Entire solutions of semilinear elliptic equations in R³ and a conjecture of De Giorgi

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS IN R 3 AND A CONJECTURE OF DE GIORGI Luigi Amb osio and Xa ie Cab  e 1. In o duc ion This pap e is conce ned wi h he s udy o b ounded solu ions o semilinea ellip ic equa ions  u ; F 0 ( u ) = 0 in he whole space R n , unde he assump ion ha u is mono one in one di ec ion, say, @ n u > 0 in R n . The goal is o es ablish he one-dimensional cha ac e o symme y o u , namely, ha u only dep ends on one a iable o , equi alen ly, ha he le el se s o u a e hyp e planes. This yp e o symme y ques ion was aised by De Gio gi in 1978, who made he ollowing conjec u e { we quo e li e ally (3), page 175 o DG]: Conjec u e (DG]). Le us conside a solu ion u 2 C 2 ( R n ) o  u = u 3 ; u such ha j u j 1  @ n u> 0 in he whole R n .Is i ue ha , o e e y  2 R , he se s u =  g a e hype planes, a leas i n  8? When n =2, his conjec u e was ecen ly p o ed by Ghoussoub and Gui GG]. In he p esen pap e we p o ei o n =3. The conjec u e, howe e , emains op en in all dimensions n  4. The p o o s o n = 2 and 3 use some echniques in he linea heo y de elop ed by Be es ycki, Caa elli and Ni enbe g BCN] in one o hei pap e s on quali a i e p op e ies o solu ions o semilinea ellip ic equa ions. The ques ion o De Gio gi is also connec ed wi h he heo ies o minimal hy- p e su aces and phase ansi ions. As we explain la e in he in o duc ion, he conjec u e is some imes e e ed o as he " - e sion o Be ns ein p oblem o min- imal g aphs". This ela ion wi h Be ns ein p oblem is p obably he eason whyDe Gio gi s a es a leas i n  8" in he ab o e quo a ion. The au ho s would like o hank Ma iano Giaquin a o se e al use ul discussions. Mos o his wo k was done while he second au ho was isi ing he Uni e si yo Pisa. He hanks he Depa men o Ma hema ics o i s hospi ali y. 2000 Ma hema ics Subjec Classica ion. P ima y 35J60, 35B05, 35B40, 35B45. 1 2 LUIGI AMBROSIO AND XAVIER CABR  E Mos a icles dealing wi h he ques ion o De Gio gi ha e also conside ed he conjec u e in a sligh ly simple e sion. I consis s o assuming ha , in addi ion, (1.1) lim x n !1 u ( x 0 x n )=  1 o all x 0 2 R n ; 1 : He e, he limi s a e no assumed o be uni o m in x 0 2 R n ; 1 . E en in his simple o m, he conjec u e was  s p o ed in GG] o n = 2, in he p esen a icle o n =3, and i emains op en o n  4. The p osi i e answe s o he conjec u e o n = 2 and 3 apply o mo e gen- e al nonlinea i ies han he scala Ginzbu g-Landau equa ion  u + u ; u 3 = 0. Th oughou he pap e , we assume ha F 2 C 2 ( R ) and ha u is a bounded solu- ion o  u ; F 0 ( u )=0 in R n sa is ying @ n u> 0in R n . Unde hese assump ions, Ghoussoub and Gui GG] ha e es ablished ha , when n = 2, u is a unc ion o one a iable only (see sec ion 2 o he p o o ). He e, he only equi emen on he nonlinea i y is ha F 2 C 2 ( R ). The ollowing a e ou esul s o n =3. We s a wi h he simple case when