Entire Solutions of a Semilinear Elliptic Equation in R3 and a Conjecture of De Giorgi

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ENTIRE

SOLUTIONS

OF

SEMILINEAR

ELLIPTIC

EQUA

TIONS

IN

R 3

AND

A

CONJECTURE

OF

DE

GIOR

GI

Luigi Ambr osio and Xa vier Cabr e Abstra ct. In 1978 De Giorgi form ulated the follo wing conjecture. Let u be asolu-tion of  u = u 3 ; u in all of R n such that j u j 1 and @ n u> 0 in R n . Is it true that all level sets f u =  g of u ar ehyp erplanes, at least if n  8? Equiv alen tly ,d oe s u dep end only on one variable? When n = 2, this conjecture was pro ved in 1997 by N. Ghoussoub and C. Gui. In the presen tpap er wep rov ei tfor n = 3. The question, ho wev er, remains op en for n  4. The results for n = 2and 3apply also to the equation  u = F 0( u )f or al arge class of nonlinearitie s F .

1.

Intro

duction

This pap er is concerned with the study of bounded solutions of semilinear elliptic equations 

u

;

F

0 (

u

) = 0 in the whole space R n ,under the assumption that

u

is monotone in one direction, say ,

@

n

u

>

0i n R n . The goal is to establish the one-dimensional character or symmetry of

u

,n amely ,that

u

only dep ends on one variable or, equiv alen tly ,that the lev el sets of

u

are hyp erplanes. This typ e of symmetry question was raised by De Giorgi in 1978, who made the follo wing conjecture {w eq uote (3), page 175 of DG]:

Conjecture

(DG]).

Let us consider as olution

u

2

C

2 ( R n )of 

u

=

u

3 ;

u

such that j

u

j 1



@

n

u>

0 in the whole R n .I si tt rue that all level sets f

u

=



g of

u

are hyp erplanes, at least if

n

 8? When

n

= 2, this conjecture was recen tly pro ved by Ghoussoub and Gui GG]. In the presen tpap er wep rov ei tf or

n

= 3. The conjecture, ho wev er, remains op en The authors would lik et othank Mariano Giaquin ta for sev eral useful discussions. Most of this work was done while the second author was visiting the Univ ersit y of Pisa. He thanks the Departmen to fMathematics for its hospitalit y. 2000 Mathematics Subje ct Classic ation. Primary 35J60, 35B05, 35B40, 35B45. Keywor ds. Nonlinear elliptic PDE Symmetry and monotonicit yprop erties Energy estimates Liouville theorems. 1

2 LUIGI AMBR OSIO AND XA VIER CABR  E in all dimensions

n

 4. The pro ofs for

n

= 2and 3u se some tec hniques in the linear theory dev elop ed by Berest yc ki, Caarelli and Niren berg BCN] in one of their pap ers on qualitativ ep rop erties of solutions of semilinear elliptic equations. The question of De Giorgi is also connected with the theories of minimal hy-pe rsurfaces and phase transitions. As we explain later in the intro duction, the conjecture is sometimes referred to as \the

"

-version of Bernstein problem for min-imal graphs". This relation with Bernstein problem is probably the reason wh yD e Giorgi states \at least if

n

 8" in the ab ov equotation. Most articles dealing with the question of De Giorgi ha ve also considered the conjecture in as ligh tly simpler version. It consists of assuming that, in addition, (1.1) lim xn !1

u

(

x

0

x

n )=  1 for all

x

0 2 R n ; 1

:

Here, the limits are not assumed to be uniform in

x

0 2 R n ; 1 .E ven in this simpler form, the conjecture was rst pro ved in GG] for

n

= 2, in the presen tarticle for

n

= 3, and it remains op en for

n

 4. The positiv ea ns wers to the conjecture for

n

= 2 and 3 apply to more gen-eral nonlinearities than the scalar Ginzburg-Landau equation 

u

+

u

;

u

3 =0 . Throughout the pap er, we assume that

F

2

C

2 ( R )and that

u

is ab ounded solu-tion of 

u

;

F

0 (

u

)=0 in R n satisfying

@

n

u>

0i n R n .Under these assumptions, Ghoussoub and Gui GG] ha ve established that, when

n

=2 ,

u

is a function of one variable only (see section 2f or the pro of) .Here, the only requiremen to n th e nonlinearit yi st ha t

F

2

C

2 ( R ). The follo wing are our results for

n

=3 .W estart with the simpler case when the solution satises (1.1).

Theorem

1.1.

