A Sharp Existence and Uniqueness Theorem for Linear Fuchsian Partial Differential Equations

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1 A Sharp Exisence and Uniqueness Theorem for Linear Fuchsian Parial Differenial Equaions Jose Ernie C. LOPE Absrac This paper considers he equaion Pu = f, where P is he linear Fuchsian parial differenial operaor m 1 P = (D ) m + j= α m j a j,α (, z)(µ()d z ) α (D ) j. We will give a sharp form of unique solvabiliy in he following sense: we can find a domain Ω such ha if f is defined on Ω, hen we can find a unique soluion u also defined on Ω. 1 Inroducion and Resul Fuchsian parial differenial operaors, deemed o be naural generalizaions of Fuchsian ordinary differenial operaors, were firs defined by Baouendi and Goulaouic in [1]. There, hey gave some generalizaions of he classical Cauchy-Kowalewski and Holmgren heorems for his ype of operaors. Their mehod has been applied and exended o various cases as can be seen, for example, in Tahara [6], Mandai [5] and Yamane [8]. In a previous paper [4], he auhor made use of he concep of weigh funcions o define a slighly larger class of operaors; exisence and uniqueness heorems similar o hose given in [1] were proved. In his paper, we shall presen a formulaion leading o an exisence and uniqueness resul sharper han he one given in [4]. The resul is sharper in he sense ha he soluion u(, z) of he equaion Pu = f has he same domain of definiion as he inhomogeneous par f(, z). We now sae he following noaions and definiions. Le N denoe he se of all naural numbers including zero. Le (, z) = (, z 1,..., z n ) R C n and denoe he corresponding derivaives by D = 1

2 / and D z = ( / z 1,..., / z n ). Le R > be sufficienly small, and for ρ (, R], denoe by B ρ he polydisk {z C n ; z i < ρ for i = 1, 2,..., n}. Le T >. We say ha a coninuous, posiive-valued funcion µ() on he inerval (, T ) is a weigh funcion if µ() is increasing and he funcion ϕ() = µ(s) s ds (1.1) is well-defined on (, T ), i.e., he inegral on he righ is finie. (See Tahara [7].) Definiion 1.1. Le τ (, T ), γ > and ϕ() be he one in (1.1). We define (i) ω τ [γ] = {z C n ; z i < R γϕ(τ) for i = 1, 2,..., n}, and (ii) Ω T [γ] = {(τ, z) R C n ; τ T and z ω τ [γ]}. Given any open subse D of C n, he se of all funcions g(z) holomorphic in D and coninuous up o D forms a Banach space A(D) wih norm g D = max z D g(z). In paricular, we will deal wih he spaces A(B ρ ) and A(ω τ [γ]). The norms in hese spaces will be denoed by Bρ and ωτ [γ], respecively. Definiion 1.2. Le p N and le C ([, τ], A(ω τ [γ])) be he se of funcions coninuous on he inerval [, τ] and valued in he space A(ω τ [γ]). (i) We say ha f(, z) belongs in K (Ω T [γ]) if for each τ [, T ], we have f() C ([, τ], A(ω τ [γ])). (ii) We say ha w(, z) belongs in C p([, τ], A(ω τ [γ])) if for all j p, we have (D ) j w() C ([, τ], A(ω τ [γ])). (iii) We say ha u(, z) belongs in K p (Ω T [γ]) if for each τ [, T ], we have u() C p([, τ], A(ω τ [γ])). Consider now he linear Fuchsian parial differenial operaor m 1 P = (D ) m + j= α m j a j,α (, z)(µ()d z ) α (D ) j. (1.2) Here, µ() is a weigh funcion and he coefficiens a j,α (, z) belong in he space C ([, T ], A(B R )), i.e., for any s [, T ], each of he funcions a j,α (s, z), 2

