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1 O Sovbiiy or Higher Order Prboic Eqios Mrí López Mores Deprme o Comper Sciece Moerrey Isie o echoogy Meico Ciy Cmps Ce de PeeNo Ejidos de HipcopCP438 Meico DF MEXICO Absrc: - We cosider he Cchy probem or higher order ier prboic eqios i yer We esbish ew priori esimes or he soio o his probem i geer isoropic Höder orm der he ssmpio h he coeicies d he idepede erm sisy he geer isoropic Höder codiio o epoe α( wih respec o he spce vribes oy I his coecio however we so obi esime or he mods o coiiy wih respec o he ime o he higher derivives wih respec o spce vribes o he correspodig soios O he bsis o or ew priori esimes or he soio o he Cchy probem or higher order ier prboic eqios we esbish he correspodig sovbiiy heorem or his probem i geer Höder isoropic spces We esbish he oc sovbiiy o he Cchy probem or higher order oier prboic eqios wih he id o he ress o he correspodig ier heory Key-Words: - esimes sovbiiy eqios prboiciy soio probem Irodcio I he prese wor we cosider he higher order ier prboic eqio ( D ( m Π i he yer [ iii codiio ϕ ( Here ( ( E wih he ( is poi o he - dimesio Ecide spce E [ i ( D i We esbish ew priori esimes or soios ( o he probem (( i geer isoropic orms der he ssmpio h he coeicies d he idepede erm re coios sisy he geer Höder i Π o epoe α ( > wih respec o he spce vribes ( oy ( See heorem I his coecio however we so obi esime or he mods o coiiy wih respec o he ime o he edig derivives D m (See heorems 3 Noe h i he wors [-[ he priori esimes o his ype hve bee obied der he ime o (geer Höder codiio wih respec o he oiy o vribes ( o he coeicies d he idepede erm o eqio ( O he bsis o or ew priori esimes or he soios o he probem ( ( we esbish he correspodig sovbiiy heorems or his probem i geer Höder isoropic spces (See heorem 4 We ssme h he coeicies o Eqio ( sisy he iorm prboiciy codiio: or y o-zero vecor ξ ξ ξ E ; Π ( (
2 ( m m ( λ cos > ξ ξ ξ > λ ξ ξ m ÀÀ We ppy or ress i he ier heory o esbish he oc sovbiiy wih respec o he ime i geer Höder isoropic spces o he Cchy probem or he oier prboic eqio (See heorem 5 m ( D D A (4 i Π wih he iii codiio ( D ( d D r is he se o derivive D r r m I he prese wor he eqio (4 is ierized direcy No codiios re imposed here o he re o he growh o he oieriy o he cio m A ( p p p (See [9 p r p which p scr ( is deied or ( Π d y r p p r m he mi ssmpio cocerig o he m cio A( p p p is he prboiciy codiio: or y o zero vecor ξ ( ξ ξ E ; ( Π m d p p p ( m A m m p ξ ξ ξ p p p m ξ > ÀÀ I mos he wor we sppose h i he eqio ( he cio ;he cios ( m d sisy he geer Hóder codiio i Π o epoe ( > wih respec o he spce vribes ( oy d sisies he geer Höder codiio i Π o epoe α( > wih respec o he spce vribes ( oy A he coeicies d he idepede erms o eqio re coios i he yer Π We reqire ess smoohess codiios rom m A p p p d he cios ( ( ϕ h i he wors b[ 6 [ 7 d [ 9 ( See heorem 5 Some cose ress or secod order prboic eqios hve bee esbished i [3 -[7 d [9 Bsic Noios Aiiry Proposiios We sh sy h he cio ( deied i he yer Π sisies he geer Höder codiio o epoe ( > i Π wih respec o he spce vribes i here eiss cos C > sch h ( y ( (y C y ((y Π he cio ( is deied d coios i < < Moreover i hs he oowig properies: I ( σ [ [ i Ib ( or i or Ic I σ he or > or ( d ( ( ( [ is siciey sm mber (we sppose h he derivive ( eiss d i is coios cio i [ [ R Noe h he codiio Ic irodces ew se o cios ( he cios ( h sisy he geer Höder codiio wih respec o spce vribes oy I his ew se o cios we wi obi he correspodig eisece d iqeess heorems or he soios o he probems (( d (4( I oows rom he codiios I Ib Ic h