he solu ion sa ises (1.1). Theo em 1.1. Le u be a bounded solu ion o (1.2)  u ; F 0 ( u )= 0 in R 3 sa is ying (1.3) @ 3 u> 0 in R 3 and lim x 3 !1 u ( x 0 x 3 )=  1 o al l x 0 2 R 2 : Assume ha F 2 C 2 ( R ) and ha (1.4) F  min F ( ; 1) F (1) g in ( ; 1  1) : Then he le el se s o u a e planes, i.e., he e exis a 2 R 3 and g 2 C 2 ( R ) such ha u ( x )= g ( a  x ) o al l x 2 R 3 : No e ha he di ec ion a o he a iable on which u dep ends is no known ap io i. Indeed, i u is a one-dimensional solu ion sa is ying (1.3), we can sligh ly" o a e co o dina es o ob ain a new solu ion s ill sa is ying (1.3). Ins ead, i we u he assume ha he limi s in (1.1) a e uni o m in x 0 2 R n ; 1 , hen we a e imp osing an ap io i choice o he di ec ion a , namely, a  x = x n .In his esp ec , i has b een es ablished in GG] o n = 3, and mo e ecen ly in BBG], BHM] and F2] o e e y dimension n , ha i he limi s in (1.1) a e assumed o be uni o m in x 0 2 R n ; 1 hen u only dep ends on he a iable x n , ha is, u = u ( x n ). This esul applies o equa ion (1.2) o a ious classes o nonlinea i ies F whichalways include he Ginzbu g-Landau mo del. Theo em 1.1 applies o F 0 ( u ) = u 3 ; u since F ( u ) = (1 ; u 2 ) 2 = 4is a double- well po en ial wi h absolu e minima a u =  1. Fo his nonlinea i y, he explici ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 3 one-dimensional solu ion (which is unique up o a ansla ion o he indep enden a iable) is gi en by anh( s= p 2). Hence, in his case he conclusion o Theo em 1.1 is ha u ( x ) = anh  a  x ; c p 2  in R 3  o some c 2 R and a 2 R 3 wi h j a j =1 and a 3 > 0. The hyp o hesis (1.4) made on F in Theo em 1.1 is a necessa y condi ion o he exis ence o a one-dimensional solu ion as in he heo em see Lemma 3.2(i). A he same ime, mos o he equa ions conside ed in Theo em 1.1 admi a one- dimensional solu ion. Mo e p ecisely, i F 2 C 2 ( R ) sa ises F>F ( ; 1) = F (1) in ( ; 1  1) and F 0 ( ; 1) = F 0 (1) = 0, hen h 00 ; F 0 ( h ) = 0 has an inc easing solu ion h ( s ) (which is unique up o a ansla ion in s ) such ha lim s !1 h ( s )=  1 see Lemma 3.2(ii). The ollowing esul es ablishes o n = 3 he conjec u e o De Gio gi in he o m s a ed in DG]. Namely, we do no assume ha u ! 1 as x 3 !1 . The esul applies o a class o nonlinea i ies which includes he mo del case F 0 ( u ) = u 3 ; u and also F 0 ( u )=sin u , o ins ance. Theo em 1.2. Le u be a bounded solu ion o  u ; F 0 ( u )= 0 in R 3 sa is ying @ 3 u> 0 in R 3 : Assume ha F 2 C 2 ( R ) and ha (1.5) F  min F ( m ) F ( M ) g in ( m M ) o each pai o eal numbe s m < M sa is ying F 0 ( m ) = F 0 ( M )= 0 , F 00 ( m )  0 and F 00 ( M )  0 . Then he le el se s o u a e planes, i.e., he e exis a 2 R 3 and g 2 C 2 ( R ) such ha u ( x )= g ( a  x ) o al l x 2 R 3 : Ou p o o o Theo em 1.1 will only equi e F 2 C 1  1 ( R ), i.e., F 0 Lipschi z. Howe e , in Theo em 1.2 we need F 0 o class C 1 . Ques ion. Do Theo ems 1.1 and 1.2 hold o e e y nonlinea i y F 2 C 2 ?Tha is, can one emo e hyp o heses (1.4) and (1.5) in hese esul s? The  s pa ial esul on he ques ion o De Gio gi was ound in 1980 by Mo dica and Mo ola MM2]. They ga e a p osi i e answe o he conjec u e o n = 2 unde he addi ional assump ion ha he le el se s o u a e he g aphs o an equi- Lipschi zian amily o unc ions. No e ha , since @ n u> 0, eachle el se o u is he g aph o a unc ion o x 0 . 