Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

Publicacions Ma emá iques, Vol 37 (1993), 443-463 . A bs ac SOME THEOREMS OF PHRAGMEN-LINDELOF TYPE FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS RAMÓN QUINTANILLA The p esen pape s udies second o de pa ial di e en ial equa ions in wo independen a iables o he o m Di (pl ¡u,, J' -l u,l , P21U,2 I n-1 u,2) = 0 . We ob ain decay es i- ma es o he solu ions in a semi-in ini e s ip . The esul s may be seen as heo ems o Ph agmen-Lindelo ype . The me hod is s ongly based on he ideas o Ho gan and Payne [5], [6], [8] . 1 . In oduc ion In [5] Ho gan and Payne s udied he asymp o ic beha iou o solu ions o he equa ions : (P(-'C, u, VU)u, a ),a = 0, on he semi-in ini e s ip R = {(x1, x2)/0 _< x2 <_ h, x1 >_ 0} unde bounda y condi ions : (1 .1)  u(x 1 , 0) = u(xi, h) = 0,  x1 > 0, (1 .2)  u and u, a  + 0 (uni o mly in x2) as x 1 --, oo, (1 .3)  u(O,x2) = (x2), 0 < x2 < h, whe e he p esc ibed unc ion is su icien ly smoo h and sa is ies (0) _ (h) = 0 . This s udy was made o wo kinds o unc ions (case 1)  O < ml :5p< M l + K,P(P 2 + q2), (case 2)  0 < m2 < P -1 < M2 + K2P(P2 + q2), 444  R . QUINTANILLA whe e p = au/Ox1 and q = 8u/¿9x2 . Ho gan and Payne ob ained expo- nen ial decay o he solu ions . Recen ly [8] he assump ion (1 .2) on he beha iou o he solu ion a in ini y has been weakened and many o he s geome ies o he domains ha e been conside ed . These esul s we e mo i a ed by he desi e o es ablish e sions o Sain -Venan 's p inciple in elas ici y . A good su ey o esea ch on his p inciple in se e al kind o p oblems may be ound in [3], [4] . Simila me hods a e cu en ly used in he s udy o cons ained elas ic cylinde s o a iable c oss sec ion [111, [12] . Recen esul s using ela ed me hods may be ound in [9], [10] . Fu he mo e, some conside a ions o he p-laplacian equa ion we e gi en in [13] and o nonlinea ou h o de equa ions in [15] . Al e na i ely, he esul s may be iewed as heo ems o Ph ag nen- Lindelo ype [2] . Following he main ideas o he wo k o Ho ganand Payne, we gene - alize he esul s o a la ge class o equa ions . F om now on, R will be a s ip lying in he hal plane x1 >_ 0 wi h ixed la e al cu es c(x1),c(x1) o x 1 >_ 0 . We may designa e R = {(XI , X2) E 118 2 /x1 > 0 and c(x1) < x2 <_ ¿(x1)} . We will suppose ha he wid h o he c oss sec ion L z = {(x1, x2) E R/x1 = z} is bounded abo e by a cons an h . We obse e ha he Di ichle homogeneous bounda y condi ion a e : (l .lb)  u(x1, c(x1)) = u(x1, ¿(x1)) = 0 o all x1 > 0, and he end condi ion may be w i en as : (1 .3b)  u(0, x2) = (X2) o all (0, x2) E Lo, whe e he p esc ibed unc ion sa is ies (c(0)) = (¿(O»= 0 .  