜✓ü☛ý✣✓✆ý✥✤✦✓✆ø✧✔✪û✁✤✌★☛✩✪✓✫✤◆ý❍ø✬✔➶ø✎✭✯✮ ÿ✠û◆ú➐ú✱✰✢ú✲✓✳✤☛✩✴★❲ú✱✵✱✭✶★❲ø✬✷ ú✒ö✎★❲ü❍ý☛ú✲★♠ý✸✔✹✤☛✑✓ú✱✓➐ø✬✩✴★❺û◆ú✺✟✫✍ ✏✒✑✓ú✻✭✠ö✬✓✳✤✼✤☛✑✪ö✎✤✯✮↕ý❍û✽★◆ú✲✾✬✩✓ü♣ö✎★✼✕✗✑✪ú✛✔❀✿↕ý❍û✯★❲ú✛✾✬✩✪ü❍ö✬★➆ÿ✠ö✎✔✏þðý✥✔❂❁❃✦✝✬❄✺✤☛✑✓ú♠þ✵ø✬✷➧ö✿ý✸✔ðý♣ûìû☛✩✓ý✥✤❲ö✿÷✪ü✥✘ ★❲ú✛✾✬✩✪ü❍ö✬★❅✟✄✑✪öPû❰ùPú✲★☛✘❳ý✸✷❆✜✟ø✬★☛✤❲ö✬✔✙✤♠ý✸✷❇✜✓ü☛ý✣✓➐ö✎✤◆ý❍ø✬✔✪û✾ö❃û❈★❲ú✛✾✙ö✎★❺þ✓û❉✤✌✑✓ú❂✓✆ø✧✔✙ù❃ú✛★✌✾Pú✲✔❊✓✆ú✗★❺ö✖✤❲ú✧ø✬✭✼✤☛✑✓ú ❋✔✓ý✥✤◆ú➶ú➐ü❍ú✛✷❳ú✛✔✙✤✱✷❳ú✳✤✌✑✓ø✵þ❡ö✬✔✪þ●✤☛✑✓ú❍✓✆ø✬✔✏û✁✤✌★☛✩❊✓✳✤◆ý❍ø✬✔❡ø✎✭✱■❆❏▲❑✫▼❖◆✖❑✫▼ðö✎✔✪þP■❇❏✴◆✖◗❅❘❚❙✛❑✫▼❖◆✖❑✫▼❛ú✛★✌★◆ø✧★ ú✒û❯✤◆ý✸✷➧ö✖✤❲ú✒û✲✍ ❱❉❲❨❳ ❩❭❬❫❪✽❴❛❵❝❜❡❞❛❞❣❢ ❤❥✐✣❦✼✐❖❤ ❧✼♠✛♥✖♦❥♣✯q✆r✯♦❊♥✖s t❥✉✙✈①✇✽②✆③ ④ø✬✷➧ö✿ý✸✔✼⑤ ⑥⑧⑦➎ÿ❖⑨✴⑩✲￾✺✟❶✍ ❷ý✥✔✏þ❸✮❫û❯✩❊✓❅✑❍✤☛✑✏ö✖✤ ❹ ✮❥❺✺❺ ⑦ ✿ ý✥✔❻⑥❼⑩ ✮æÿ❖⑨✙✟❽⑦ ⑨❾⑩ ✮▲❺✪ÿ✁￾✠✟❽⑦ ❿➀⑩ ✭✯ø✬★✗✾❃ý☛ù❃ú✛✔➁✿❥⑩❯❿➂✍ ❤❥✐✣❦✼✐✣❦ ➃➅➄❖♣➆➄❖s➇➄❨➈✬➉✢♠✺➄❖♦❥♣⑧❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍➌ ④ú❋✔✓ú ➎➐➏❼➑✧➒➔➓❾→●✡❿ÿ➣⑥✗✟↕↔➙➒✟ÿ❨⑨✧✟✄⑦❫⑨❈➛❾✍ ❷ý✥✔✏þ ✮❍⑦❚ö✎★✌✾➇✷❳ý✥✔ ➜➆➝✬➞❝➟ ÿ❖➠❛✟ ✕✗✑✓ú✲★◆ú ➟ ÿ❖➠❛✟✻⑦ ￾ ✵ ➡✡ ➢ ➠❺✦➤✧➥ ✝ ❹ ➡✡ ➢ ✿❀➠ ➤✬➥ ❹ ❿❛➠✱ÿ✁￾✠✟❡➦ ➧▲➨▼✣◗➔➩✳◆✖➫①➫✸◆✖➭❡◗❍➩❅❑☛◆✎➯➲❘➨ ❙➳❏❥❑☛❙✛➵✺▼❖◆✖➸➙◗✆➫✥❙❅➺✳❘➣➸✢❑☛❙✳➻❍➼▲◆✖❘➣▼①➼✙➽❫❘➨ ■✖❘❆➾✲➚✧➪➶❿✹➒ ➤✙➹ ▼✣◗ ➨ ❙✛❑☛❙➴➘✫➸➙◗❅❘ ❿✒➒✟ÿ❯￾✺✟✛➷➴➬➳❙❀➺❅■✖➼➀■✎➫➮◗✛◆❇◗➨ ◆✎➭⑧❘➨▼✣◗❂❙☛➱✳❏▲➫✃▼❖➺✳▼①❘➣➫✃❐➴❒✛❐❇▼①➼❊❘❚❙➣➽✬❑☛■✎❘❨▼①➼➙➽➔❒✳❐❛❏❊■✖❑✫❘❨◗❶❘❚◆✒❮✄➼▲❰✞Ï ➟✧Ð ÿ✶✮❭✟✄⑦ ➾ ✡ ➢ ➒●Ñ ❹ ✮▲❺✲❺ ❹ ✿✯Ò ➤✧➥ ✂➇➒✟ÿ❯￾✺✟✫Ñ✲✮▲❺✓ÿ✁￾✠✟ ❹ ❿❥Ò❛⑦P⑨✴➻▲Ó❀➒➔➓❍➎⑧➷ ❤❥✐✣❦✼✐✸Ô ÕÖ➊✙➉✴×Ø❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍Ù ❷ý✥✔✏þ❸✮➳➓❾➎ û❯✩✪✓❅✑❍✤✌✑✪ö✖✤ Ï ➟Ð ÿ✶✮❭✟✄⑦❫⑨❍⑩ Ó❀➒➔➓✹➎ Ú ➡✡ ➢ ✮❥❺✒➒✎❺ ➤✧➥ ⑦ ➡✡ ➢ ✿❀➒ ➤✧➥ ✂✆❿❣➒✭ÿ✁￾✺✟❶⑩ Ó✱➒➴➓✹➎Û➦ Ü