4 LUIGI AMBROSIO AND XAVIER CABR  E In 1985, Mo dica M1] p o ed ha i F  0 in R hen e e y bounded solu ion u o  u ; F 0 ( u )=0 in R n sa ises he g adien b ound (1.6) 1 2 j u j 2  F ( u )in R n : In 1994, Caa elli, Ga o alo and Segala CGS] gene alized his bound o mo e gene al equa ions. They also showed ha , i equali y o ccu s in (1.6) a some poin o R n hen he conclusion o he conjec u e o De Gio gi is ue. Mo e ecen ly, Ghoussoub and Gui GG] ha e p o ed he conjec u e in ull gene ali y when n =2 (see also F3], whe e weake assump ions han @ 2 u > 0and mo e gene al ellip ic op e a o s a e conside ed). Unde he addi ional assump ion ha u ( x 0 x n ) ! 1as x n !1 uni o mly in x 0 2 R n ; 1 , i is known ha u only dep ends on he a iable x n  he e, he hyp o hesis @ n u > 0 is no needed. This esul was  s p o ed in GG] o n = 3, and mo e ecen ly in any dimension n by Ba low, Bass and Gui BBG], Be es ycki, Hamel and Monneau BHM], and Fa ina F2]. Thei esul s apply o a ious classes o nonlinea i ies F , which always include he Ginzbu g-Landau mo del. These pap e s also con ain ela ed esul s whe e he assump ion on he uni o mi y o he limi s u ! 1 is eplaced by a ious hyp o heses on he le el se s o u . The pap e BBG] uses p obabilis ic me ho ds, BHM] uses he sliding me ho d, and GG] and F2] a e based on he mo ing planes me ho d. Using a one-dimensional a angemen a gumen , Fa ina F1] p o ed he con- clusion u = u ( x n ) p o ided ha u minimizes he ene gy unc ional in an inni e cylinde !  R (wi h ! b ounded) among he unc ions sa is ying ( x 0 x n ) ! 1as x n !1 uni o mly in x 0 2 ! . Ou p o o o he conjec u e o De Gio gi in dimension 3 p o ceeds as he p o o gi en in BCN] and GG] o n = 2. Tha is, o e e y co o dina e x i , we conside he unc ion  i = @ i u=@ n u .The goal is o show ha  i is cons an ( hen he conjec u e ollows immedia ely) and his will be achie ed using a Liou ille yp e esul (P op osi ion 2.1 b elow) o a degene a e ellip ic equa ion sa ised by  i . The ollowing ene gy es ima e is he key esul ha will allow us o apply such Liou ille yp e heo em when n = 3. This ene gy es ima e holds, howe e , in all dimensions and o a bi a y C 2 ( R ) nonlinea i ies. Theo em 1.3. Le u be a bounded solu ion o  u ; F 0 ( u )=0 in R n  whe e F is an a bi a y C 2 ( R ) unc ion. Assume ha @ n u> 0 in R n and lim x n ! + 1 u ( x 0 x n )= 1 o al l x 0 2 R n ; 1 : Fo e e y R> 1 , le B R = j x j <R g . Then, Z B R  1 2 j u j 2 + F ( u ) ; F (1)  dx  CR n ; 1 ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 5 o some cons