In his pape we p o e he exponen ial decay o classical solu ions o he equa ion : (1 .4)  DZ (pllpln -1 p,P21gin -1 q) =0,  n > 0, whe e p i = pi(x, u, Vu) in he semi-in ini y s ip R subjec o he bound- a y condi ions (l .lb), (1 .3b) and unde sui able hypo hesis a in ini y (H .s) which is he na u al ansla ion o ou case o he hypo hesis used in [8] . Ou esul s may be seen as analogues o Ph agmen-Lindelo he- o em o he equa ions and domains we s udy . We es ablish ou esul s o ou clases o equa ions depending on he unc ions p1 and p2 . The main esul is o es ablish an exponen ial decay es ima e o he ene gy : (1 .5)  E s (z) = R z IDI'( +1) (PlIPI" 1 +P2Igj +1 )dA (H .0) Type I Type II Type III Type IV SOME THEOREM OF PHRAGMEN-LINDELOF TYPE  445 con ained in he subdomain R z = {(x1, x2) E 1[8 2 /0 < z < x1 < oo} n R p o ided ha he o al ene gy E S (0) is ini e o s = 0, 1 . Ne e heless he me hod may be used o all na u al numbe s s . Te ensu e ha he unc ion Eo is well de ined we need o impose a in ini y he condi ion ha he solu ions sa is y : lim in z - 1 1  (P1 IPIn +1 + P2I qI n+1)dx2 = 0, L z while e ensu e ha E S (s > 0) is well de ined we suppose ha (H .0) is sa is ied oge he wi h he condi ion en he beha io o he solu ions a in ini y : (H .s)  lzm in z - 1 I  Iul s(n+1) (Pl lpln+1 + P2I qI n + 1 )dx2 < OO . ~ We may obse e ha whene e p1 and P2 a e bounded abo e by a con- inuous unc ion o u, p and q hen condi ion (1 .2) implies (H .s) o all sEN . We sepa a e he hypo heses in o he ollowing ou ypes : 0 < m1 < P2 Pi 1 , and 0 < m2 < P2 < M2 + K2(P1IPIn+1 + P2IgIn+1) . 0 < m1< p= 1P2 1/n, 0 < p21 < (M2 + K2(P1IpIn+1 + P2Igjn+1))n . 0<M2 < piP2, (1 .8)  0 < p1 < M1 +K1(Plipi n+1 +P2IgI n+1 ) . 0 <m2<p2P1 n > 0 < m1 < p- 1 < M1+ K1(PlIpin+1 + P2IgIn+1) . 44 6  R . QUINTANILLA We obse e ha when pI = p2 Types I and III become iden ical and na u ally gene alize Case 1 in [5] . We can say he same hing abou Types II and IV in he case n = 1 . Then ou hypo heses a e a na u al gene aliza ion o Case 2 in [5] . A amily o unc ions sa is ying he o egoing condi ions is gi en by : (1 .10) pl = (1+ llpl n+l + 2lgl n+I )yand p2 = (1+ ,Ipln+I+ 2lgln+1)x whe e l and 2 a e wo posi i e numbe s . I 0 <_ y < x <_ y + 1 hen an easy calcula ion p o es ha hese unc ions a e o he Type I by aking m l = m2 = M2 = 1 and K 2 = max(x l, x 2) . In he case 0 <_ x <_ -ny, he unc ions (1 .10) a e o Type II by aking ml = M 2 = 1 and K 2 = 0 . In a simila way ou unc ions a e o Type III i 0 <_ -y <_ x/n by aking m2 = Ml = 1 and K l = 0 . Fo he Type IV we can ake m l = m2= M 2 = 1 and Kl = max( l, 2) whene e -z < y < 0 and x > -y . In Sec ion 2 we s a e wo lemmas which a e use ul in he s udy o he exponen ial decay o he ene gies E S o he Type I o IV . In Sec ion 3 we p o e he exponen ial decay o E S o all he Types . We gi e a comple e p oo o Type I, bu p o ide only a ske ch o he o he h ee ypes . We also gi e in Sec ion 4 a heo em o decay o he ene gies wi hou he es ic ion ha he c oss sec ion is bounded abo e, bu he hypo heses on asymp o ic beha io (H .0) and (H .s) mus be modi ied o ano he (H' .s) . Sec ion 5 is de o ed o ob ain Ln+ l -es ima es in he whole s ip . We make a simila s udy in Sec ion 6 bu o he c oss-sec ion . In Sec ion 7 we ob ain an es ima e o Eo o he Neumann bounda y p oblem . Sec ion 8, which concludes he pape , summa ises ou esul s by lis ing he a ious Types and measu es o which exponen ial decay ha e been es ablished . An appendix o he pape desc ibes a me hod o es ima ing an uppe bound o he o al ene gies E S (0) in e ms o he end condi ions . Acknowledgmen s . I am g a e ul o P o . C . O . Ho gan and P o . R . J . Knops o hei commen s on an ea lie e sion . 