AAin + Vlx )U* If For some f- 70 - All * in @ . Ev (4) x S 5- - PowerPoint PPT Presentation

2. Energy space DIED . Assume I uxi > 0 : AAin + Vlx )U* If For some f- 70 - All * in @ . Ev (4) x S 5- fleece ? (d) 42 Then - k¥ . 52 ake k*=foglxD± on IR2 ) Example C- is T . = 'T µ q¥ µ ⇒ =G , - but 52=1177 BT . Skate 5¥42 =§t5 ④ ' 42 si F ⇒ TD s -

Assume 7 a E L eoe (D) , A > 0 ' Theorem a. e. : . G) Eu (a) = 9 a G) 42 dx tee CIGS . - ¥ • the completion of CI (D) w.int . Then = LEAD 't is the Hilbert space ② (D) , 1144 = Soda 4 + SV44 - scalar prod , ( ie , 4) v D D @ i (D) a. L4D , a ) dx . - Lambda property ( * ) - .

⇒ ⇒ ⇒ on CE1R ) ( hehe ⇒ ⇐ 4=0 ) c) Heller is a norm a Cauchy sequence 2) ten ) w.int . to hell u is ) ) is a Cauchy seq . in L4D , a ten dx a an = him Elan ) Ev ( ie ) : ) h - Ooo ⇐ DIED a L the HDD K - and " ( § an - so

Examples : IRN , 1) N73 - D ou . - ¥ (1+1 × 12) alent function - T T ake U* = e (1+1 × 12) = ↳ 1¥ in IRN - - All * = = ¥ = llelxif " * * ⇒ a a lHxtjtdxjs@i.la ⇒ 0 ! LIRN ) as L2 4RN , . @ loaf )% - completion of CE4R ' ) w.int )

a LIE IN ) , 0 ! I 4 L4H ) ② . LIRN ) ~ ⑥ ! ( RN ) ⇒ H ( Rn ) ! ' - does not satisfy A- property 2) on IR2 - D is the only positive super harmonic ! ) ( const e) is not weel defined ! ⇒ ②

IR2 1BT 3) - is on = fog 1 × 1 ) 't in IR7B , T ake U * > 0 a. G) = ¥ = I i × ¥g ° 7 BY a LY1R ? By + G) dx ) ② ! .

Remark so teethe ) Assume Ev : does not satisfy but Ev a- property . → + V has just one luptoseaear ) Then = solution positive and super solution is eaeeed critical operator :b TV . - property ⇒ Ev satisfies A suberitio.ae operator - At V is JFA 2006 ] [ Pinch over , Tintarev ,

i ) on IR2 - critical Examples : - b - Ist Dirichlet eigenvalue D- bounded , 2) ai is critical oud - b- Ii is the only positive solution ! ) ( Qi = aisle ' Sleep tee case f- b- A) 4 , = 0

ttfe@tlsl.bt 7 use @ HR ) Theorem . , - 9- - D Ust Vas in 52 . Minimize the functional & - ( fie ) on ② (d) { Heli → min - B Iz Ev

( weak Maximum principle ) Theorem assume UE8Y (D) is a super solution , A. i. e. , for some fe ② dR))* = f- 70 - DUTVU in 52 ⑦ Then U70 . e- DIED . u=u+ let , u - u - - ¥-1 , U - 4 = OE5 ounce t9Vuy=f µ )nitfVluhi)u= - aw - Strewth - SV4Y - Evti ) so - - - 0k¥ ⇒ Ev (a) = 0 ⇒ u

Corollary ( Weak comparison principle ) u ,VeH' eoelsD.nl 'ealR,KDdx ) are Let sub and super solutions : - AV1-4DV SO in @ - but Vlx )u=0 , e ② (D) ( u - v ) and on 052 - I U7U . Then in 52 U > V . + KD4V ) > 0 in @ - o Luv ) . Osten → HID 'v ake Que CE (D) , T F- ③ repeat previous argument

Three possibilities for - Dtv in 52 : 1) 74=10 : Efa ) no positive superset a 0 ⇒ Er (4) 30 and 2) → + V is critical ⇒ exactly one positive ( ) solution super ④ a- property to -N is subcriticaei 3) Ev (e) 70 ⇒ large cone of positive super solutions ④ variational principle in @ I (D) comparison principle ④

- Ep in RN ) ( Hardy operator Exercise . - D - D- ¥2 satisfies A- property in IRN Show that ' ee ↳ etc ) , eH=f¥ ) t and N =3 . 4 × 1=1 × 11 ¥2 & It - but u* - -" ⇒ ¥A¥Io > 0 - roots of skin - D= , t.at -1 C- EL , Lt ) the Ht E- HI , n )

- All + IT U =§g 1 × 1+2 = Eu = IT xx ) LZGRN , II. dx ) , ⇐ ( Rn ) ' ② a t tea CH N73 .