an C independen o R . The ene gy unc ional in B R , E R ( u )= Z B R  1 2 j u j 2 + F ( u ) ; F (1)  dx has  u ; F 0 ( u ) = 0 as Eule -Lag ange equa ion. In 1989, Mo dica M2] p o ed a mono onici y o mula o he ene gy. I s a es ha i F  F (1) in R and u is a b ounded solu ion o  u ; F 0 ( u )=0 in R n , hen he quan i y E R ( u ) R n ; 1 is a nondec easing unc ion o R . Theo em 1.3 es ablishes ha his quo ien is, in addi ion, b ounded om ab o e. Mo eo e , he mono onici y o mula shows ha he upp e b ound in Theo em 1.3 is op imal: indeed, i E R ( u ) =R n ; 1 ! 0 as R ! 1 hen we would ob ain ha E R ( u )= 0 o any R> 0, and hence ha u is cons an in R n . No e ha he es ima e o Theo em 1.3 is clea ly ue assuming ha u is a one- dimensional solu ion see (3.7) in Lemma 3.2(i). The es ima e is also easy o p o e o u as in Theo em 1.3 unde he addi ional assump ion ha u is a lo cal minimize o he ene gy see Rema k 2.3. In his case, he es ima e al eady app ea s as a lemma in he wo k o Caa elli and Co doba CC] on he con e gence o in e media e le el su aces in phase ansi ions. The p o o o he es ima e o u as in Theo em 1.3 in ol es a new idea. I o igina ed om he p o o o lo cal minimize s and om a ela ion be ween he key hyp o hesis @ n u > 0and he second a ia ion o ene gy see sec ion 2. Finally, we ecall he heu is ic a gumen ha connec s he conjec u e o De Gio gi wi h Be ns ein p oblem o minimal g aphs. Fo simplici y le us supp ose ha F ( u ) = (1 ; u 2 ) 2 = 4. Wi h u as in he conjec u e, conside he blown-down sequence u " ( y )= u ( y=" ) o y 2 B 1  R n  and he p enalized ene gy o u " in B 1 : H " ( u " )= Z B 1  " 2 j u " j 2 + 1 " F ( u " )  dy : No e ha H " ( u " )is a b ounded sequence, by Theo em 1.3. As " ! 0, he unc ion- als H " ;-con e ge o a unc ional whichis ni e only o cha ac e is ic unc ions wi h alues in ; 1  1 g and equal (up o he mul iplica i e cons an 2 p 2 = 3) o he a ea o he hyp e su ace o discon inui y see MM1] and LM]. Heu is ically, he sequence u " is exp ec ed o con e ge o a cha ac e is ic unc ion whose hyp e su ace 6 LUIGI AMBROSIO AND XAVIER CABR  E o discon inui y S has minimal a ea o is a leas s a iona y. The se S desc ib es he beha io a inni y o he le el se s o u , and S is exp ec ed o be he g aph o a unc ion dened on R n ; 1 (since he le el se s o u a e g aphs due o hyp o hesis @ n u > 0). The conjec u e o De Gio gi s a es ha he le el se s a e hyp e planes. The connec ion wi h he Be ns ein p oblem (see chap e 7 o G] o a comple e su ey on his opic) is due o he ac ha e e y minimal g aph o a unc ion dened on R m = R n ; 1 is known o be a hyp e plane whene e m  7, i.e., n  8. On he o he hand, Bombie i, De Gio gi and Gius i ga e in BDG] an example o minimal g aph o a unc ion o 8 a iables die en han a hyp e plane. In a o hcoming wo k AAC] wi h Alb e i, wewill use new a ia ional me ho ds o s udy he conjec u e o De Gio gi in highe dimensions. In sec ion 2 we p o e Theo ems 1.1 and 1.3. Sec ion 3 is de o ed o es ablish Theo em 1.2. 