Viscosity Solutions with Asymptotic Behavior of Hessian Quotient Equations. Limei Dai +

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1 Itertio oferee o ompter d Atomtio Egieerig IAE IPIT vo 44 IAIT Pre igpore OI: 776/IPIT44 ioit otio ith Amptoti ehvior of Hei Qotiet Eqtio Limei i hoo of Mthemti d Iformtio iee Weifg Uiverit Weifg 66 The Peope epbi of hi Abtrt I thi pper e e the Perro method to prove the eitee of vioit otio ith mptoti behvior t ifiit to Hei qotiet eqtio Keord: vioit otio mptoti behvior Hei qotiet eqtio Itrodtio I thi pper e td the Hei qotiet eqtio i \ Ω Here Ω i trit ove boded domi < deote the Hei mtri of the ftio d i defied to be the th eemetr mmetri ftio of the eigeve λ λ λ of tht i Whe e deote σ λ i i < < i λ λ Hei qotiet eqtio i importt of f oier eipti eqtio hih i oe reted to geometri probem ome e-o eqtio be regrded it pei e Whe it i -Hei eqtio I prtir it i Poio eqtio if hie it i Moge-Ampère eqtio if Whe tht i det Δ rie from pei Lgrgi geometr[]: if i otio of the grph of over i i pei Lgrgi bmifod i tht i it me rvtre vihe everhere d the ompe trtre o ed the tget pe of the grph to the orm pe t ever poit Therefore h dr mh ttetio ee [-5] The etire otio of PE hve bee tdied b m thor ee [67] I [L] ffrei d Li hve ivetigted the eitee of otio ith mptoti behvior to Moge-Ampère eqtio i eterior domi i i [] h proved the eitee of vioit otio ith mptoti behvior to Hei eqtio i eterior domi I thi pper e td the vioit otio ith mptoti behvior t ifiit to Hei qotiet eqtio i \ Ω To or i the rem of eipti eqtio e hve to retrit the of ftio Let Γ λ σ λ > i { } orrepodig thor E-mi ddre: imeidi@hooom 65

2 A ftio \ Ω i ed -ove iform -ove if λ Γ Γ here λ λ λ λ i the eigeve of the Hei mtri From [] d [] e o tht i eipti d i ove ftio of the eod derivtive of if i iform -ove It i tr for the otio of to be oidered i the of iform -ove ftio A eteive td of vioit otio of eod order prti differeti eqtio be fod i [8] d [9] For the reder' oveiee e re the defiitio of vioit otio to Hei qotiet eqtio A ftio \ Ω i ed vioit botio of if for \ Ω ξ \ Ω tifig e hve ξ \ Ω d ξ ξ A ftio \ Ω i ed vioit perotio of if for \ Ω - ove ftio ξ \ Ω tifig e hve ξ \ Ω d ξ ξ A ftio \ Ω i ed vioit otio of if i both vioit botio d vioit perotio of A ftio \ Ω i ed -ove if i the vioit ee σ λ i \ Ω \ Ω i -ove if d o if i ove Mi et bhrmoi; i -ove if d o if i From Propoitio i [9] e o the premm of et of botio i ti botio Moreover omprio priipe of vioit otio to Hei qotiet eqtio hod ee Theorem i [8] The e tte the fooig eitee d iqee ret ee Propoitio i [9] Lemm Let be b i d f be oegtive ppoe re repetive vioit botio d perotio of f i d tif ϕ the there eit iqe -ove ftio tifig d ϕ o Lemm Let be b i d f be oegtive ppoe tif i the vioit ee f i The the irihet probem f i o h iqe -ove vioit otio Lemm Let be ope et i d f be oegtive Ame -ove ftio v tif repetive v f f 66

3 Moreover et The tifie i the vioit ee v v v \ f Lemm 4 Let Ω be boded trit ove domi Ω v Ω The there eit ott o depedet of Ω v h tht for Ω there eit tifig Ω < v Ω \{ } here v Or mi ret i the fooig theorem Theorem Let For there eit ott h tht for > there eit - ove vioit otio \ Ω of hih tifie im p b < 4 Ω here d Ω i ove domi i Proof btrtig ier ftio from e o eed to prove the theorem for b We divide the proof ito three tep I the firt tep e otrt vioit botio of Let Ω be boded trit ove domi i α ppoe Φ Ω ee [] i -ove ftio tifig Φ Ω > Φ Ω the omprio priipe Φ i Ω Ad b Lemm 4 for eh Ω there eit h tht < Φ Ω \{ } here d p < Therefore Th Ω Φ Ω p Ω 67

4 tifie Ω Φ 5 d from Propoitio i [9] efie \ Ω Ω Φ The 5 d Lemm tifie i the vioit ee F Fi ome > h tht Ω d et For > defie if d A diret tio give for > i i i δ here rottig the oordite e m et r therefore here o Γ λ Let the - the defiitio of oeqet if if d < 6 68

5 Fi ome > tifig > We hooe > h tht for if d > The b 6 the defiitio of if if if d d d d d Let if d μ The μ i otio d mootoi ireig for d he μ Ad μ Moreover he O μ 7 For et μ d defie for > { } m The Ω d b 7 he O 8 hooe ffiiet rge h tht for if d Therefore 69

6 Lemm tifie i the vioit ee I the eod tep e defie the Perro otio of For et deote the et of ftio v tifig v v Ω v er Hee φ efie i p{ v v } the defiitio of i vioit botio of d tifie Ω I the fooig e prove i vioit perotio of I the third tep e prove i vioit otio of tifig 4 For \ Ω ε > hooe b ε \ Ω Lemm the irihet probem h vioit otio From the omprio priipe efie ψ \ Ω Lemm ee here Hee ψ \ Ω g g g from the omprio priipe g ψ the defiitio of ψ i oeqet ee i rbitrr e o i vioit perotio of the defiitio of o from 8 tifie 4 Theorem i proved Aoedgemet g i A ret The reerh pported b hdog Provie Yog d Midde-Aged ietit eerh Ard FdF5 hdog Provie iee d Tehoog eveopmet ProetY6 hdog Provie Ntr iee FodtioZAL8 4 eferee 7

7 [] Hrve d H Lo ibrted geometrie At Mth 98 48: [] L ffrei L Nireberg L d J pr The irihet probem for oier eod-order eipti qtioiii Ftio of the eigeve of the Hei At Mth : 6- [] N Trdiger O the irihet probem for Hei eqtio At Mth : 5-64 [4] J G o J Y he G et Liovie propert d regrit of Hei qotiet eqtio Am J mth 5: -6 [5] M Li d J G o The o regrit for trog otio of the Hei qotiet eqtio J Mth A App 5 : [6] L ffrei L d Y Y Li A eteio to theorem of Jörge bi d Pogoreov omm Pre App Mth 56: [7] L M i d J G o Mti-ved otio to f oier iform eipti eqtio J Mth A App 89: 4- [8] M G rd H Ihii d P L Lio Uer' gide to vioit otio of eod order prti differeti eqtio Am Mth o 99 7: -67 [9] H Ihii O iqee d eitee of vioit otio of f oier eod-order eipti PE omm Pre App Mth 989 4:

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