GLOBAL ANALYTIC REGULARITY FOR NON-LINEAR SECOND ORDER OPERATORS ON THE TORUS

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 12, Page 3783 3793 S 0002-9939(03)06940-5 Aricle elecronically publihed on February 28, 2003 GLOBAL ANALYTIC REGULARITY FOR NON-LINEAR SECOND ORDER OPERATORS ON THE TORUS CHIARA BOITI AND LUISA ZANGHIRATI (Communicaed by David S. Tarakoff) Abrac. Auming a ubellipic a-priori eimae we prove global analyic regulariy for non-linear econd order operaor on a produc of ori, uing he mehod of majoran erie. 1. Inroducion Hypoellipiciy for linear parial differenial operaor ha been largely inveigaed by many auhor. In he non-linear cae, on he conrary, here are ill few reul and many open queion. Some reul abou C -hypoellipiciy for non-linear parial differenial equaion have been obained in [X] and [G], uing he para-differenial calculu of Bony [B]. We are inereed in analyic hypoellipiciy for non-linear econd order p.d.e.. Local analyic regulariy for a model operaor given by um of quare of non-linear vecor field ha been proved in [TZ]. Here we prove global analyic regulariy on he oru for non-linear econd order operaor conruced from rigid vecor field, generalizing he reul obained for he linear cae in [T]. The problem of regulariy of oluion on he oru in he linear cae ha been udied by many oher auhor in he framework of C, Gevrey and analyic funcion (ee, for inance, [GPY] and he reference here). 2. Noaion and main reul Le T N be he N-dimenional oru and pli T N T m T n. Le u hen conider, for u C (T N ) and for ome ineger n, he operaor (1) P = P u = P (x, u, D) = j,k=1 a jk (u(, x))x j X k + b j (u(, x))x j + X 0 + c(u(, x)) Received by he edior July 4, 2002. 2000 Mahemaic Subjec Claificaion. Primary 35B65, 35B45; Secondary 35H10, 35H20. Key word and phrae. Analyic regulariy, non-linear, um of quare of vecor field, oru. 3783 c 2003 American Mahemaical Sociey Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

3784 CHIARA BOITI AND LUISA ZANGHIRATI defined for (, x) T m T n, where he real analyic coefficien a jk (u), b j (u) and c(u) are complex valued, bu he real analyic rigid vecor field (2) X j = n k=1 d jk (x) x k + m k=1 e jk (x) k, j =0,...,, are real valued (rigid mean ha he coefficien d jk,e jk do no depend on ). Aume alo ha, for every x T n, he field n (3) X j = d jk (x), j =1,...,, x k k=1 pan he angen pace T x (T n ). Le u now denoe by A(T N ) he pace of real analyic funcion on T N, and fix aoluionu C (T N ) of he equaion Pu = f for f A(T N ). We hall aume in he equel ha he following a-priori eimae i aified for ome δ, C > 0andforallv C (T N ): (4) i, X i X j v µ + X j v µ + v µ+δ C( P u v µ + v µ ), where µ i a fixed ineger wih µ>n/2, o ha he Sobolev pace H µ (T N )ian algebra and fg µ Λ f µ g µ f,g H µ (T N ), for a poiive Λ depending only on N. Before giving he analyic regulariy reul, we fir give an example of an operaor of ype (1) aifying he required aumpion, and in paricular he a-priori eimae (4). Example 2.1. Le (, x) T 2 and conider he operaor P = 2 x +in 2 x (1 + a 2 (u(, x))) 2, where a(u) i a real analyic funcion. Thi operaor i of he form (1) wih X 1 = X 1 = x,x 2 =inx, a 11 (u) 1, a 12 (u) a 21 (u) 0, a 22 (u) =1+a 2 (u). We mu prove he a-priori eimae (4). From [RS] i eaily follow ha 2 v 2 µ+δ c X j v 2 µ + v 2 µ v C (T 2 ), δ =1/2. Thi implie, by andard argumen, he following a-priori eimae for he operaor P = 2 x +in 2 x 2 : (5) 2 2 X i X j v 2 µ + X j v 2 µ+δ + v 2 µ+2δ c Pv,v µ + v 2 µ, i, for ome c > 0andforallv C (T 2 ). Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