2. P o o o Theo em 1.1 To p o e he conjec u e o De Gio gi in dimension 3, we will use he ene gy es ima e o Theo em 1.3. I is his es ima e ha will allow us o apply, when n = 3, he ollowing Liou ille yp e esul o he equa ion  ( ' 2  )=0 (whe e ' = @ n u ) sa ised by  = @ i u=@ n u . P op osi ion 2.1. Le ' 2 L 1 lo c ( R n ) be a posi i e unc ion. Suppose ha  2 H 1 lo c ( R n ) sa ises (2.1)   ( ' 2  )  0 in R n in he dis ibu ional sense. Fo e e y R> 1 ,le B R = j x j <R g and assume ha (2.2) Z B R ( ' ) 2  CR 2  o some cons an C independen o R .Then  is cons an . The s udy o his yp e o Liou ille p op e y, i s connec ions wi h he sp ec um o linea Sch odinge op e a o s, as well as i s applica ions o symme y p op e ies o solu ions o nonlinea ellip ic equa ions, we e de elop ed by Be es ycki, Caa elli and Ni enbe g BCN]. In he pap e s BCN] and GG], his Liou ille p op e y was shown o hold unde a ious decay assump ions on ' . These hyp o heses, which we e mo e es ic i e han (2.2), could no b e e ied when ying o es ablish he conjec u e o De Gio gi o n  3. We hen ealized ha hyp o hesis (2.2) could be e ied when (and only when) n  3 and ha , a he same ime, (2.2) was su cien o ca y ou he p o o o he Liou ille p op e y gi en in BCN]. Fo con enience, we include b elow hei p o o o P op osi ion 2.1. See Rema k 2.2 o ano he ques ion ega ding his Liou ille p op e y. Be o e p o ing Theo em 1.3 and P op osi ion 2.1, we use hese esul s o gi e he de ailed p o o o Theo em 1.1. Fi s , we es ablish some simple b ounds and ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 7 egula i y esul s o he solu ion u . We assume ha u is a b ounded solu ion o  u ; F 0 ( u )= 0 in he dis ibu ional sense in R n .I ollows ha u is o class C 1 , and ha u is b ounded in he whole R n ,i.e., (2.3) j u j2 L 1 ( R n ) : Indeed, applying in e io W 2 p es ima es, wi h p>n , o he equa ion  u = F 0 ( u ) 2 L 1 in e e y ball B 2 ( y )o adius 2 in R n , we nd ha k u k W 2 p ( B 1 ( y ))  C  k u k L 1 ( B 2 ( y )) + k F 0 ( u ) k L p ( B 2 ( y ))   C wi h C indep enden o y . Using he Sob ole embedding W 2 p ( B 1 ( y ))  C 1 ( B 1 ( y )) o p>n , we conclude (2.3) and ha u 2 C 1 . Nex , we e i y ha u 2 W 3 p lo c ( R n ) o all 1  p< 1  in pa icula , we ha e ha u 2 C 2  ( R n ) o all 0 << 1 : Indeed, since F 0 is C 1 , and u and u a e b ounded, weha e ha F 0 ( u ) 2 W 1 p lo c ( R n ), F 0 ( u )= F 00 ( u ) u , and (2.4)  @ j u ; F 00 ( u ) @ j u =0 in he weak sense, o e e y index j . Since F 00 ( u ) @ j u 2 L 1 ( R n )  L p lo c ( R n ), we ob ain @ j u 2 W 2 p lo c ( R n ). P oo o Theo em 1.1 . Fo each i 2 1  2 g ,we conside he unc ions ' = @ 3 u and  i = @ i u @ 3 u : No e ha  i is well dened since @ 3 u > 0. We also ha e ha  i is C 1  (see he ema ks made ab o e ab ou he egula i y o u ) and ha ' 2  i = @ 3 u @ i u ; @ i u @ 3 u: No e ha he igh hand side o he las equali y belongs o W 1 p