Let

u

bea bounde dsolution of (1.2) 

u

;

F

0 (

u

)=0 in R 3 satisfying (1.3)

@

3

u>

0 in R 3 and lim x3 !1

u

(

x

0

x

3 )=  1 for all

x

0 2 R 2

:

Assume that

F

2

C

2 ( R )and that (1.4)

F

 min f

F

( ; 1)

F

(1) g in ( ; 1



1)

:

Then the level sets of

u

are planes, i.e., ther ee xist

a

2 R 3 and

g

2

C

2 ( R )such that

u

(

x

)=

g

(

a



x

) for all

x

2 R 3

:

Note that the direction

a

of the variable on whic h

u

dep ends is not kno wn apriori. Indeed, if

u

is ao ne-dimensional solution satisfying (1.3), we can \sligh tly" rotate

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 3 co ordinates to obtain a new solution still satisfying (1.3). Instead, if we further assume that the limits in (1.1) are uniform in

x

0 2 R n ; 1 ,t he n we are imp osing an apriori choice of the direction

a

,namely ,

a



x

=

x

n . In this resp ect, it has been established in GG] for

n

= 3, and more recen tly in BBG], BHM] and F2] for ev ery dimension

n

,t hat if the limits in (1.1) are assumed to be uniform in

x

0 2 R n ; 1 then

u

only dep ends on the variable

x

n ,t ha ti s,

u

=

u

(

x

n ). This result applies to equation (1.2) for various classes of nonlineariti es

F

whic h alw ays include the Ginzburg-Landau mo del. Theorem 1.1 applies to

F

0 (

u

)=

u

3 ;

u

since

F

(

u

)=( 1 ;

u

2 ) 2

=

4 is a double-well po ten tial with absolute minima at

u

=  1. For this nonlinearit y, the explicit one-dimensional solution (whic h is unique up to at ranslation of the indep enden t variable) is giv en by tanh(

s=

p 2). Hence, in this case the conclusion of Theorem 1.1 is that

u

(

x

)= tanh

a



x

;

c

p 2  in R 3



for some

c

2 R and

a

2 R 3 with j

a

j =1 an d

a

3

>

0. The hyp othesis (1.4) made on

F

in Theorem 1.1 is a necessary condition for the existence of ao ne-dimensional solution as in the theorem see Lemma 3.2(i). At the same time, most of the equations considered in Theorem 1.1 admit aone-dimensional solution. More precisely ,i f

F

2

C

2 ( R )satises

F>

F

( ; 1) =

F

(1) in ( ; 1



1) and

F

0 ( ; 1) =

F

0 (1) = 0, then

h

00 ;

F

0 (

h

)= 0has an increasing solution

h

(

s

)( whic h is unique up to atranslation in

s

)s uc h that lims !1

h

(

s

)=  1 see Lemma 3.2(ii). The follo wing result establishes for

n

= 3t he conjecture of De Giorgi in the form stated in DG]. Namely ,w ed on ot assume that

u

! 1a s

x

3 ! 1 .The result applies to aclass of nonlineariti es whic h includes the mo del case

F

0 (

u

)=

u

3 ;

u

and also

F

0 (

u

)= sin

u

,for instance.

Theorem

1.2.

Let

u

bea bounde dsolution of 

u

;

F

0 (

u

)=0 in R 3 satisfying

@

3

u>

0 in R 3

:

Assume that

F

2

C

2 ( R )and that (1.5)

F

 min f

F

(

m

)

F

(

M

) g in (

m

M

) for each pair of real numb ers

m<M

satisfying

F

0 (

m

)=

F

0 (

M

)= 0,

F

00 (

m

)  0 and

F

00 (

M

)  0. Then the level sets of

u

are planes, i.e., ther eexist

a

2 R 3 and

g

2

C

2 ( R )such that

u

(

x

)=

g

(

a



x

) for all

x

2 R 3

:

Our pro of of Theorem 1.1 will only require

F

2

C

1  1 ( R ), i.e.,

F

0 Lipsc hitz. Ho wev er, in Theorem 1.2 wen eed

F

0 of class

C

1 .

4 LUIGI AMBR OSIO AND XA VIER CABR  E Question. Do Theorems 1.1 and 1.2 hold for ev ery nonlinearit y

F

2

C

2 ? That is, can one remo veh yp otheses (1.4) and (1.5) in these results? The rst partial result on the question of De Giorgi was found in 1980 byM od ica and Mortola MM2]. They ga ve a po sitiv e answ er to the conjecture for

n

= 2 under the additional assumption that the lev el sets of

u

are the graphs of an equi-Lipsc hitzian family of functions. Note that, since

@

n

u>

0, eac hl ev el set of

u

is the graph of af unction of

x

0 . In 1985, Mo dica M1] pro ved that if

F

 0i n R then ev ery bounded solution

u

of 

u

;

F

0 (

u

)= 0i n R n satises the gradien tb ou nd (1.6) 1 2jr

u

j 2 

F

(

u

) in R n

:

In 1994, Caarelli, Garofalo and Segala CGS] generalized this bound to more general equations. They also sho wed that, if equalit yo ccurs in (1.6) at some poin t of R n then the conclusion of the conjecture of De Giorgi is true. More recen tly , Ghoussoub and Gui GG] ha vep rov ed the conjecture in full generalit ywhen

n

=2 (see also F3], where weak er assumptions than

@

2

u>

0and more general elliptic op erators are considered). Under the additional assumption that

u

(

x

0

x

n ) ! 1a s

x

n ! 1 uniformly in

x

0 2 R n ; 1 ,iti sk no wn that

u

only dep ends on the variable

x

n h ere, the hyp othesis

@

n

u>

0i snot needed. This result was rst pro ved in GG] for

n

= 3, and more recen tly in an y dimension

n

by Barlo w, Bass and Gui BBG], Berest yc ki, Hamel and M onneau BHM], and Farina F2]. Their results apply to various classes of nonlinearities

F

,whic ha lw ays include the Ginzburg-Landau mo del. These pap ers also con tain related results where the assumption on the uniformit y of the limits

u

! 1i sr eplaced byv arious hyp otheses on the lev el sets of

u

.The pap er BBG] uses probabilistic metho ds, BHM] uses the sliding metho d, and GG] and F2] are based on the mo ving planes metho d. Using a one-dimensional arrangemen targumen t, Farina F1] pro ved the con-clusion

u

=

u

(

x

n )p rovided that

u

minimizes the energy functional in an innite cylinder

!

R (with

!

bo unded) among the functions satisfying

v

(

x

0

x

n ) ! 1a s

x

n ! 1 uniformly in

x

0 2

!

. Our pro of of the conjecture of De Giorgi in dimension 3p ro ceeds as the pro of giv en in BCN] and GG] for

n

=2 .That is, for ev ery co ordinate

x

i ,w econsider the function



i =

@

i

u=@

n

u

. The goal is to sho wt ha t



i is constan t(then the conjecture follo ws immediately) and this will be ac hiev ed using a Liouville typ e result (Prop osition 2.1 be low) for anon-uniformly elliptic equation satised by



i . The follo wing energy estimate is the key result that will allo w us to apply suc h Liouville typ et heorem when

n

=3 . This energy estimate holds, ho wev er, in all dimensions and for arbitrary

C

2 ( R )n onlinearities.

Theorem

1.3.

Let

u

bea bounde dsolution of 

u

;

F

0 (

u

)=0 in R n



ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 5 wher e

F

is an arbitr ary

C

2 ( R )function. Assume that

@

n

u>

0 in R n and lim xn ! + 1

u

(

x

0

x

n )=1 for all

x

0 2 R n ; 1

:

For every

R>

1,l et

B

R = fj

x

j

<R

g .T hen, Z BR  1 2jr

u

j 2 +

F

(

u

) ;

F

(1) 

dx



CR

n ; 1 for some constant

C

indep endent of

R

. The energy functional in

B

R ,

E

R (

u

)= Z BR  1 2jr

u

j 2 +

F

(

u

) ;

F

(1) 

dx

has 

u

;

F

0 (

u

)= 0a sEuler-Lagrange equation. In 1989, M od ica M2] pro ved a monotonicit yf orm ula for the energy .I tstates that if

F



F

(1) in R and

u

is ab ounded solution of 

u

;

F

0 (

u

)=0 in R n ,then the quan tit y

E

R (

u

)

R

n ; 1 is anondecreasing function of

R

.T heorem 1.3 establishes that this quotien tis, in addition, bounded from ab ov e. Moreo ver, the monotonicit yform ula sho ws that the upp er bound in Theorem 1.3 is optimal: indeed, if

E

R (

u

)

=R

n ; 1 ! 0a s

R

!1 then wew ould obtain that

E

R (

u

)= 0for an y

R>

0, and hence that

u

is constan t in R n . _Note that the estimate of Theorem 1.3 is clearly true assuming that

u

is ao ne-dimensional solution see (3.7) in Lemma 3.2(i). The estimate is also easy to pro ve for

u

as in Theorem 1.3 under the additional assumption that

u

is al ocal minimizer of the energy see Remark 2.3. In this case, the estimate already app ears as al emma in the work of Caarelli and Cordoba CC] on the con vergence of intermediate lev el surfaces in phase transitions. The pro of of the estimate for

u

as in Theorem 1.3 inv olv es an ew idea. It originated from the pro of for loc al minimizers and from a relation be tw een the key hyp othesis

@

n

u>

0a nd the second variation of energy see section 2. Finally ,w e recall the heuristic argumen tthat connects the conjecture of De Giorgi with Bernstein problem for minimal graphs. For simplicit y let us supp ose that