GLOBAL ANALYTIC REGULARITY 3785 Since P = P + a 2 (u)x2 2,wehaveha Pv,v µ Pv,v µ + a 2 (u)x2 2 v, v µ 1 2 Pv 2 µ + 1 2 v 2 µ + ε a2 (u)x 2 2 v 2 µ + 1 4ε v 2 µ 1 2ε +1 2 Pv 2 µ + v 2 µ + εk X 2 4ε 2v 2 µ for ome conan K>0. Subiuing in (5) we obain he deired eimae (4), for ε>0 mall enough. Le u now ae he main reul of hi paper. Theorem 2.2. Le P be he operaor defined in (1), and aume ha he vecor field {X j } j=0,..., are rigid and ha for every fixed x T n he {X j },..., pan T x (T n ). Aume moreover ha u C (T N ) i a oluion of he equaion P (x, u, D)u = f, for ome f A(T N ), and ha he a-priori eimae (4) i aified. Then alo u A(T N ). Remark 2.3. We can follow [X] o obain from he a-priori eimae (4) and he ue of para-differenial operaor a reul of C -hypoellipiciy for he operaor (1) wih he given aumpion on he X j : if f C (T N )andu C µ+3 (T N )ia oluion of Pu = f, henu C (T N ). Before giving he proof of Theorem 2.2, we fir need ome noaion. Define, for u C (T N ), u µ = i, X i X j u µ + X j u µ + u µ+δ, and conider he equence m q = cq!/(q +1) 2, where he conan c i uch ha (ee [AM]) ( ) α (6) m β m α β m α. β 0 β α Then e M q = ε 1 q m q for ε>0andq 1. The relaion (6) implie ha ( ) α (7) M β M α β εm α β 0<β<α and hence, if we conider he formal power erie (8) θ(y )= α>0 M α Y α, α! for Y =(, x) R N, we obain ha θ q (Y ) ε q 1 θ(y ) q 1, Y R N, meaning ha each coefficien of he formal power erie on he lef i le han or equal o he correponding coefficien of he formal power erie on he righ-hand ide. Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

3786 CHIARA BOITI AND LUISA ZANGHIRATI (9) I follow ha, if we chooe A, R > 0 aifying for every ineger q 0 i, ij H µ (u(t N )) + b (q) j H µ (u(t N )) + c (q) H µ (u(t N )) AR q q! a (q) (which i poible becaue of he analyiciy of he coefficien), and we define he formal power erie (10) hen, for every ρ>0, (11) φ(w) = + q=1 AR q w q, for w R, φ(ρθ(y )) A + ε θ(y ) (ρrε) q = for all ε>0 uch ha ερr < 1. q=1 ARρ 1 ερr θ(y ) Proof of Theorem 2.2. From he given aumpion on he vecor field X j,ii ufficien o prove he analyic eimae for b k u µ for k =1,...,m and for every b 1. We fix k and denoe, for impliciy, = k and T = = k. Then we define T q u µ [u],r = up, 0<q r M q and prove by inducion on r 1 ha here exi ε, M > 0 uch ha for all r 1, (12) [u],r M. We claim ha we can ake { M =max 1, 4 } (13) c max T q u 1 q 3 µ, wherea ε will be choen in he following. For p =1, 2, 3 we clearly have ha [u],p M for ε mall enough. Aume ha (12) i aified for all 3 r<bandle u prove i for r = b (he above reque b 3 will be underood in he following). By he a-priori eimae (4) we have ha T b u µ = j,k=1 X j X k T b u µ + X j T b u µ + T b u µ+δ (14) C( PT b u µ + T b u µ ). For every ε 1,δ 1 > 0 we can find a poiive conan C ε1,δ 1 > 0 uch ha (15) Moreover (16) T b u µ ε 1 T b u µ+δ + C ε1,δ 1 T b u µ δ1. PT b u µ [P, T b ]u µ + T b Pu µ. Since ε 1 T b u µ+δ will be aborbed in he lef-hand ide of (14), T b u µ δ1 will be eimaed by inducion and T b Pu µ will no give any problem becaue of he Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