3 ( II is moooicy icresig cio or R ( ( q IIb σ ( σ i d oy i or iormy respec o q < q b < We deoe by Γ he se o cios > or which ( ( ( Λ Λ ( For he cios ( Γ d < we irodce he Λ ( cios B ( Λ ( he cio B ( is cio o he ype ( III B ( > IIIb here eis coss C C sch h or R d or [ ( B ( < C B > < C or IIIc here eiss cos C 3 sch h or ( ( [ ( ρ ρ dρ C 3 Λ ( ρ Now we give wo empes o cios o he ype ( : ( cos < < b ( < ep( ( b b ( b > ep( < < cos < (see [8 For he cios ( deied d Höder coios( i he geer sese o epoe b ( > i he yer Π wih respec o spce vribes we irodce he oowig orms [ sp [ sp (6 ( ( sp ( E H ( (7 ( ( ( ( y H ( sp ( y y E y (8 y For he cios ( h hve coios derivives wih respec o p o he order ( m m icsivey i he yer Π d sisyig he geer Höder codiio o epoe ( > wih respec o spce vribes i he yer Π we deie he orms m ( D ( m [ m ( m (9 sp m ( m ( We wi deoe by ( C Π m m ( he Bch spce o cios ( h re coios i Π [ E ogeher wih derivives respec o p o he order ( m m icsivey d hve iie orm ( Wih respec o he coeicies o he eqio ( we ssme h [ ( C ( Π m 3
4 d m ( [ B< ( Lemm Sppose h he cio ( C ( Π mb he or < ε < he oowig ieqiy hods [ [ ε m B mb [ C( ε For he proo o his emm see [7 For eqio (4 we cosider i ddiio o he prboiciy codiio (5 h here eiss domi H {( Π ; M m M r p M i r m M cos} m which he cio A ( p p p ogeher wih is derivives wih respec o p r r r p ( p r m p o he secod order icsivey is coios sisies he Lipschiz codiio wih respec o p r r m d he geer Höder codiio o epoe ( > wih respec o d wih he cos C M [ α ( Moreover A( C ( Π [ ( C A α ( A he meioed derivives re boded i H M by he cos C M Now we sh cosider he eqio ( wih he iii zero codiio: ( 3 Bods or soios o he Cchy probem or ier prboic eqio [ heorem Le ( C ( Π mb be soio o he probem ( ( i he yer Assme h Π [ ( C ( Π [ α ( C ( Π ( α( Γ ( σ α( σ i or σ < σ < Frhermore he codiios ( 3 d ( hod he here eiss cos K depedig oy o m λ B α( ( d B ( sch h or σ σ [ [ [ K[ m mb ( ( α( (3 Remr We c redce he Cchy probem wih o-zero iii codiio ( o he Cchy probem ( ( by mes o he rsormio ( ( [ ( C ( E m ( Now we sh esbish esime o he mods o coiiy wih respec o he ime o he derivives D m or he soios o he eqio ( [ heorem Le ( C ( Π be mb soio o he eqio ( i he cyidric domi Q [ Ω Ω boded domi i E Assme h [ ( C ( Q ( Γ σ ( i or σ < Frhermore he codiios ( 3 d ( hod he i he pois ( ( < < < δ δ Q ( δ Ω δ δ { Ω ; dis( Ω δ } here Ω > 4
5 eiss cos K depedig oy o m λ B α( ( d B ( sch h or y derivive D m o he cio ( he oowig esime hods or j m D m j m j ( D ( m B m j ( [ [ K[ he proo o he heorem is simir o he proo o he heorem i [6 b resoig s i he proo o heorem o he prese pper heorem 3Sppose h ssmpios o he heorem hod he or every ( ( Π < here eiss cos K depedig oy o m λ B α( ( d B ( sch h or y derivive D m o he soio ( o he Cchy probem ((he oowig esimes hod or j m ( (4 D ( K M m j K M( (5 M [ ( D m m j B ( ( m j σ σ [ [ m α( he proo o his heorem is simir o he proo o he heorem 3 i [6 b resoig s i he proo o heorem o he prese pper 4 Eisece d iqeess heorems heorem 4Sppose h codiios o heorem re re he here eis iqe soio [ ( C ( Π mb o he ( Cchy probem ( ( wih coios derivives i Π We c ge he proo o his heorem o he bsis o he ew priori esimes esbished i his wor d wih he id o he mehod o coiiy i prmeer (see [4 d [ We proceed ow o orme he oc eisece heorem or soios o he oier probems or he eqio (4 Here we cosider h he cio m ( D D A L( F( D A( L( (A p (A A( ( D m p D m m ( D heorem 5 Sppose h ssmpios wih respec o he cio m A p p p hod Moreover ( σ < σ < he here eiss deermied by he bove ssmpios sch h he probem (4 ( hs i he Π E iqe soio yer [ [ ( C mb ( Π wih coios derivive i [ Π E Remr We c redce he Cchy probem wih o-zero iii codiio ϕ( o he Cchy probem wih he zero iii codiio by mes 5
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