3 when viewed as a funcion of z, is holomorphic in B R and coninuous up o B R. Define he characerisic polynomial associaed wih he operaor P by C(λ, z) = λ m + a m 1, (, z)λ m a, (, z). (1.3) Is roos λ 1 (z),..., λ m (z) will be referred o as characerisic exponens. In his paper, we will assume ha here exiss a posiive number L such ha Rλ j (z) L < for all z B R and 1 j m. (1.4) Under he above assumpions, we have he following main resul. Theorem 1.3. Le P be he operaor given in (1.2). Then here exis T > and γ > depending on P such ha for any f(, z) K (Ω T [γ ]), he equaion Pu = f in Ω T [γ ] (1.5) has a unique soluion u(, z) in K m (Ω T [γ ]). Moreover, he soluion saisfies he a priori esimae m 1 p= max (D ) p u C max f, (1.6) where is he closure of Ω T [γ ] and C > is some consan dependen on he above equaion and on he domain Ω T [γ ]. We remark ha he domains of definiion of f(, z) and u(, z) are he same. Since boh f(, z) and u(, z) are defined on Ω T [γ ], we can resae our heorem in he following manner: for any T, γ >, le X T,γ and Y T,γ be he spaces K m (Ω T [γ]) and K (Ω T [γ]), respecively. Le W T,γ be he subspace of X T,γ consising of funcions u X T,γ such ha Pu belongs in Y T,γ. Define a linear operaor Ψ from X T,γ o Y T,γ wih domain W T,γ by Ψu = Pu. Le T,γ denoe he maximum norm in he closure of Ω T [γ]. Then X T,γ and Y T,γ are Banach spaces; given u X T,γ and f Y T,γ, we define heir norms by m 1 p= (D ) p u T,γ and f T,γ, respecively. Moreover, he operaor Ψ is a closed linear operaor from X T,γ o Y T,γ. Then he above heorem can now be saed as follows. Theorem 1.3. There exis T, γ > depending on P such ha he operaor Ψ is a one-one, closed linear operaor from X T,γ ono Y T,γ. Since Ψ is an injecion, Ψ 1 exiss and is also closed. The Closed Graph Theorem furher implies ha Ψ 1 is coninuous. The esimae given in (1.6) is jus a consequence of he coninuiy of Ψ 1. 3

4 2 Preliminary Discussion We can rewrie he operaor P as m 1 P = Q + j= α m j where he operaor Q is defined by and c j,α (, z)(µ()d z ) α (D ) j, Q = (D ) m + a m 1, (, z)(d ) m a, (, z) (2.1) c j,α (, z) = { a j,α (, z) if α, a j,α (, z) a j,α (, z) if α =. Noe ha he coefficiens of he ordinary differenial operaor Q are holomorphic funcions of z in B R. Noe furher ha he characerisic exponens of Q are he same as ha of P, and hence saisfy (1.4). Lemma 2.1. Fix τ > and le g() C ([, τ], A(ω τ [γ])). Then he equaion Qu = g has a unique soluion u() Cm([, τ], A(ω τ [γ])). This unique soluion is given by u() = 1 m! σ S m sm s2 ( sm ) λσ(m) ( sm 1 ) λσ(m 1) s m ( s1 ) λσ(1) g(s 1 ) ds 1 ds 2 ds m. (2.2) s 2 s 1 s 2 s m Here, S m is he group of permuaions of {1, 2,..., m}. A resul in symmeric enire funcions assers ha he soluion is holomorphic wih respec o z. The fac ha u(, z) belongs in C m([, γ], A(ω τ [γ])) is seen in he inegral expression above, bu may acually be obained a priori. (See Baouendi-Goulaouic [1].) To faciliae compuaion, we define for λ = (λ 1,..., λ m ) he funcion Gθ def (λ) = 1 m! ( sm ) λσ(m) ( sm 1 σ S m s m ) λ σ(m 1) ( θ s 2 ) λσ(1), (2.3) for some dummy variables s 2,..., s m. Define, oo, he inegral operaor [;θ] g def = sm s2 g(θ) dθ θ ds 2 s 2 ds m s m (2.4) 4