GLOBAL ANALYTIC REGULARITY 3787 analyiciy of f = Pu,wefireimae [P, T b ]u µ. To hi aim we compue [T b,p]= T b P PT b (17) = b j,k=1 + b b T b (a jk (u(, x))) X j X k T b b b T b (b j (u(, x))) X j T b b + b b T b (c(u(, x))) T b b ince he X j do no depend on he variable. Le u denoe by a u he generic coefficien a jk (u(, x)) or b j (u(, x)) or c(u(, x)), and wrie (X 2 ) for he generic erm of he form 1, or X j or X j X k.thenweeimae b (18) b T b (a u)(x 2 )T b b u Λ T b (a u) µ (X 2 )u µ µ +Λ b 1 b T b (a u) µ (X 2 )T b b u µ. I can be eaily proved by inducion on p 1 ha he derivaive T p of he compoie funcion a(u(, x)) can be wrien a T p (a u) = r i N\{0} r 1+...+r q=p = a (u)t p u + for ome C q,r > 0, and herefore T b (a u) µ a (u)t b u µ + C q,r a (q) (u) r1 r 1+...+r q=p 0<r i<p 0<r i<b u rq u C q,r a (q) (u) r1 u rq u C q,r Λ q a (q) (u) µ r1 u µ rq u µ (19) Λ a (u) µ T b u µ + 0<r i<b C q,r a (q) (u) µ (Λ[u],b 1 ) q r1 θ(0) rq θ(0) ince r h u µ [u],b 1 M rh =[u],b 1 r h θ(0) for 1 r h b 1, h =1,...,q, where θ(y ) i he formal power erie defined in (8). Wih he choice made for A, R > 0in(9),wehaveha a (q) (u) µ AR q q!and hence, ubiuing in (19), (20) T b (a u) µ ΛAR T b u µ + 0<r i<b C q,r AR q q!(λ[u],b 1 ) q r1 θ(0) rq θ(0). Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

3788 CHIARA BOITI AND LUISA ZANGHIRATI Le u now remark ha, for φ given by (10) and ρ>0, T b (φ(ρθ)) = C q,r φ (q) (ρθ)ρ q r1 θ rq θ and herefore = φ (ρθ)ρ b θ + T b (φ(ρθ(y ))) Y =0 = ARρM b + 0<r i<b 0<r i<b ince α θ(0) = M α and φ (q) (0) = AR q q!. Subiuing in (20) wih ρ =Λ[u],b 1, C q,r φ (q) (ρθ)ρ q r1 C q,r AR q q!ρ q r1 θ rq θ θ(0) θ(0) rq (21) T b (a u) µ ΛAR T b u µ + T b (φ(λ[u],b 1 θ(y ))) Y =0 ΛAR[u],b 1 M b. From (11) we deduce ha and hence T b (φ(ρθ(y ))) ARρ 1 ερr T b θ(y ) if ερr < 1, T b (φ(λ[u],b 1 θ(y ))) Y =0 ARΛ[u],b 1 1 εrλ[u],b 1 M b if εrλ[u],b 1 < 1. By he inducive aumpion we can ake ε = ε o /(MRΛ), wih 0 <ε o < 1obe choen in he following, o ha εrλ[u],b 1 ε o MRΛ RΛM = ε o < 1. We hu obain from (21) and (15) ha T b (a u) µ ΛAR(ε 1 T b u µ+δ + C ε1,δ 1 T b u µ δ1 )+ ΛAR 1 ε o [u],b 1 M b ΛAR[u],b 1 M b = ε 1 ΛAR T b u µ+δ +ΛARC ε1,δ 1 T b u µ δ1 + ε o (22) ΛAR[u],b 1 M b. 1 ε o Thi eimae will be ubiued in (18). In a imilar way we obain he following eimae for 1 b b 1: (23) T b (a u) µ b 1 C q,r AR q q!(λ[u],b 1 ) q r1 θ(0) θ(0) rq = T b (φ(λ[u],b 1 θ(y ))) Y =0 ΛAR 1 ε o [u],b 1 M b. Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

GLOBAL ANALYTIC REGULARITY 3789 Subiuing (22) and (23) in (18), (24) b b { T b (a u)(x 2 )T b b u Λ (X 2 )u µ ε 1 ΛAR T b u µ+δ µ +ΛARC ε1,δ 1 T b u µ δ1 + +Λ b 1 ε } o ΛAR[u],b 1 M b 1 ε o ΛAR b [u],b 1 M b (X 2 )T b b u µ. 1 ε o Le u e m = u µ (o ha (X 2 )u µ m), and eimae he norm T b u µ δ1 and (X 2 )T b b u µ : T b u µ δ1 T b 1 u µ+δ [u],b 1 M b 1 if δ 1 1 δ, δ 1 > 0, (X 2 )T b b u µ T b b u µ [u],b 1 M b b ince b 1. b Then, from (24) and he inducive aumpion, b T b (a u)(x 2 )T b b u µ + ε o mλ 2 ARMM b + Λ2 AR b 1 M 2 ε 1 mλ 2 AR T b u µ+δ + mλ 2 ARC ε1,δ 1 MM b 1 ε 1 mλ 2 AR T b u µ+δ + mλ 2 ARC ε1,δ 1 MεM b + ε o mλ 2 ARMM b + Λ2 AR M 2 εm b b M b M b b becaue of (7) and of he eimae M b 1 εm b for b 3. (Here i he only reaon why we ar wih r = 3 in he inducion.) By he choice of ε = ε o /(MRΛ) we have (25) b ( ) b b T b (a u)(x 2 )T b b u ε 1 mλ 2 AR T b u µ+δ µ ( +ε o mλac ε1,δ 1 + mλ2 ARM + ΛAM ) M b. From (17) and (25) we finally obain he deired eimae for [P, T b ]u µ : [P, T b ]u µ ε 1 mλ 2 AR( 2 + +1) T b u µ+δ ( +ε o mλac ε1,δ 1 + mλ2 ARM + ΛAM ) ( 2 (26) + +1)M b. Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