F

(

u

)= (1 ;

u

2 ) 2

=

4. With

u

as in the conjecture, consider the blo wn-do wn sequence

u

" (

y

)=

u

(

y=

"

) for

y

2

B

1 R n



6 LUIGI AMBR OSIO AND XA VIER CABR  E and the pe nalized energy of

u

" in

B

1 :

H

" (

u

" )= Z B1 

"

jr 2

u

" j 2 + 1 (

u

"F

" ) 

dy

:

Note that

H

" (

u

" )i sa bo unded sequence, by Theorem 1.3. As

"

! 0, the function-als

H

" ;-con verge to a functional whic h is nite only for characteristic functions with values in f; 1



1 g and equal (up to the multiplicati ve constan t2 p 2

=

3) to the area of the hyp ersurface of discon tin uit y see MM1] and LM]. Heuristically ,the sequence

u

" is exp ected to con verge to ac haracteristic function whose hyp ersurface of discon tin uit y

S

has minimal area or is at least stationary .The set

S

describ es the beha vior at innit y of the lev el sets of

u

,a nd

S

is exp ected to be the graph of af unction dened on R n ; 1 (since the lev el sets of

u

are graphs due to hyp othesis

@

n

u>

0). The conjecture of De Giorgi states that the lev el sets are hyp erplanes. The connection with the Bernstein problem (see chapter 17 of G] for acomplete surv ey on this topic) is due to the fact that ev ery minimal graph of af unction de-ned on R m = R n ; 1 is kno wn to be ah yp erplane whenev er

m

 7, i.e.,

n

 8. On the other hand, Bom bieri, De Giorgi and Giusti BDG] established the existence of asmo oth and en tire minimal graph of af unction of 8v ariables dieren tt ha n a hyp erplane. Inaf orthcoming work AA C] with Alb erti, we will use new variational metho ds to study the conjecture of De Giorgi in higher dimensions. In section 2 we pro ve Theorems 1.1 and 1.3. Section 3i sdev oted to establish Theorem 1.2.

2.

Pro

of

of

Theorem

1.1

Top rov e the conjecture of De Giorgi in dimension 3, we will use the energy estimate of Theorem 1.3. It is this estimate that will allo wu sto apply ,when

n

= 3, the follo wing Liouville typ er esult for the equation r (

'

2 r



)= 0, where

'

=

@

n

u

,



=

@

i

u=@

n

u

,a nd r denotes the div ergence op erator.

Prop

osition

2.1.

Let

'

2

L

1 lo c ( R n )b e a positive function. Supp ose that



2

H

1 lo c ( R n )satises (2.1)



r (

'

2 r



)  0 in R n in the distributional sense. For every

R>

1,l et

B

R = fj

x

j

<R

g and assume that (2.2) Z BR (

'

) 2 

CR

2



for some constant

C

indep endent of

R

.T he n



is constant. The study of this typ eo fL iouville prop ert y, its connections with the sp ectrum of linear Sc hr odinger op erators, as well as its applications to symmetry prop erties

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 7 of solutions of nonlinear elliptic equations, were dev elop ed by Berest yc ki, Caarelli and Niren be rg BCN]. In the pap ers BCN] and GG], this Liouville prop ert yw as sho wn to hold under various deca y assumptions on

'

.These hyp otheses, whic h were more restrictiv ethan (2.2), could not be veried when trying to establish the conjecture of De Giorgi for

n

 3. W et hen realized that hyp othesis (2.2) could be veried when (and only when)

n

 3and that, at the same time, (2.2) was su!cien t to carry out the pro of of the Liouville prop ert ygiv en in BCN]. For con venience, we include be low their pro of of Prop osition 2.1. See Remark 2.2 for another question regarding this Liouville prop ert y. Before pro ving Theorem 1.3 and Prop osition 2.1, we use these results to giv e the detailed pro of of Theorem 1.1. First, we establish some simple bo unds and regularit y results for the solution

u

. W ea ssume that

u

is a bounded solution of 

u

;

F

0 (

u

)= 0in the distribution al sense in R n .It follo ws that

u

is of class

C

1 , and that r

u

is bo unded in the whole R n ,i. e., (2.3) jr

u

j2

L

1 ( R n )

:

Indeed, applying interior

W

2 p estimates, with

p>n

,t ot he equation 

u

=

F

0 (

u

) 2

L

1 in ev ery ball

B

2 (

y

)o fradius 2in R n ,w e nd th at k

u

k W 2 p ( B1 ( y )) 

C

 k

u

k L 1 ( B2 ( y )) + k

F

0 (

u

) k L p ( B2 ( y ))  

C

with

C

indep enden to f

y

.U sing the Sob olev em bedding

W

2 p (

B

1 (

y

))

