SPD bili fvdA,u∈X b Exercise 2 prove that a is indeed an spd bilinear form over X hint. ou must use the boundary conditions. (Note a is SPD because the underlying Minimization prine gm要 k sta (u,0)=(u),VU∈X (a) Show that if J:Y+R is defined by J(w)=sa(w, w)-t(w)for any SPD bilinear for form e over Y, then the mi ay, g “ anti-variation” to find J) (b) Take Y= R, and thus show, by appropriate choice of a and e, that the minimizer u E y of J()=5wGw-wTF- for any SPD matrix G∈R ndF∈R ∈X ince a involves only first derivatives {u∈H2(9) 0}≡H0() H()={1/v2dA,/v2d➐P➑➓➒❩➔⑧→✾➣✣↔✔↕➙↔✔➛✏➜✸➝✻➞✄➟✣➠P➞✥➡ ➢❁➤ ➐P➑❫➥ ➦✚➧✶➨❼➩✜➫❙➭❼➯⑥➲s➨❉➳➸➵❖➺➼➻s➨☞➽➎➾ ➐❺↕➙↔✔➛✏➜✸➝✻➞✄➟✣➠P➞✥➡➪➚ ➶➘➹❲➴❡➷✻➬➸➮❜➱✶✃✚➷➘❐❮❒✜❰●Ï✻Ð✚Ñ❖Ò❢Ó➐Ò❖Ô❇Õ❷ÖsÕ➑✧➥Ñ✲Ñ➥❮➐P➑✞×❒❁ØÚÙ❑Õ❷ÛÜÕ➑Ñ➐❰✁Ý❊Ï✚❰●ÞßÏ✻Ð✚Ñ✰❰ ➾ ➚❚às↔✔➛✧á✣â ãÏ➸ä☞Þ✝ä✧ÖåÒæä❫ÖçÑ❉Ò❢Ó✧Ñ❉Ù✏Ï✚ä➑✧➥❑➐❰ã☞èÏ➑✧➥ÕÜÒ❢Õ❷Ï➑Ö ➚ ➧✶éÏPÒ●Ñ✁Ô☛Õ❷Ö ×❒❁ØêÙ✏Ñè➐ä✧Ö❢Ñ✐Ò●Ó❑Ñ☎ä➑✧➥Ñ✰❰❢ÛãÕ➑✧ë Ï✚ì✏Ñ✲❰➐Ò●Ï✚❰■ÕíÖ ×❒✄Ø ➚ ➩ î ï✒ð✔ñ✖ò✘ó✒ô õÕ➑Õ❷Þ❧Õ❷ö➐Ò❢Õ❷Ï➑ ❒✜❰●Õ➑èÕ❷ì❑Û❷Ñ â÷ ➫ ➐❰ë Þ❧Õ➑ ø✈ù✚úüû➤ Ô ➧❊ý❖➺çý❲➩✜þÿ➦✚➧✶ý❲➩ ￾ ✁✄✂ ☎ ✆✞✝ø✠✟ ✡ ☛Ñ➐✌☞➎×Ò ➐Ò❢Ñ✲Þ❤Ñ➑Ò â ÷ ➽❺➾ ✍ Ô ➧÷ ➺❢➨❼➩✜➫⑥➦✚➧❊➨❩➩ ￾ ✁✄✂ ☎ ✎✑✏ ✆✓✒✔✝✖✕✟✘✗✠✙ ➺ ➻☎➨❺➽❺➾ ✡ ➢✛✚ ➶✠➹❲➴❡➷✻➬➸➮❜➱✶✃✚➷✢✜ ➧ ➐ ➩ ×Ó❑Ï✤✣✞Ò●Ó➐Ò✜ÕÜÝ✦✥ â★✧✪✩✬✫✭ Õ❷Ö ➥Ñ✓✮➑Ñ➥ Ùã ✥ ➧❊ý❉➩❁➫ ✯ ✰ Ô ➧❊ýs➺❢ý❲➩✒þ✝➦✚➧❊ý❲➩ Ý❊Ï➸❰ ➝✻➛✲✱❖×❒✄Ø Ù❑ÕÜÛ❷Õ➑Ñ➐❰✁Ý❊Ï✚❰●Þ➪Ô ➐✚➑✧➥ ÛÜÕ➑Ñ➐❰✁Ý❊Ï✚❰●Þ ➦ Ï✻Ð✚Ñ✲❰ ✧ ✍ Ò❢Ó✧Ñ➑ Ò❢Ó❑Ñ☞Þ❧Õ➑Õ❷Þ❤ÕÜö✰Ñ✲❰ ÷ Ö ➐Ò●Õ❷Ö✳✮✧Ñ✰Ö Ô ➧÷ ➺❢➨❼➩s➫❅➦✚➧❊➨❩➩ ✍ ➻❚➨ ➽ ✧❚➚ ➧ ✫➑ Ò❢Ó❑ÕíÖ✴✣➐ã ✍ ëÕ❷Ð✚Ñ➑➘➐ ✣tÑ➐✵☞ ÖçÒ ➐Ò●Ñ✲Þ❤Ñ➑Ò ✍ Ï➑Ñ è➐✚➑ ✶➐P➑Ò●Õ✸✷❵Ð➐❰●Õ➐Ò❢Õ❷Ï➑✺✹ Ò●Ï✻✮➑✧➥ ✥ ➚ ➩ ➧Ù ➩ ✼➐✵☞Ñ ✧ ➫ ✫✭✾✽ ✍ ➐✚➑✧➥ Ò❢Ó❼ä✧Ö☛Ö❢Ó❑Ï✤✣✍ Ùã ➐ì❑ì❑❰●Ï✚ì❑❰●Õ➐Ò❢Ñ èÓ❑Ï➸ÕèÑ➎Ï✚Ý❉Ô ➐P➑✧➥ ➦ ✍ Ò●Ó➐Ò Ò❢Ó❑Ñ❧Þ❤Õ➑Õ❷Þ❤ÕÜö✰Ñ✲❰ ÷ ➽ ✧ Ï✚Ý✿✥ ➧❊ý❲➩✁➫ ✯ ✰ ý ❀❂❁ý þ ý ❀✦❃✑❄ Ý❊Ï✚❰ ➐P➑ã ×❒✄Ø❅Þ➐Ò❢❰●Õ✸❅ ❁ ➽ ✫✭✽✲❆❇✽ ➐P➑❫➥ ❃ ➽ ✫✭✽ ❄ Ö ➐Ò●Õ❷Ö✳✮✧Ñ✰Ö ❁ ÷ ➫ ❃ ➚ ❈❊❉●❋❍❉✖■ ❏▲❑✤▼✲◆✛❖P❑❘◗❊◆❂❙❯❚❱❖✞❲✵❳ ÷ ➽✠➾ î ï✒ð✔ñ✖ò✘ó✵❨ ×Õ➑èÑsÔ☛Õ➑Ð✚Ï➸ÛÜÐ➸Ñ✰Ö ➠✻➛❫↕❩✱✿❬✜➞❪❭✥á❴❫✚➜✲➞✥↔❛❵❜➝Pá❪↔❛❵✻➜✄❭ ➾ß➫ ✍ ❜ ➨☛➽❘❝✯ ➧❡❞②➩❣❢❩➨❤❢ ✐ ➫❦❥❇❧ ♠ ❝ ✯ ✙ ➧♥❞②➩ â ❝ ✯ ➧❡❞②➩ ♠ ❜ ➨♦❢❘➭❑➯ ➨ ✰ ➳✒➵✞➺❲➭❼➯☎➨ ✰♣ ➳✒➵✞➺❲➭❼➯ ➨ ✰ q ➳➸➵ ✮➑Õ➙Ò●Ñ ❧ r s