3790 CHIARA BOITI AND LUISA ZANGHIRATI Then, from (14), (15), (16) and (26), [ T b u µ C ε 1 mλ 2 AR( 2 + +1) T b u µ+δ ( +ε o mλac ε1,δ 1 + mλ2 ARM + ΛAM ) ( 2 + +1)M b ] +B b+1 b! f (b +1) 2 + ε 1 T b u µ+δ + C ε1,δ 1 [u],b 1 M b 1 ε 1 [CmΛ 2 AR( 2 + +1)+C] T b u µ+δ ( +ε o CM mλac ε1,δ 1 + mλ2 AR + ΛA ) ( 2 + +1)M b +B b+1 b! f (b +1) 2 + CC ε 1,δ 1 M ε o MRΛ M b, where B f > 0 i given by he analyiciy of f. We now chooe 0 <ε 1 < 1wih A ε1 = ε 1 [CmΛ 2 AR( 2 + +1)+C] < 1 and hen 0 <ε o < 1 ufficienly mall o ha, for b 3, and C ε o A ε1 B b+1 f b! A ε1 (b +1) 2 1 2 M b = c 2 ε 1 b o b! (MRΛ) 1 b (b +1) 2 [( mλac ε1,δ 1 + mλ2 AR + ΛA ) ( 2 + +1)+ C ] ε 1,δ 1 1 RΛ 2. Wih uch choice we finally have ha T b u µ MM b and hence, by he inducive aumpion, { T q u µ T q u µ [u],b = up =max up, T } b u µ M. 0<q b M q 0<q<b M q M b The heorem i herefore proved. 3. The cae of coefficien alo depending on (, x) T m T n Le u now conider he cae in which he operaor P ha real analyic complex valued coefficien which depend alo on (, x) T m T n T N : (27) P = P (, x, u, D) = j,k=1 a jk (, x, u(, x))x j X k + b j (, x, u(, x))x j + X 0 + c(, x, u(, x)), where he real valued rigid vecor field {X j } 0 j are defined a in (2) and aify he ame aumpion a in 2. Then we can prove he analogue of Theorem 2.2: Theorem 3.1. Le P be he operaor defined in (27) and aume ha he vecor field {X j } j=0,..., are rigid and ha for every fixed x T n he {X j },..., pan T x (T n ). Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

GLOBAL ANALYTIC REGULARITY 3791 Aume moreover ha u C (T N ) i a oluion of he equaion P (, x, u, D)u = f, for ome f A(T N ) and ha he a-priori eimae (4) i aified. u A(T N ). Then alo Proof. I i analogou o ha of Theorem 2.2, and herefore we give here only he kech of i. Following he ame ouline a in he proof of Theorem 2.2, we replace (9) and (10) defining A, R > 1 and he formal power erie φ by he following formula: i, r q u a ij H µ (T N u(t N )) + r q u b j H µ (T N u(t N )) + r q u c H µ (T N u(t N )) AR r+q (r + q)! r, q 0 φ(w) = + q=1 A(2R) q w q for w R, o ha we have (11) wih R =2R inead of R. Wehene { M =max 1, 4 } c max T q u 1 q 3 µ, 2A inead of he choice made for M in (13). Following he proof of Theorem 2.2, we mu eimae he H µ (T N )-norm of T b a(, x, u(, x)). To hi aim we fir recall he following formula, which can be eaily proved by inducion on p 1: (28) p a(, x, u(, x)) = ( p a)(, x, u(, x)) p ( ) p + C q,r ( q u p a)(, x, u(, x)) r1 r 1+...+r q= for ome C q,r > 0. Then T b a(, x, u(, x)) = ( b a)(, x, u(, x)) + b a(τ,x,u(, x)) τ = b 1 (29) + C q,r ( q u b a)(, x, u(, x)) r1 r 1+...+r q= u(, x) rq u(, x) u(, x) rq u(, x), for b 3. Le u denoe by a u he compoie funcion a(, x, u(, x)). The fir erm on he righ-hand ide of (29) i hen eaily eimaed by b a u µ AR b b! M 2 M b, if 0 <ε ε wih ε mall enough in order ha R p p! M p = ε 1 p p! (30) c (p +1) 2 p 2. The econd erm on he righ-hand ide of (29) i eimaed a in (22) wih R =2R inead of R and 0 < ε o < min{1, εm RΛ} inead of ε o,akingε = ε o /(M RΛ). Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