C

1 (

B

1 (

y

)) for

p>n

,w ec onclude (2.3) and that

u

2

C

1 . Next, wev erify that

u

2

W

3 p lo c ( R n ) for all 1 

p<

1  in particular, weh av ethat

u

2

C

2  ( R n ) for all 0

< <

1

:

Indeed, since

F

0 is

C

1 ,a nd

u

and r

u

are bounded, weh av et ha t

F

0 (

u

) 2

W

1 p lo c ( R n ), r

F

0 (

u

)=

F

00 (

u

) r

u

,and (2.4) 

@

j

u

;

F

00 (

u

)

@

j

u

=0 in the weak sense, for ev ery index

j

. Since

F

00 (

u

)

@

j

u

2

L

1 ( R n )

L

p lo c ( R n ), we obtain

@

j

u

2

W

2 p lo c ( R n ). Pr oof of The orem 1.1 .F or eac h

i

2f 1



2 g ,w ec onsider the functions

'

=

@

3

u

and



i =

@

i

u @

u:

3

8 LUIGI AMBR OSIO AND XA VIER CABR  E Note that



i is well dened since

@

3

u>

0. W ea lso ha vet ha t



i is

C

1  (see the remarks made ab ov ea bout the regularit yo f

u

)and that

'

2 r



i =

@

3

u

r

@

i

u

;

@

i

u

r

@

3

u:

Since the righ th and side of the last equalit yb elongs to

W

1 p lo c ( R 3 ), we can use that

@

i

u

and

@

3

u

satisfy the same linearized equation 

w

;

F

00 (

u

)

w

= 0 to conclude that r (

'

2 r



i )=0 in the weak sense in R 3 . Our goal is to apply to this equation the Liouville prop ert y of Prop osition 2.1. Since

'

i =

@

i

u

condition (2.2) will be established if we sho w that, for eac h

R>

1, (2.5) Z BR jr

u

j 2 

CR

2 for some constan t

C

indep enden to f

R

. Recall that, by assumption,

F

 min f

F

( ; 1)

F

(1) g in ( ; 1



1). Supp ose rst that min f

F

( ; 1)

F

(1) g =

F

(1). In this case weh av e

F

(

u

) ;

F

(1)  0i n R 3 . Hence, applying Theorem 1.3 with

n

= 3 (it is here and only here that we use

n

=3 ), we conclude that 1 2 Z BR jr

u

j 2  Z BR  1 2jr

u

j 2 +

F

(

u

) ;

F

(1)  

CR

2

:

This pro ves (2.5). In case that min f

F

( ; 1)

F

(1) g =

F

( ; 1), we obtain the same con-clusion by applying the previous argumen tw ith

u

(

x

0

x

3 )r eplaced by ;

u

(

x

0



;

x

3 ) and with

F

(

v

)r eplaced by

F

( ;

v

). By Prop osition 2.1, weh av ethat



i is constan t, that is

@

i

u

=

c

i

@

3

u

for some constan t

c

i . Hence,

u

is constan ta long the directions (1



0



;

c

1 )and (0



1



;

c

2 ). W e conclude that

u

isaf un cti on of the variable

a



x

alone, where

a

=(

c

1

c

2



1). When carried out in dimension 2, the previous pro of is essen tially the one giv en in GG] to establish their extended version of the conjecture of De Giorgi for

n

=2 . The pro of ab ov esho ws that ev ery bounded solution

u

of 

u

;

F

0 (

u

)= 0i n R 2 , with

@

2

u>

0and

F

2

C

2 ( R ), is a function of one variable only . Here, no other assumption on

F

is required, since there is no need to apply Theorem 1.3. Indeed, when

n

= 2, (2.5) is ob viously satised since r

u

is bounded.

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 9 Remark 2.2 .In BCN], the authors raised the follo wing question: do es Prop osi-tion 2.1 hold for

n

 3 under the assumption

'

2

L

1 ( R n ){ instead of (2.2)? If the answ er were yes, then the previous pro of would establish the conjecture of De Giorgi in dimension

n

,since weh av et ha t

'

i =

@

i

u

is bounded in R n .H ow ev er, it has been established by Ghoussoub and Gui GG] for

n

 7, and later byB arl ow B] for

n

 3, that the answ er to the ab ov eq uestion is negativ e. W et ur n no wt ot he Pr oof of The orem 1.3 .W ec onsider the functions

u

t (

x

)=

u

(

x

0

x

n +

t

)



dened for

x

=(

x

0

x

n ) 2 R n and

t

2 R .F or eac h

t

,w eh av e 

u

t ;

F

0 (

u

t )=0 in R n and j

u

t j+ jr

u

t j

C

in R n



by (2.3) throughout the pro of,

C

will denote dieren tp ositiv ec onstan ts indep en-den to f

R

and

t

.Note also that _lim t ! + 1

u

t (

x

)=1 for all

x

2 R n

:

Denoting the deriv ativ eo f

u

t (

x

)w ith resp ect to

t

by

@

t

u

t (

x

), weh av e

@

t

u

t (

x

)=

@

n

u

(

x

0

x

n +

t

)

>

0 for all

x

2 R n

:

W econsider the energy of

u

t in the ball

B

R =

B

R (0) dened by

E

R (

u

t )= Z BR  1 2jr

u

t j 2 +

F

(

u

t );

F

(1) 

dx:

Note that (2.6) lim _t! + 1

E

R (

u

t )=0

:

Indeed, the term R BR f

F

(

u

t );

F

(1) g tends to zero as

t

! + 1 by the Leb esgue dominated con vergence theorem. To see that the term R BR (1

=

2) jr

u

t j 2 also tends to zero, wem ultiply 

u

t ;

F

0 (

u

t )=0 by

u

t ; 1and wei ntegrate by parts in

B

R . W eo btain Z BR jr

u

t j 2 = Z @BR

@u

t

@

(

u

t ; 1) ; Z BR

F

0 (

u

t )(

u

t ; 1)

:

10 LUIGI AMBR OSIO AND XA VIER CABR  E Clearly ,t he last tw oi ntegrals con verge to zero, again by the dominated con vergence theorem. Next, we compute and bound the deriv ativ eo f

E

R (

u

t )with resp ect to

t

.W euse the equation 

u

t ;

F

0 (

u

t )=0 ,t he

L

1 bo unds for

u

t and r

u

t ,and the crucial fact

@

t

u

t

>

0. W e nd that

_@

E

tR (

u

t )= Z BR r

u

t r (

@

t

u

t )+ Z BR

F

0 (

u

t )

@

u

t t = Z @BR

@u

t

@

@

t

u

t ;

C

Z @BR

@

t

u

t

:

(2.7) Hence, for eac h

T>

0, weh av e

E

R (

u

)=

E

R (

u

T ) ; Z T 0

dt

@

t

E

R (

u

t ) 

E

R (

u

T )+

C

Z T 0

dt

Z @BR

d

(

x

)

@

t

u

t (

x

) =

E

R (

u

T )+

C

Z @BR

d

(

x

) Z T 0

dt

@

t 

u

t (

x

)] =

E

R (

u

T )+

C

Z @BR

d

(

x

)(

u

T ;

u

)(

x

) 

E

R (

u

T )+

C

j

@B

R j =

E

R (

u

T )+

CR

n ; 1

:

(2.8) Letting

T

! + 1 and using (2.6), we obtain the desired estimate. No w that Theorems 1.3 and 1.1 are pro ved, wec an verify that these results only require

F

0 Lipsc hitz {instead of

F

2

C

2 . The only delicate poin tt ob ec hec ked is the linearized equation (2.4), whic h is then used to deriv et he equation satised by



i in the weak sense. Tov erify (2.4), we use that

u

2

W

1 p lo c ( R n ) \

L

1 ( R n )and that

F

0 is Lipsc hitz. It follo ws (see Theorem 2.1.11 of Z]) that

F

0 (

u

) 2

W

1 p lo c ( R n ) and r

F

0 (

u

)=

F

00 (

u

) r

u

almost ev erywhere. Using this, we deriv e(2.4). Remark 2.3 .Recall that

u

is said to be a loc al minimizer if, for ev ery bo unded domain ",

u

is an absolute minimizer of the energy in " on the class of functions agreeing with

u

on

@

". It is easy to pro ve estimate

E

R (

u

) 

CR

n ; 1 ,for

R>

1, whenev er

u

2

L

1 ( R n )isa local minimizer. W ej ust compare the energy of

u

with the energy of af unction satisfying

v

 1i n

B

R; 1 and

v

=

u

on

@B

R . Tak e, for instance,

v

=



+( 1 ;



)

u

,where 0 



 1h as compact supp ort in

B

R and



 1 in

B

R; 1 .T hen

E

R (

u

) 

E

R (

v

)= Z BR n BR ; 1  1 2jr

v

j 2 +

F

(

v

) ;

F

(1) 

dx



C

j

B

R n

B

R; 1 j

CR

n ; 1



ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 11 with

C

indep enden to f

R

. This pro of suggested to look for an appropriate path connecting

u

with the constan tfunction 1, in the general case of Theorem 1.3. W eh av eseen that this is giv en by the solution

u

itself. Indeed, sliding

u

in the direction

x

n ,w eobtain the path

u

t (

x

)=

u

(

x

0

x

n +

t

)connecting

u

for

t

= 0and the function 1f or

t

=+ 1 in the ball

B

R =

B

R (0). Moreo ver, this path is made by functions whic h are all solutions of the same Euler-Lagrange equation. At the same time, it is interesting to observ et hat the condition