2 . Two p e ious lemmas In his sec ion we es ablish wo lemmas abou in eg a ion which a e use ul in p o ing he esul s o Sec ion 3 . Fi s , we gene alize a weigh ed Poinca é inequali y, due o Ho gan and Payne [5, p . 314] . Lemma 2 .1 . Le p : [0, h] , l[8+ - {0} be a con inuous unc ion and u : [0, h] -> R a di e en iable unc ion such ha u(0) = u(h) = 0 .  Then SOMETHEOREM OF PHRAGMEN-LINDELOF TYPE  447 he e exis wo cons an e 0 +1 and B such ha : h  h Plul +ldx <_ /3 +1B +1 1 p- Idu/dxl +ldx, 0  0 whe e Q +1 depends on and B = h pdx . P oo . We conside he change o a iable : so ha , d = pdx, hence we see : = x p(e)da, 0 oh PI uI +l dx = l a I4dl +l ( )d . 0  0 Now, we can apply Poinca é inequali y [1], o ob ain : o  I UI +ld < P +1B +1 J B I(du/d )I-+ld , whe e Q +1 is a cons an ha depends on " + 1" . Upon ecalling ha : we ob ain he desi ed inequali y . F om Lemma 2 .1 we may ob ain an inequali y use ul in he s udy o he beha iou o E s : ph (2 .1)  PI uI ( +1)(s+l)dx < 0 whene e u(0) = u(h) = 0 . This inequali y may be p o ed by applying he o me lemma o he unc ion us+1(s + 1) -l . Now, we ecall he ollowing esul which is p o ed in [5, p . 315] . Lemma 2 .2 . Le p : [0, l) -> ll8+ - {0} be a con inuous unc ion such ha : Then : du/d = (du/dx)(dx/d ) = p-1du/dx, h < (s+1) +1(0 +l)-lB +1 p- luls( +1)I(du/dx)I'+ldx, 0 o Zpdx<Az+B  whe e A,B>0 . J zp -l dx>_ Á-Á 2 , o allz>0 . o 448 In his sec ion we s a e and p o e some heo ems o he exponen ial decay o he ene gy (1 .5) o each o he ypes conside ed in Sec ion 1 . We gi e a comple e p oo only in he case o Type I . We ske ch he p oo in he o he cases . In o de o p o e he heo ems, we need o in oduce he unc ion : (3 .3) whe e and R . QUINTANILLA 3 . Decay o he ene gy G, (z) = s(n + 1 1) + 1  L  P1IPI n-1 PI UI , (n +1 )udx2 . z o he di e gen e heo em, his may be ew i en as : Because (3 .2)  G, (z) = G,(zo) + o all z > zo > 0 . Di ec di e en ia ion gi es : Gs(z) = IUI '(n+1) (PlIPI' +1 +P2IgIn+1 )dx2 . L . Holde 's inequali y applied o (3 .1) leads o he inequali y : (3 .4)  IG,(z)I <_ 1 P1IuI n+1  s(n+1)  +i  (s+1)(n+1)dxz] n4-1 . < s(n + 1) + 1 [ p1I pj  I uI  dx2~ nn  ~IL . Now, we s a e he ollowing esul . Theo em 3 .1 . Le u E C 2 (R)nC 1 (RUóR) be a unc ion ha sa is ies equa ion (1 .4), he bounda y condi ions (1 .1b) and he end condi ions (1 .3b), (H .0) and (H .s) . We suppose ha p, and P2 sa is y condi ions o Type 1(1 .6) . Then o all z > 0 we ha e : (3 .5)  ES(z) < ES(0)eQle- .alz, x uls(n+1) (P1 IpI n+1 + P2IgI n+1 )dA a L I  Q  K2Eo(0)(n+ 1)(s(n+ 1) + 1)mi 1 =  n M2 h 2 (s + 1)nnnn (n+ 1)(s(n+ 1) + 1)m 1 - " 0 1+1 m2 1  M2h(s + 1)nnn+~ nn1 On+1 m2 SOME THEOREM OF PHRAGMEN-LINDELOF TYPE  449 