5 Using he above, we can now wrie he soluion u() of he equaion Qu = g as u() = [;s] G s(λ) g. In our proof of he main heorem, i will be necessary o consider he acion of he differenial operaor (D ) p on inegral expressions similar o he one in (2.2). One can easily verify he following Lemma 2.2. Le u() be he soluion of Qu = g. Then for a naural number p less han m, we have m (i) { 1 ( (D ) p si ) λσ(i) u = g h i (σ, λ) i=m p [;s 1 ] m! σ S m ) λσ(i 1) ( s1 ) } λσ(1). (2.5) s 2 ( si 1 s i The funcions h i (σ, λ) are polynomial funcions of he characerisic exponens λ 1 (z),..., λ m (z), and hence are bounded on B R. For breviy, le us se, for a naural number k, Hθ (k, λ) = 1 ( sk ) λσ(k) ( sk 1 ) λσ(k 1) ( θ ) λσ(1) h k (σ, λ). (2.6) m! s k s 2 σ S m By symmery, he funcions H s(k, λ) are holomorphic wih respec o z and hus belong in A(B R ). The following lemma is useful in evaluaing some inegral expressions in he proof. Lemma 2.3. Le k be naural number. Then following hold: sk sk 1 s1 ( s ) L ds (a) ds k 1 = 1 s k s s k 1 L k (b) sk s1 µ(s k ) s k µ(s k 1 ) s k 1 µ(s 1) s 1 ( s ) L s 1 [ϕ() ϕ(s )] k ds... ds k = 1 L k! The firs equaliy is obvious. The second can be proved by reversing he order of inegraion and recalling ha ϕ () = µ(). To esimae he derivaives wih respec o z, we have he following lemma. (For a proof, see Hörmander [3], Lemma ) 5

6 Lemma 2.4. Le he funcion v(z) be holomorphic in B R, and suppose here are posiive consans K and c such ha v ρ K (R ρ) c for every ρ (, R). (2.7) Then we have D α z v ρ Ke α (c + 1) α (R ρ) c+ α for every ρ (, R). (2.8) In he above, we define (c) p = (c)(c + 1) (c + p 1). 3 Proof of Main Theorem Le γ > and f C ([, ], A(ω [γ])). We will use he mehod of successive approximaions o solve he equaion Pu = f. For breviy, le S() = m 1 j= α m j c j,α(, z)(µ()d z ) α (D ) j. Define he approximae soluions as follows: and for k 1, u () = u k () = [;s] [;s] G s(λ) f (3.1) G s(λ) [ f S(s)u k 1 ]. (3.2) Noe ha for all k, he approximae soluions u k (, z) are defined on he region {(τ, z); τ and z ω τ [γ]}. Furhermore, hey are coninuous wih respec o and holomorphic wih respec o z on his region. For each naural number k, we also define he sequence of funcions v k () = u k () u k 1 (), where we have se u 1. Then he funcions v k (, z) are also defined on he same region as u k (, z), and are also coninuous wih respec o and holomorphic wih respec o z. Using he expression for u k (), we have v () = [;s] G s(λ) f (3.3) 6