3792 CHIARA BOITI AND LUISA ZANGHIRATI We have o eimae he hird erm of he righ-hand ide of (29). To hi aim we fir recall ha (k + j)! 2 k+j k!j! for all k, j N, and hence u q b a u µ AR q+b (q + b )! A R q+b q!(b )!. Therefore b 1 b 1 r 1+...+r q= R b (b )! C q,r q u b r 1+...+r q= RbT b 2 b 1 φ(λ[u],b 1 θ(y )) Y =0 + A RΛ [ [u],b 1 1 ε o 3A R 2 Λ [u],b 1 εm b, 1 ε o becaue of (30), of b 2 RbM b 1 + a u r1 u rq u µ C q,r A R q q!(λ[u],b 1 ) q r1 ( b θ(0) rq θ(0) M b T φ(λ[u],b 1 θ(y )) Y =0 ] )M b M T p φ(λ[u],b 1 θ(y )) Y =0 A RΛ [u],b 1 M p p 1, 1 ε o of (7) and of bm b 1 2εM b for b 3. The ame argumen can be hold o eimae T b a u µ for 1 b b 1. We can hu proceed a in he proof of Theorem 2.2 o obain he deired eimae (12). Acknowledgemen The auhor are graeful o Profeor P. Popivanov for hi uggeion abou he ubjec of hi paper and for hi helpful remark. Reference [AM] S. Alinhac - G. Meivier, Propagaion de l analyicié deoluiondeyème hyperbolique non-linéaire, Inven. Mah., 75 (1984), pp. 189-204 MR 86f:35010 [B] J.M. Bony, Calcul ymbolique e propagaion de ingularié pour le équaion aux dérivée parielle non linéaire, Ann. Scien. Éc. Norm. Sup., 4 e érie (1981), pp. 209-246 MR 84h:35177 [GPY] T. Gramchev - P. Popivanov - M. Yohino, Global properie in pace of generalized funcion on he oru for econd order differenial operaor wih variable coefficien, Rend. Sem. Ma. Univ. Pol. Torino, vol. 51, n. 2 (1993) MR 95k:35047 [G] P. Guan, Regulariy of a cla of quailinear degenerae ellipic equaion, Advance in Mahemaic, 132 (1997), pp. 24-45 MR 99a:35068 [RS] L.P. Rohchild - E.M. Sein, Hypoellipic differenial operaor and nilpoen group, Aca Mah., 137 (1976), pp. 247-320 MR 55:9171 [T] D.S. Tarakoff, Global (and local) analyiciy for econd order operaor conruced from rigid vecor field on produc of ori, Tran. of A.M.S., Vol. 348, n. 7 (1996), pp. 2577-2583 MR 96i:35018 Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue

GLOBAL ANALYTIC REGULARITY 3793 [TZ] [X] D.S. Tarakoff - L. Zanghirai, Analyiciy of Soluion for Sum of Square of Non-linear Vecor Field, o appear in Proc. A.M.S. C.J. Xu, Regulariy of oluion of econd order non-ellipic quailinear parial differenial equaion, C.R. Acad. Sci. Pari, Sér. I, n. 8,. 300 (1985), pp. 235-237 MR 86m:35073 Diparimeno di Maemaica, Via Machiavelli n.35, 44100 Ferrara, Ialy E-mail addre: boii@dm.unife.i Diparimeno di Maemaica, Via Machiavelli n.35, 44100 Ferrara, Ialy E-mail addre: zan@dn.unife.i Licene or copyrigh rericion may apply o rediribuion; ee hp://www.am.org/journal-erm-of-ue