Estimating constants in generalised Wente-type estimates Asama Qureshi Supervised by Dr. Yann Bernard, Dr. Ting-Ying Chang Monash University
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1 Estimating constants in genealised Wente-tye estimates Asama Queshi Suevised by D Yann Benad D Ting-Ying Chang Monash Univesity Vacation Reseach Scholashis ae funded jointly by the Deatment of Education and Taining and the Austalian Mathematical Sciences Institute
2 Intoduction Let Ω be a bounded simly connected domain in R with a smooth bounday Ω Given two functions a and b such that a L (Ω) and b L (Ω) then let φ be the uniue solution in L (Ω) to the Diichlet oblem φ a b a b x y y x φ in Ω () on Ω whee Ω is aameteised by x and y and subscit x and y denote the atial deivatives of a and b The Wente (969) ineuality states that thee exists a constant C (Ω) such that kφkl (Ω) +k φkl (Ω) C (Ω)k akl (Ω) k bkl (Ω) () In this ae we examine the following genealisation of the Wente ineuality on D whee D a disk of adius in R centeed at the oigin kφkl (D ) +k φkl (D ) C ( D )k akl (D ) k bkl (D ) with + (3) < < We find the best constant C ( D ) in the following esult Theoem - φ satisfies the genealised Wente ineualities: kφkl (D ) C ( D )k akl (D ) k bkl (D ) and (4) k φkl (D ) C ( D )k akl (D ) k bkl (D ) fo two otimal constants C ( D ) and C ( D ) satisfying: K C ( D ) C ( D ) K whee K sin(π/) ( )/ < < This is a significant esult because () initially only tells us that φ is in L but this does not imly that φ L o that φ L The div-cul stuctue of () with the esence of the negative sign intoduces a comensation henomenon allowing us to make the estimates in () Baaket (996) and He lein () found the otimal constant C (Ω) + indeendently of the domain Ω That is Baaket found this constant fo simly connected Ω and He lein found this fo any Ω
3 Poof of esults Hee we will ove the esults stated in (4) Fist we find C ( D ) and then we will find C ( D ) Theoem - Let φ be the solution to the Diichlet oblem φ φ ax by ay bx in D on D (5) Then it holds: kφkl (D ) C ( D )k akl (D ) k bkl (D ) with C ( D ) K whee K sin(π/) ( )/ < < Poof By Geen s Reesentation Theoem we have the following exession fo φ E(x x ) φ φ(x ) φ E(x x ) ds + E(x x ) φdx v v D D fo fixed x D E(x x ) log x (6) x and v is the exteio nomal vecto to D Choosing x in (6) gives us φ() E(x) φdx (7) B Lemma - Pefoming a change of vaiables on (5) into ola coodinates gives φ whee x + y and θ Actan y x (a bθ aθ b ) (8) Poof We stat out with (5) and use the chain ule ax by ay bx (a x + aθ θx )(b y + bθ θy ) (a y + aθ θy )(b x + bθ θx ) a bθ x θy + aθ b y θx a bθ y θx aθ b x θy a bθ (x θy y θx ) aθ b (x θy y θx ) (a bθ aθ b )(x θy y θx ) (9)
4 Noting that x θy y θx x x +y + y x x y yx x +y + y x x 3 y 3 and substituting into (9) the oof is comlete Substituting (8) back into (7) we have φ() log (a bθ aθ b ) dθd log (a bθ aθ b ) dθd log ((abθ ) (ab )θ ) dθd log (abθ ) dθd Using the oduct ule φ() φ() Since ab () ab () φ() then using integation by ats abθ dθd () Now obseve that (a a)bθ dθ abθ dθ whee a() a( σ)dσ Hence (a a)bθ dθ abθ dθ ka akl () kbθ kl () () whee the last ineuality is tue by Ho lde s ineuality By the Poincae ineuality we have that ka akl () kbθ kl () K kaθ kl () kbθ kl () whee K sin(π/) ( )/ () see Aendix A 3
5 Now substituting () into () we have that K K φ() d kaθ kl kbθ kl kaθ kl kbθ kl d (3) With anothe alication of Ho lde s ineuality Note that K φ() K and kaθ kl! d kbθ kl aθ dθ R R d! d bθ dθ d (4) so we have K φ() aθ dθd bθ dθd (5) Now obseve that aθ dθd aθ aθ! dθd + a a dθd dθd!! Thus: aθ dθd k akl (D ) (6) Similaly bθ dθd k bkl (D ) (7) Substituting (6) and (7) back into (5) we have φ() K k akl (D ) k bkl (D ) (8) This gives us the ue bound of φ at the cente of the disk To find the ue bound fo φ ove the whole disk we intoduce the confomal tansfomation T : D D given by T (z) z + z + z z with fixed z D (9) 4