lo c ( R 3 ). Using ha @ i u and @ 3 u sa is y he same linea ized equa ion  w ; F 00 ( u ) w = 0, we conclude ha  ( ' 2  i )= 0 in he weak sense in R 3 . Ou goal is o apply o his equa ion he Liou ille p op e y o P op osi ion 2.1. Since ' i = @ i u 8 LUIGI AMBROSIO AND XAVIER CABR  E condi ion (2.2) will be es ablished i we show ha , o each R> 1, (2.5) Z B R j u j 2  CR 2 o some cons an C indep enden o R . Recall ha , by assump ion, F  min F ( ; 1) F (1) g in ( ; 1  1). Supp ose  s ha min F ( ; 1) F (1) g = F (1). In his case we ha e F ( u ) ; F (1)  0 in R 3 . Hence, applying Theo em 1.3 wi h n = 3 (i is he e and only he e ha we use n =3), we conclude ha 1 2 Z B R j u j 2  Z B R  1 2 j u j 2 + F ( u ) ; F (1)   CR 2 : This p o es (2.5). In case ha min F ( ; 1) F (1) g = F ( ; 1), we ob ain he same con- clusion by applying he p e ious a gumen wi h u ( x 0 x 3 ) eplaced by ; u ( x 0  ; x 3 ) and wi h F ( ) eplaced by F ( ; ). By P op osi ion 2.1, we ha e ha  i is cons an , ha is @ i u = c i @ 3 u o some cons an c i . Hence, u is cons an along he di ec ions (1  0  ; c 1 ) and (0  1  ; c 2 ). We conclude ha u is a unc ion o he a iable a  x alone, whe e a =( c 1 c 2  1).  When ca ied ou in dimension 2, he p e ious p o o is essen ially he one gi en in GG] o es ablish hei ex ended e sion o he conjec u e o De Gio gi o n =2. The p o o ab o e shows ha e e y b ounded solu ion u o  u ; F 0 ( u ) = 0 in R 2 , wi h @ 2 u > 0 and F 2 C 2 ( R ), is a unc ion o one a iable only. He e, no o he assump ion on F is equi ed, since he e is no need o apply Theo em 1.3. Indeed, when n = 2, (2.5) is ob iously sa ised since u is b ounded. Rema k 2.2 . In BCN], he au ho s aised he ollowing ques ion: do es P op osi- ion 2.1 hold o n  3 unde he assump ion ' 2 L 1 ( R n ) { ins ead o (2.2)? I he answe we e yes, hen he p e ious p o o would es ablish he conjec u e o De Gio gi in dimension n , since weha e ha ' i = @ i u is b ounded in R n . Howe e , i has b een es ablished by Ghoussoub and Gui GG] o n  7, and la e by Ba low B] o n  3, ha he answe o he ab o e ques ion is nega i e. We u n now o he P oo o Theo em 1.3 . We conside he unc ions u ( x )= u ( x 0 x n + )  dened o x =( x 0 x n ) 2 R n and 2 R . Fo each , we ha e  u ; F 0 ( u )=0 in R n ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 9 and j u j + j u j C in R n  by (2.3) h oughou he p o o , C will deno e die en p osi i e cons an s indep en- den o R and . No e also ha lim ! + 1 u ( x )=1 o all x 2 R n : Deno ing he de i a i e o u ( x ) wi h esp ec o by @ u ( x ), we ha e @ u ( x )= @ n u ( x 0 x n + ) > 0 o all x 2 R n : We conside he ene gy o u in he ball B R = B R (0) dened by E R ( u )= Z B R  1 2 j u j 2 + F ( u ) ; F (1)  dx: No e ha (2.6) lim ! + 1 E R ( u )= 0 : Indeed, he e m R B R F ( u ) ; F (1) g ends o ze o as ! + 1 by he Leb esgue domina ed con e gence heo em. To see ha he e m R B R (1 = 2) j u j 2 also ends o ze o, we mul iply  u ; F 0 ( u )=0 by u ; 1and we in eg a e by pa s in B R . We ob ain Z B R j u j 2 = Z @B R @u @ ( u ; 1) ; Z B R F 0 ( u )( u ; 1) : Clea ly, he las woin eg als con e ge o ze o, again by he domina ed con e gence heo em. Nex , we compu e and b ound he de i a i eo E R ( u ) wi h esp ec o . We use he equa ion  u ; F 0 ( u ) = 0, he L 1 b ounds o u and u , and he c ucial ac @ u > 0. We nd ha @ E R ( u )= Z B R u ( @ u )+ Z B R F 0 ( u ) @ u = Z @B R @u @ @ u ; C Z @B R @ u : (2.7) 16 LUIGI AMBROSIO AND XAVIER CABR  E o some cons an C indep enden o R . Le m =in R 3 u and M =sup R 3 u and conside he unc ions u ( x 0 )= lim x 3 !;1 u ( x 0 x 3 )and u ( x 0 )= lim x 3 ! + 1 u ( x 0 x 3 ) : No e ha u < u in R 2 , m =in R 2 u , and M =sup R 2 u . We apply Lemma 3.1. I u is cons an hen necessa ily u  M , F 0 ( M )= 0 by (3.2), and F 00 ( M )  0 as s a ed in Lemma 3.1. In case (b) o Lemma 3.1, we see ha he unc ion h sa ises (3.4). Hence, we can apply Lemma 3.2(i) wi h m 1 = in u<m 2 = M =sup u ,and we ob ain again ha F 0 ( M ) = 0 and, using (3.6), ha F 00 ( M )  0. Hence, we ha e p o ed ha we always ha e F 0 ( M )= 0 and F 00 ( M )  0 : In an analogous way, a guing wi h u (o simply eplacing u ( x 0 x 3 )by ; u ( x 0  ; x 3 ), and F ( )by F ( ; )), wesee ha F 0 ( m )= 0 and F 00 ( m )  0 : By he hyp o hesis made on F , i ollows ha F  min F ( m ) F ( M ) g in ( m M ). Supp ose  s ha min F ( m ) F ( M ) g = F ( M ) ( he o he case educes o his one, again by he same change o u and F as b e o e). Then, F ( u ) ; F ( M )  0in R 3 . Hence, he heo em will be p o ed i we show ha Z B R  1 2 j u j 2 + F ( u ) ; F ( M )  dx  CR 2 o each R> 1. To es ablish his, we p o ceed as in he p o o o Theo em 1.3. Tha is, we conside he unc ions u ( x )= u ( x 0 x n + ) dened o x =( x 0 x n ) 2 R n and 2 R , and he ene gy o u in he ball B R = B R (0), dened now by E R ( u )= Z B R  1 2 j u j 2 + F ( u ) ; F ( M )  dx: We need o show ha E R ( u ) = E R ( u 0 )  CR 2 .The compu a ions leading o inequali ies (2.7) and (2.8) a e s ill alid he e { since he ex a hyp o hesis o The- o em 1.1, lim x 3 !1 u ( x 0 x 3 )=  1, was only used in he p o o o Theo em 1.3 o es ablish (2.6), i.e., lim ! + 1 E R ( u ) = 0. Using (2.8) we see ha E R ( u )  CR 2 will hold i we e i y lim sup ! + 1 E R ( u )  CR 2 : ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 17 This inequali y is an easy consequence o Lemmas 3.1 and 3.2(i). Indeed, using s anda d ellip ic es ima es and ha u ( x ) inc eases in B R o u ( x 0 )as ! + 1 ,we ha e lim ! + 1 E R ( u )= Z B R  1 2 j u ( x 0 ) j 2 + F ( u ( x 0 )) ; F ( M )  dx  CR Z B 0 R  1 2 j u ( x 0 ) j 2 + F ( u ( x 0 )) ; F ( M )  dx 0  whe e B 0 R = j x 0 j <R g R 2 .Bu he las in eg al R B 0 R (1 = 2) j u ( x 0 ) j 2 + F ( u ( x 0 )) ; F ( M ) g dx 0 ,whichis compu ed in a wo-dimensional ball, is bounded by CR , since u is a unc ion o one a iable only (by Lemma 3.1), and in his a iable he ene gy is in eg able on all he eal line, by (3.7). The p o o is now comple e.  