P oo . Le G,(z) be de ined by (3 .1) . On using he i s inequali y o (1 .6) in he inequali y (3 .4) we ob ain : whe e nn}1 lGs(z)l  s(n + l l) + 1 [1, Pl lpln+l luls(n+l)dx2] n+l x x ~~ Pzlul (S+l)(n+l) dx2]n+ 1 G L z (s + 1)B(z)  n+l  S n+l  Pl I pl  I ul (  ) dx2] n+1 x (s(n + 1) + 1)mi +1 ~3n+i  L= x  P2 nluls(n+1) lgln+ldx ] 1 2 n+1 z (s + 1)nn+l 11  B(z) x (n + 1) (s(n + 1) + 1)mi +1 ~ 3 n+i m2 x~ lul s(n+l) (Pllpl n+l +P2[4l n+l )dx2, whe e B (z) = Lz P 2 (z, x2)dx2 . We obse e ha he second inequali y ollows om (2 .1) and he hi d one om he second inequali y in (1 .6) . Thus we may conclude ha : (3 .6)  lG,(z)1 C k,B(z)Gs(z), k -  (s + 1)nn +1 S -  _1  1  (n + 1) (s(n + 1) + ')Mi +1  +i m2 and B(z) has been de ined p e iously . F om inequali y (3 .6) deduce he ollowing wo i s -o de di e en ial inequali ies : (3 .7)  G, 5 (z) < k,B(z)Gs(z), (3 .8)  - GS(z) < ksB(z)Gs(z) . z G,(z) > G S(z*) x exp CkS  d l  7  . £Z i,, P2 (77,x2)dx2 we may We now show ha G S (z) G 0 o all z >_ 0 . To his end, le us suppose ha he e exis s z* > 0 such ha G s (z*) > 0 . Then in eg a ion o (3 .7) gi es : 45 0  R . QUINTANILLA Bu om (3 .7) and he second inequali y in (1 .6), we ha e : J = IUI'(+ 1 )(PlIPI" 1 +P2Igj'+ 1 )dx2 > ks 1 G s (z*) (  x M2h + K2 L> (P1Ipin+1 + P2IgIn+1)dx2) (3 .9)  x exp (k,, -1 Iz  d l . M2h + K2 i  (p1Ipin+1 + P21qln+1)dX2 and since condi ion (H .0) is sa is ed, we ob ain : Z_ 1 1  Iuls(n+1) (P1 Ipln +l + P2I gI n + 1 )dx2 > ~ k -1 G s (z*) >  exp(ks l e -1 K2 l ln(M2h + cK2 7) I z*) _ z(M2h + K2ez) ks l G,5(z * )  M2h + eK2z -  ]eK2ks z(M2h + K2ez) M2h + eK2z* o all e small and o all z > z(e) > 0 . I hen ollows ha : (3 .10)  z-1~ luls(n+1)(PlIpin+1 + P21ql n+1 )dX2 > L z -1 > ks l Gs(z * )[M2h + eK2z] eksK2 _ 2 [M2h + eK2z*] ek .9K 2. On le ing z --> oo, we obse e ha o e suicien ly small he igh -hand side ends o in ini y, which con adic s condi ion (H .s) . Hence we ha e p o ed ha G, (z) <_ 0 o all z >_ 0 . On ecalling inequali y (3 .8), we ha e : (3 .11)  E S (z) < E S (0) exp (- ks 1  oz B(97)-ld77/ J 0 The uppe bound o P2 in (1 .6) gi es he inequali y : while om lemma 2 .2, we conclude : z Á,, p 2 dA < M 2 hz + K2Eo(0), o B(  z _ K2Eo(0) z  l)  d~ > M 2 h  MZh2 Thus, lle desi ed es íma e is es ablished . (3 .12) whe e and whe e and SOMETHEOREM OF PHRAGMEN-LINDELOF TYPE  451 Theo em 3 .2 . Le uE C 2 (R)nCI(RUOR) be a unc ion ha sa is ies equa ion ( 1 . 4), he bounda y condi ions (1 .1b) and he end condi ions (1 .3b), (H .0) and (H .s) . We suppose ha p l and p2 sa is y condi ions o Type 11(1 .7) . Then o all z > 0 we ha e : ES ( z ) < ES (0)eQ2 e-UZZ, K2Eo(0)(n + 1)(s(n + 1) + 1)mi +1 13n+ Q2 =  i M2 h 2 (s+1)n n n+1 ' a2 - (n + 1) (s(n + 1) + 1)M1 + 1 ~n+i . M2h(s + 1)nn+1 Ske ch o P oo .: Rom inequali y (3 .4) we ob ain a simila o (3 .6) bu wi h B (z) = L, p2 l / n (z, x2)dx2 and ~_n_- k, =  (s + 1)n +l (n + 1) (s(n + 1) + 1)m i+ 1 Qn+i Theo em 3 .3 . Le uE C 2 (R)nc 1 (RUOR) be a unc ion ha sa is ies equa ion ( 1 .4 +), he bounda y condi ions (1 .1b) and he end condi ions (1 .3b), (H .0) and (H .s) . We suppose ha pl and p2 sa is y condi ions o Type 111(1 .8) . Then o all z > 0 we ha e : (3 .13)  E .