6 T is a smooth ma that mas the bounday of D to itself and T () z Let a a T b b T φe φ T Lemma - Euation (5) is confomally invaiant unde T namely: φe a x b y a y b x φe () Poof Let (u v) T (x y) e y) φ(u v) φ(u v)xx φ(u v)yy φ(x (φu ux + φv vx )x (φu uy + φv vy )y (φux ux + φu uxx ) (φvx vx + φv vxx ) (φuy uy + φu uyy ) (φvy vy + φv vyy ) Since T is a confomal tansfomation then u and v satisfy the Cauchy-Riemann euations (ie ux vy and uy vx ) giving us φe (ux (φux + φvy ) + uy (φuy φvx ) + φu (uxx + uyy ) + φv (vxx + vyy )) Futhemoe since T is confomal u and v ae hamonic functions (ie uxx + uyy and vxx + vyy ) so φe (ux (φux + φvy ) + uy (φuy φvx )) (ux (φuu ux + φuv vx + φuv uy + φvv vy ) + uy (φuu uy + φuv vy φuv ux φvv vx )) (φuu ux + φvv ux vy + φuu uy φvv uy vx ) (φuu + φvv )(ux + uy ) φ(ux + uy ) () Now evaluating the ight hand side of () a x b y a y b x (au ux + av vx )(bu uy + bv vy ) (au uy + av vy )(bu ux + bv vx ) au ux bv vy + av vx bu uy au uy bv vx av vy bu ux (au bv av bu )(ux vy uy vx ) (au bv av bu )(ux + uy ) () 5
7 Combining () and () we see that e y) a x b y a y b x φ(x φ(u v)(ux + uy ) (au bv av bu )(ux + uy ) Howeve ux + uy and since T 6 then ux + uy > thus we have φ(u v) au bv av bu Using Geen s Reesentation Theoem on () similaly as in (7) we get e a b θ dθ d φ() (3) T induces a diffeomohism of D We aamateize D T ( D ) by θ [ ] Thee exists a constant γ : Actan(T ()) deending only on z such that a b θ dθ +γ abθ dθ (4) γ The Poincae ineuality is invaiant unde tanslation (mutatis mutandis Aendix A) +γ abθ dθ K kaθ kl () kbθ kl () (5) γ hence a b θ dθ K kaθ kl () kbθ kl () Hence as in (8) e φ() K k akl (D ) k bkl (D ) φ(z ) K k akl (D ) k bkl (D ) Thus: As this holds fo all z in D with the same constant K we find kφkl (D ) K k akl (D ) k bkl (D ) (6) Thus C ( D ) K theeby comleting the oof 6
8 Now we will ove the claim fo the constant C ( D ) in (4) Theoem - Let φ be the solution to (5) Then we have: k φkl (D ) C ( D )k akl (D ) k bkl (D ) with K C ( D ) and K sin(π/) ( )/ Poof Let ( y x ) ( x y ) Then a b ax by ay bx (7) By Ho lde s ineuality a b L (D ) k akl (D ) k bkl (D ) (8) whee + < < Now we find an ue bound fo k φkl k φkl (D ) φ ddθ φ )ddθ θ φ ddθ + (φ + φ ddθ θ (9) Let A φ ddθ and B φ ddθ θ Fist we simlify A With an integation by ats we get: A dθ φφ φφ ddθ φφ ddθ Then because φ on D A φφ ddθ φφ ddθ 7
9 Now we simlify B Using integation by ats gives us B φφθ θ d θ φφθθ dθd φφθ ( ) φφθ ( ) so we have: B φφθθ dθd Substituting A and B back into (9) we get k φkl (D ) φφ + φφ + φφθθ ddθ φ φddθ (3) By (7) k φkl (D ) φ a b ddθ and by Ho lde s ineuality φ a b ddθ kφkl (D ) a b L (D ) (3) By (6) and (8) k φkl (D ) kφkl (D ) k akl (D ) k bkl (D ) K k akl (D ) k bkl (D ) Hence k φkl (D ) K k akl (D ) k bkl (D ) (3) Theefoe we see that C ( D ) K as claimed 3 Futhe wok Ou esults give the best constant fo the disk but we do not know whethe this is tue fo any Ω unlike the standad Wente ineuality [Toing (997)] Futhe it is not clea whethe the constant emains valid fo Ω simly connected unlike the standad Wente ineuality [Baaket (996)] Insecting ou oof we conjectue: 8
10 Let φ be the uniue solution to () whee Ω is a gah ove a cicle then it holds: kφk L (Ω) C ( Ω)k akl (Ω) k bkl (Ω) kφk C ( Ω)k ak k bk L (Ω) L (Ω) L (Ω) whee C ( Ω) and C ( Ω) ae as in (4) This is because thee exists a smooth bijective confomal ma between all simly connected domains in C by the Riemann maing theoem taking the bounday of D to the bounday of Ω Since D is maed to Ω in a one-to-one fashion and all oints on Ω can be aameteised along [ ] uniuely this should not affect ou constant found in (4) Aendix A Poincae Ineuality The Poincae ineuality in one dimension on ( ) is given by a(x) a x whee a x R L ( ) G a (x) L ( ) (33) a(x)dx Stanoyevitch (99) found on the inteval ( ) the Poincae constant G to be G sin(π/) π( )/ < < We will show that on ( ) a(y) a y whee a y L () πg a (y) L () (34) a(y)dy (35) Poof Let y (x + )π dy π dx We have a(y) a y L () a(y) a(u)du dy a(π(x + )) π a(π(u + ))du dx 9
11 Let A(x) a(π(x + )) and A x a(y) a y R A(u)du then L () A(x) π π A(x) A(u)du dx A x L ( ) By the Poincae ineuality π A(x) A x L ( ) πg A (x) L ( ) a (π(x + )) dx πg Now subsituting y in πg a (π(x + )) dx G da dy dy dy dx G π a (y) L () G π a (y) dy Thus as claimed a(y) a y L () G π a (y) L () Refeences Baaket S (996) Estimations of the best constant involving the L nom in Wente s ineuality Annales de la faculte des sciences de Toulouse 5(3) He lein F () Hamonic Mas Consevation Laws and Moving Fames nd edn Cambidge Univesity Pess chate Stanoyevitch A (99) Geomety of Poincae Domains PhD thesis The Univesity of Michigan Toing P (997) The otimal constant in Wente s L estimate Comment Math Helv Wente H (969) An existence theoem fo sufaces of constant mean cuvatue J Math Anal Al
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