@

n

u>

0forces the second variation of energy in

B

R at

u

(and hence, also at eac h function

u

t in the path) to be nonnegativ eu nder perturbations vanishing on

@B

R .Indeed,

@

n

u

is ap ositiv esolution of the linearized equation 

@

n

u

;

F

00 (

u

)

@

n

u

= 0. Byaw ell kno wn result in the theory of the maxim um principle, this implies that the rst eigen value of the op erator ; +

F

00 (

u

)i ne very ball

B

R is nonnegativ e( this result will be needed and established in the pro of of Lemma 3.1). Therefore, Z BR jr



j 2 +

F

00 (

u

)



2  0 for all



2

C

1 c (

B

R )



whic hm eans that the second variation of energy is nonnegativ eu nder perturbations vanishing on

@B

R . Finally ,w ep res en tthe pro of of the Liouville prop ert yexactly as giv en in BCN]. Pr oof of Pr op osition 2.1 .Let



be a

C

1 function on R + suc ht ha t0 



 1and



=  1 if 0 

t

 1, 0 if

t

 2. For

R>

1, let



R (

x

)=



j

x

j

R

 for

x

2 R n

:

Multiplying (2.1) by



2 R and integrating by parts in R n ,w eo bt ain Z



2 R

'

2 jr



j 2 ; 2 Z



R

'

2



r



R r



 2 " Z f R< j x j < 2 R g



2 R

'

2 jr



j 2 # 1 = 2 Z

'

2



2 jr



R j 2 1 = 2 

C

" Z f R< j x j < 2 R g



2 R

'

2 jr



j 2 # 1 = 2 1 2

_R

Z B2 R (

'

) 2 1 = 2



for some constan t

C

indep enden to f

R

.Using hyp othesis (2.2), we infer that (2.9) Z



2 R

'

2 jr



j 2 

C

" Z f R< j x j < 2 R g



2 R

'

2 jr



j 2 # 1 = 2



12 LUIGI AMBR OSIO AND XA VIER CABR  E again with

C

indep enden to f

R

.This implies that R



2 R

'

2 jr



j 2 

C

and, letting

R

!1 ,w eo btain Z R n

'

2 jr



j 2 

C:

It follo ws that the righ th and side of (2.9) tends to zero as

R

!1 ,and hence Z R n

'

2 jr



j 2 =0

:

W ec onclude that



is constan t.

3.

Pro

of

of

Theorem

1.2

Top rov eT heorem 1.2, we pro ceed as in the previous section. W eneed to es-tablish the energy estimate

E

R (

u

) 

CR

2 . In the denition of

E

R (

u

), wen ow replace the term

F

(1) of the previous section by

F

(sup

u

). Lo oking at the pro of of Theorem 1.3, we see that the di!cult y arises when trying to sho w (2.6), i.e., limt ! + 1

E

R (

u

t )= 0{ since we no longer assume limx 3 ! + 1

u

(

x

0

x

3 )= sup

u

for all

x

0 .H en ce, we consider the function 0

_u

(

x

)= lim _x3 ! + 1

u

(

x

0

x

3 )



whic hi sa solution of the same semilinear equation, but no wi n R 2 .U sing am etho d dev elop ed by Berest yc ki, Caarelli and Niren be rg BCN] to study symmetry of solutions in half spaces, we establish a stabilit y prop ert yf or

u

whic h will imply that

u

is actually asolution dep ending on one variable only .A sa consequence, we will obtain that the energy of

u

in at wo-dimensional ball of radius

R

is bo unded by

CR

and, hence, that lim sup t ! + 1

E

R (

u

t )

CR

2

:

Pro ceeding exactly as in the pro of of Theorem 1.3, this estimate will su!ce to establish

E

R (

u

) 

CR

2 and, under the assumptions made on

F

,the conjecture. The rest of this section is dev oted to giv ethe precise pro of of Theorem 1.2. W e start with a lemma that states the stabilit y prop ert yo f

u

and its conse-quences.

Lemma

3.1.

Let

F

2

C

2 ( R )and let

u

bea bounde d solution of 

u

;

F

0 (

u

)=0 in R n satisfying

@

n

u>

0in R n .T hen, the function (3.1)

u

(

x

0 )= lim xn ! + 1

u

(

x

0

x

n )

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 13 is ab ounde ds olution of (3.2) 

u

;

F

0 (

u

)= 0 in R n ; 1 and, in addition, ther eexists ap ositive function

'

2

W

2 p lo c ( R n ; 1 )for every

p<

1 , such that (3.3) 

'

;

F

00 (

u

)

'

 0 in R n ; 1

:

As a conse quenc e, if

n

=3 then

u

is a function of one variable only. Mor e pre cisely, either (a )

u

is equal to ac onstant

M

satisfying

F

00 (

M

)  0,o r (b )ther eexist

b

2 R 2 ,with j

b

j =1 ,a nd a function

h

2

C

2 ( R )such that

h

0

>

0in R and

u

(

x

0 )=

h

(

b



x

0 ) for all

x

0 2 R 2

:

The follo wing lemma, whic h is elemen tary ,i sc oncerned with one-dimensional solutions. W ew ill use its rst part.