(z) < ES(0)eQ3e-13Z, Q3 = K,Eo(0)(n+ 1)(s(n+ 1) + 1)m2+1 On+i M1 h 2 (s + 1)n nn+1 1 1 as - (n + 1)(s(n + 1) + l)mz +1 ~n+i . M i h(s + 1)nn+1 inequali y Ske ch o P oo . : As be o e, inequali y (3 .4) leads o an inequali y sim- ila o (3 .6) bu wi h B (z) = Lz p l (z,x2)dx 2 and ks =  (s + 1)n- ++1 (n + 1)(s(n + 1) + 1)m2+1 Qn+i 45 8  R . QUINTANILLA sec ion is o ske ch simila esul s o Neumann condi ions, whene e he c oss- sec ion is he in e al [0,h] . A deep s udy o his p oblem o he case p1 = P2 and n = 1 may be ound in [10] . Thus we examine he equa ions (1 .4) subjec o he condi ions a in ini y (H .0), and he end condi ion : and he bounda y condi ions : implies : (7 .2) inequali y o Eo : Pl¡PI' -l p = 9(x2) on x1 = 0, q(x1, 0) = q(xl, l) = 0 o all  xl > 0 . We also make an assump ion on he end condi ion (1 .3) which is ela ed o he condi ion ha he load be sel -equilib a ed o an elas ic ma e ial (see among o he s [3], [7]) : (7 .1)  I LO  = 0 . Lo On using he di e gen e heo em, we see ha he o me assump ion Plipl n-l pdx2 = 0 o all x1 > 0 . Lx Now, we ake (xl, x2) = u(x1, x2) - ü(x1), whe e , will be explici ly de ined la e . We ema k ha q(u) = q( ) . Because o (7 .2) we ha e : (7 .3)  Eo(z) = -  Plipl n-1 p dx2 . L x Di ec di e en ia ion o (1 .5), wi h s = 0, yields : (7 .4)  Eó(z) = - I Lz (Pl IPI+1 + P2IgIn+1)dx2 . In a simila way o he de i a ion o (3 .11), we ob ain he ollowing (7 .5) Eó(z)+e¿(z)Eo(z) <- Lx P2Igln+1dx2+0 .n+1(z) Áz p1 n+ldx2 . SOMETHEOREM OF PHRAGMEN-LINDELOF TYPE  459 Now, we may ob ain simila es ima es o (3 .5), (3 .12), (3 .13) and (3 .14) wi h s = 0 by aking espec i ely : L, UP2dx2 ú(X1) =  o he Type I, L .P2dx2 ic(xl)= L z uP2 i/ndx 2 o he Type II, L z P2 1/ndx2 L, upldx2 ú(X1) =  o he Type III, L, pidx2 i= upi ldx2 ú(X 1 ) =  1  o he TypeIV . L~ Pi dx 2 8 . Summa y To conclude we summa ize b ie ly he cases whe e we ha e es ablished exponen ial decay . In Sec ion 3, we ha e p o ed he exponen ial decay o he ene gies E, s o all he Types (I o IV) whene e (H .0) and (H .s) a e sa is ied and he c oss-Sec ion is bounded . In Sec ion 5, we ha e p o ed ha i (H .0) is sa is ied and he c oss Sec ion is bounded, hen he e is exponen ial decay o J(z) o he unc ions o Type I . We may say he same hing o many unc ions o Type III . We ha e also es ablished he exponen ial decay o J(z) o solu ions o he equa ions con aining he unc ions o Type II, 111 and IV whene e (H .0) and (H .s) a e sa is ied and he c oss-sec ion is bounded . In Sec ion 6, we ha e p o ed ha i (H .0) is sa is ied and he c oss- sec ion is bounded, we may conclude he exponen ial decay o H(z) o unc ions o Type 1 . We ha e also p o ed he exponen ial decay o H(z) o he solu ions o he equa ions ha ing unc ions o Type IV whene e (H .0) and (H .s) a e sa is ied and he c oss-sec ion is bounded . Some commen s on he beha io o he unc ion L(z) a e also s a ed . In o de o ob ain a good es ima e o he decay o J(z) and H(z) we may use inequali ies (5 .1), (5 .2), (6 .1) and (6 .2) . No e howe e ha we ha e no ound es ima es o he alues o he k i in (5 .2) no when we es ima e H(z) in (6 .2) . Ne e heless, i should be possible o ob ain hese alues . In Sec ion 4, we ha e ex ended he exponen ial decay esul s o he ene gy o an enla ged caass o domains . 