7 and for k 1, v k () = [;s] G s(λ)s(s)v k 1. (3.4) To prove ha he approximae soluions converge o he real soluion, we need o esimae he funcions v k (). Le C be he bound on [, T ] B R of all c j,α (, z), and K be he bound in Ω T [γ] of f(, z). As for he funcions G s(λ) and H s(k, λ), we have he following esimaes: sup z B R G s(λ) and here exiss a consan D such ha for 1 k m, ( s ) L (3.5) sup Hs(k, λ) s ) L. D( (3.6) z B R We can easily see ha v () ω is bounded by KL m for any T. Here, we have wrien for convenience ω in place of ω[γ]. For general k, noe ha v k () is given by he following ieraed inegral: v k () = ( 1) k [;s k ] [s 2 ;s 1 ] Gs k (λ)s(s k ) G s 2 s 1 (λ)s(s 1 ) [s k ;s k 1 ] [s 1 ;s ] G s k sk 1 (λ)s(s k 1 ) G s 1 s (λ)f(s ). (3.7) The expression above can be expanded using Lemma 2.2, and hus obain a finie sum whose number of erms is less han (mj) k, where J is he cardinaliy of he se {(j, α); j m 1 and α m j}. Each erm of he finie sum has he form I = ( 1) k Gs k (λ) c jk,α k (µd z ) α k [;s k ] (ik ) H s k [s k ;s k 1 ] s k 1 (i k, λ) c jk 1,α k 1 (µd z ) α k 1 (i2 ) (i1 H s 2 s 1 (i 2, λ) c j1,α 1 (µd z ) α ) 1 H s 1 s (i 1, λ) f(s ), (3.8) [s 2 ;s 1 ] [s 1 ;s ] where for each p, he relaions m j p i p m and α p m j p hold. 7

8 This is furher equal o I = ( 1) k [;s k ] (ik ) [s k ;s k 1 ] (i1 ) [s 1 ;s ] G s k c jk,α k (s k )(µ(s k )D z ) α k H s k s k 1 c jk 1,α k 1 (s k 1 )(µ(s k 1 )D z ) α k 1 H s 2 s 1 c j1,α 1 (s 1 )(µ(s 1 )D z ) α 1 H s 1 s f(s ). (3.9) Le F k (s) denoe he inegrand of he above inegral. Le R s = R γϕ(s ). Then all he funcions above, when viewed as a funcion of z, belong in A(ω s [γ]). (This explains he necessiy of he assumpion ha he coefficiens be defined up o B R, for all in he inerval [, T ].) We can herefore apply Lemma 2.4 repeaedly, saring from he righmos expression, o obain he following esimae for he inegrand: for any ρ (, R s ), ) L ( F k (s) Bρ K(CD) k µ(s 1 ) α 1 µ(s k ) α s k ( e ) α1 + +α k α1 + + α k!. (3.1) R s ρ If α α k =, hen for small T, he bound for any c j, (, z) = a j, (, z) a j, (, z) is acually small, since a j, (, z) is coninuous wih respec o. Thus, by choosing a sufficienly small T = T, we could find a consan δ such ha he following holds: Going back o he inegral, we have F k (s) ω Kδ k( s ) L. (3.11) I ω (ik ) [;s k ] [s k ;s k 1 ] (i1 ) [s 1 ;s ] Kδ k( s ) L δ k = K L m+i (by (a) of Lemma 2.3) 1+ +i k δ ) k, K( L for some consan L dependen on L. This is possible since i p 1 for all p. 8

9 If α α k, hen se he ρ in (3.1) o be equal o R γϕ(). This gives ( F k (s) ω K(CD) k µ(s 1 ) α 1 µ(s k ) α s k ( α α k! ) L e γ[ϕ() ϕ(s )] ) α1 + +α k. (3.12) By renaming if necessary, assume ha for p = 1,..., q, α p. Noe ha q 1. We will again use he coninuiy of a j, (, z) o esimae hose expressions which are no aced upon by D z, i.e., he k q cases when α p =. Jus like before, we can show ha for small δ, ( F k (s) ω K(CD) q δ k q µ(s 1 ) α 1 µ(s q ) αq s ( α α q! e γ[ϕ() ϕ(s )] Thus, he inegral I can now be esimaed as follows: I ω K(CD) q δ k q( e ) α1 + +α q α1 + + α q! γ (ik ) [;s k ] [s k ;s k 1 ] (i1 ) [s 1 ;s ] ( s ) L ) L µ(s1 ) α 1 µ(s q ) αq ) α1 + +α q. (3.13) [ϕ() ϕ(s )] α 1+ +α q. (3.14) Le d = m + i i k and b = α α q. Noe ha b q. Since for each p, we have α p m j p i p, and using he fac ha boh ϕ() and µ() are increasing on (, T ), we have I ω K (CD) q δ k q( e ) b b! ξb ξ1 ξ η1 γ µ(ξ b ) µ(ξ 1) ξ b ηd b 2 ξ 1 ( ξ ) L 1 dξ [ϕ() ϕ(ξ )] b dξ 1 dξ b ξ ( s ξ ) L ds s dη 1 η 1 (3.15) By (a) of Lemma 2.3, he second inegral is equal o L d+b+1. Thus, he above simplifies ino I ω K(CD) q δ k q( e ) bl d+b+1 b! γ ξb ξ1 µ(ξ b ) µ(ξ ( 1) ξ ) L ξ 1 ξ b ξ 1 [ϕ() ϕ(ξ )] b dξ dξ b. (3.16) 9