Re e ences AAC] G. Alb e i, L. Amb osio and X. Cab e, o hcoming. B] M. T. Ba low, On he Liou il le p ope y o di e gence o m ope a o s , Canad. J. Ma h. 50 (1998), 487{496. BBG] M. T. Ba low, R. F. Bass and C. Gui, The Liou il le p ope y and a conjec u e o De Gio gi , p ep in . BCN] H. Be es ycki, L. Caa elli and L. Ni enbe g, Fu he quali a i e p ope ies o el lip ic equa ions in unbounded domains ,Ann. Scuola No m. Sup. Pisa Cl. Sci. (4) 25 (1997), 69{94. BHM] H. Be es ycki, F. Hamel and R. Monneau, One-dimensional symme y o bounded en i e solu ions o some el lip ic equa ions , p ep in . BDG] E. Bombie i, E. De Gio gi and E. Gius i, Minimal cones and he Be ns ein p oblem , In en . Ma h. 7 (1969), 243{268. CC] L. Caa elli and A. Co doba, Uni o m con e gence o a singula pe u ba ion p oblem , Comm. Pu e Appl. Ma h. 48 (1995), 1{12. CGS] L. Caa elli, N. Ga o alo and F. Segala, A g adien bound o en i e solu ions o quasi- linea equa ions and i s consequences , Comm. Pu e Appl. Ma h. 47 (1994), 1457{1473. DG] E. De Gio gi, Con e gencep oblems o unc ionals and ope a o s , P o c. In . Mee ing on Recen Me ho ds in Nonlinea Analysis (Rome, 1978), Pi ago a, Bologna (1979), 131{188. F1] A. Fa ina, Some ema ks on a conjec u e o De Gio gi , Calc. Va . Pa ial Die en ial Equa ions 8 (1999), 233{245. F2] A. Fa ina, Symme y o solu ions o semilinea el lip ic equa ions in R N and ela ed conjec u es , Rice che di Ma ema ica XLVI I I (1999), 129{154. F3] A. Fa ina, o hcoming. GG] N. Ghoussoub and C. Gui, On a conjec u e o DeGio gi and some ela edp oblems , Ma h. Ann. 311 (1998), 481{491. G] E. Gius i, Minimal Su aces and Func ions o Bounded Va ia ion , Bi khause Ve lag, Basel-Bos on (1984). LM] S. Luckhaus and L. Mo dica, The Gibbs{Thompson ela ion wi hin he g adien heo y o phase ansi ions ,A ch. Ra ional Mech. Anal. 107 (1989), 71{83. 18 LUIGI AMBROSIO AND XAVIER CABR  E M1] L. Mo dica, Ag adien bound and a Liou il le heo em o nonlinea Poisson equa ions , Comm. Pu e Appl. Ma h. 38 (1985), 679{684. M2] L. Modica, Mono onici y o he ene gy o en i e solu ions o semilinea el lip ic equa ions , Pa ial die en ial equa ions and he calculus o a ia ions, Vol. II., P og . Nonlinea Die en ial Equa ions Appl. 2, Bi khause , Bos on (1989), 843{850. MM1] L. Mo dica and S. Mo ola, Un esempio di ; ; -con e genza , Boll. Un. Ma . I al. B (5) 14 (1977), 285{299. MM2] L. Mo dica and S. Mo ola, Some en i e solu ions in he plane o nonlinea Poisson equa- ions , Boll. Un. Ma . I al. B (5) 17 (1980), 614{622. Z] W. P. Zieme , Weakly Die en iable Func ions , Sp inge Ve lag, New Yo k (1989). L. Amb osio. Scuola No male Supe io e di Pisa. Piazza dei Ca alie i, 7. 56126 Pisa. I aly. E-mail add ess : luigi@amb osio.sns.i X. Cab  e. Depa amen de Ma em  a ica Aplicada 1, Uni e si a Poli  ecnica de Ca alunya. Diagonal, 647. 08028 Ba celona. Spain. E-mail add ess : [email p o ec ed] c.es