46 0  R . QUINTANILLA Sec ion 7 is de o ed o ob aining exponen ial decay o Eo o he Neu- mann p oblem, whene e he c oss sec ion is cons an , o he Types I o IV . The o al ene gies : Appendix A : To al ene gy bounds (A .1)  ES (0) =  Iuls(n +1) (PlIPIn +1 +P21gjn+ 1 )dA, R con ained in he s ip appea s in he es ima es . We now ob ain uppe bounds o E,, (0) in e ms o he end da a o unc ions o Type I . Simila es ima es may be ob ained o he o he ypes . We will conside ha R is de e mined by wo s aigh lines : c(x 1 ) = clxl and c(xl) = C2x1 + h . Fo an a bi a y C l unc ion O(xl, x2), de ined on R, Holde 's inequal- i y yields : (A .2) (A .5) R (P1IPIn -lp0,1 +P2Igjn - lq0,2 )IulsnosdA < < ES(0)~+i ~~ (Pl10,1 In+1 +P210,2In+1)101-9(71+1)dAl +1 . R I 0 is chosen o sa is y (l .lb), (1 .2) and (1 .3), he di e gence heo em shows ha he le hand e m o (A .2) is : (A .3)  1  1 ,~  e(u)P11PI n-1PI ul s(n+1)udx2 = E,5(0) . s(n+)+ La Thus, we ob ain : (A .4)  E S (0) < R ( .110,1 In+1 +P2I0,2 I n +l)IOIs(n +1) dA . Fo he unc ions o Type I we ha e om (A .4) : ES(o) <_ J  Pz(ml 1 10,1 In+1 + 10,2 In+ l )I0Is(n +1) dA< R < _~(M2+K2(P1p n+1 +P2g n+l ))(m1 1 10,1 I n+1 +10,2 In+l)101s(n+1)dA < R I R(ml llo,1 In+1 +10,2 I n+l )10Is(n+ 1 )dA+ +K2 mR (m l 110,1 In+1 + 10,2 I n + l )IOIs(n +1) &(O) . Thus, on se ing F(0) = 1 - K2maXR(mi 110,1 In+l + 10,2 In+1) > 0 , we ob ain Le us ake 0 =  (52-clxl)h  e -"Ixl . Easy calcula ions lead o : ( (c2 - el)x1+h ) and On supposing ha Thus : Eo(0) < SOMETHEOREM OF PHRAGMEN-LINDELOF TYPE  46)1 Eo( 0 ) < F(0) -' M2  (m1 1 10,1 1'+' + 10,2 n + 1 )dA . IR 10,1 1 <- e-751 [yI I + I 'l  CA  ], (C2 - CI)Xl + h 0,2 = e -751 '  (x2 - cixl)h  h ( (C2- Cl)xl+h) (c2 - Cl)xl+h K2 1 > [ o , aX(I ' (X2)I n+1 +mi 1 [71 (X2)I +C2I . (X2)1] n+1 ), we ha e an uppe bound : Eo(0) < M2 ¡' R 1-(-+1)-Y-1 (Mi -1[yL I +Iili  c2h  ]n +1 + I lI n+1 ( h  ) n + l dA <  (~2-~l)xl+h  (c2-cl)xl+h 1-K2 max[ o h](I ' (5z)In +l +TY l 1 [ - Y¡ ( 5 2)~-2L ' ( 5 2)Il n+1 ) M2  h  c2-cl  h  1  i n+l  i n+l < h [(n+1) -y + (n+1) 2 . 7 2  ( 0 (ml  [71 l + C2l I  + I I  )dX2) 1- K2max[o,hl(I '(X2)In+l+mil[-Y1 (X2)I+c21 '(X2)I]n+1)  , which a e inse ion in o inequali y (A .5), yields he es ima e Es (0) < K 2 O'aX s(n+1) (I '(X2)In+1+ +mi 1 [Y 1 (X2)I + C21 ~(X2)IIn+l)Eo(0)+ M2 h(n + 1)7(1 + s(n + 1)) + _  (C2 - cl) + h [  (n + 1) 2 7 2 (1 + s(n + 1))2  ]X X ( h 1 . I ls(n +1) (mi l [7l l +C2l ll]n+l + I l in +l )dX2 0 462  R . QUINTANILLA Re e ences 1 .  H . BREZIS, "Analyse onc ionnelle," Masson, Pa is, 1983 . 2 . J . B . 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