10 The las inegral is equal o (Lb!) 1, by (b) of Lemma 2.3. Meanwhile, since d > k, we can find a consan L 1, depending on L, such ha L d L k 1. Subsiuing hese resuls ino he above equaion, we ge I ω K(CD) q δ k q( el ) 2 bl k γ 1 ( CD ) q(δl1 = K ) k( el ) 2 b (3.17) δ γ Since T can be made sufficienly small, here exiss a consan δ such ha δl 1 and δl 1 are boh less han (mj) k. Now, since q b, we can choose and fix a sufficienly large γ = γ o make he remaining expression less han 1. To summarize, we have shown ha we could find consans T, δ < 1 and γ large enough, such ha for all k, v k () ω[γ ] Kδ k for any [, T ]. (3.18) I follows ha he series k= v k(, z) is majorized by a convergen geomeric series, and hence is iself convergen in C ([, τ], A(ω τ [γ ])) for all τ [, T ]. This means ha u k () converges uniformly o u() on Ω T [γ ]. By following he seps above, we can also show ha for 1 p m 1, he sequence (D ) p u k () converges uniformly o (D ) p u() on Ω T [γ ]. Thus, i follows ha on a compac subse of Ω T [γ ], he sequence D z (D ) p u k () converges o D z (D ) p u(). This implies he convergence of he approximae soluions o he rue soluion u(). Uniqueness may be proved in a similar manner. References [1] Baouendi, M. S. and C. Goulaouic, Cauchy Problems wih characerisic iniial hypersurface, Comm. Pure. Appl. Mah. 26 (1973), [2] Fischer, E., Inermediae Real Analysis, Springer-Verlag, New York- Heidelberg-Berlin, [3] Hörmander, L., Linear Parial Differenial Operaors, Springer-Verlag, Berlin-Heidelberg-New York, [4] Lope, J. E. C., Exisence and Uniqueness Theorems for a Class of Linear Fuchsian Parial Differenial Equaions, J. Mah. Sci. Univ. Tokyo 6 (1999),

11 [5] Mandai, T., Characerisic Cauchy problems for some non-fuchsian parial differenial operaors, J. Mah. Soc. Japan 45 (1993), [6] Tahara, H., On a Volevič sysem of singular parial differenial equaions, J. Mah. Soc. Japan 34 (1982), [7] Tahara, H., On he uniqueness heorem for nonlinear singular parial diferenial equaions, J. Mah. Sci. Univ. Tokyo 5 (1998), [8] Yamane, H. Singulariies in Fuchsian Cauchy Problems wih Holomorphic Daa, Publ. RIMS, Kyoo Univ. 34 (1998), Presen Address: Deparmen of Mahemaics Sophia Universiy Kioicho, Chiyoda-ku Tokyo 12, Japan j-lope@mm.sophia.ac.jp Permanen Address: Deparmen of Mahemaics Universiy of he Philippines Diliman, 111 Quezon Ciy, Philippines 11