Lemma

3.2.

Let

F

bea

C

2 ( R )function. (i) Supp ose that ther eexists ab ounde df unction

h

2

C

2 ( R )satisfying (3.4)

h

0 0 ;

F

0 (

h

)=0 and

h

0

>

0 in R

:

Let

m

1 =i nfR

h

and

m

2 = sup R

h

.Then, we have (3.5)

F

0 (

m

1 )=

F

0 (

m

2 )=0



(3.6)

F>

F

(

m

1 )=

F

(

m

2 ) in (

m

1

m

2 )



and (3.7) Z + 1 ;1  1 20

h

(

s

) 2 +

F

(

h

(

s

)) ;

F

(

m

2 ) 

ds

<

+ 1

:

(ii )Conversely, assume that

m

1

<m

2 aret wo real numb ers such that

F

satises (3

:

5) and (3

:

6). Then ther eexists an incr easing solution

h

of

h

00 ;

F

0 (

h

)=0 in R , with lims !;1

h

(

s

)=

m

1 and lims ! + 1

h

(

s

)=

m

2 .S uch solution is unique up to a translation of the indep endent variable

s

. W es tart with the pro of of Lemma 3.1. Here, wee mp loy sev eral ideas tak en from section 3of BCN]. Pr oof of Lemma 3.1 .The fact that

u

is solution of 

u

;

F

0 (

u

)= 0i n R n ; 1 is easily veried viewing

u

as afunction of

n

variables, limit as

t

! + 1 of the functions

14 LUIGI AMBR OSIO AND XA VIER CABR  E

u

t (

x

0

x

n )=

u

(

x

0

x

n +

t

). By standard elliptic theory ,

u

t !

u

uniformly in the

C

1 sense on compact sets of R n . Toc hec kt he existence of

'>

0s atisfying (3.3), we use that (3.8)

@

n

u>

0 and 

@

n

u

;

F

00 (

u

)

@

n

u

=0 in R n

:

It is well kno wn in the theory of the maxim um principle that (3.8) leads to (3.9) Z R n jr



j 2 +

F

00 (

u

)



2  0 for all



2

C

1 c ( R n ) that is, the rst eigen value (with Diric hlet bo undary conditions) of ; +

F

00 (

u

)i n ev ery bounded domain is nonnegativ e. Indeed, (3.9) can be easily pro ved multiply-ing the equation in (3.8) by



2

=@

n

u

{r ecall that

@

n

u

2

C

1  {a nd integrating by parts to obtain Z 2

 @

u

n r

@

n

u

r



+

F

00 (

u

)



2 = Z



2 (

@

n

u

) 2 jr

@

n

u

j 2

:

Then, (3.9) follo ws by the Cauc hy-Sc hw arz inequalit y. Next, we claim that (3.10) Z R n ; 1 jr



j 2 +

F

00 (

u

)



2  0 for all



2

C

1 c ( R n ; 1 )

:

To sho w this, we tak e

>

0a nd



 2

C

1 ( R )with 0 



  1, 0 



0   2,



 =0 in ( ;1



) (2



+2



+ 1 ), and



 =1 in (



+1



2



+ 1), and we apply (3.9) with



(

x

)=



(

x

0 )



 (

x

n ). W eo btain, after dividing the expression by

 = R



2 ,t hat Z R n ; 1 jr



(

x

0 ) j 2 + Z R n ; 1



2 (

x

0 ) Z R (



0 ) 2 (

x

n )

 + Z R n ; 1



2 (

x

0 ) Z R

F

00 (

u

(

x

0

x

n ))



2 (

x

n )

 is nonnegativ e. Passing to the limit as



! + 1 ,and using

F

2

C

2 and that

u

(

x

0

x

n )c on verges to

u

(

x

0 )a s

x

n ! + 1 uniformly in compact sets of R n ; 1 ,w e conclude (3.10). This is the crucial po int where we need

F

2

C

2 ,a nd not only

F

2

C

1  1 . No w, (3.10) implies that the rst eigen value



1R of ; +

F

00 (

u

)i n the ball

B

0 R = f

x

0 2 R n ; 1 : j

x

0 j

<R

g is nonnegativ efor ev ery

R>

1. Let

'

R

>

0b et he corresp onding rst eigenfunction in

B

0 R : ( 

'

R ;

F

00 (

u

)

'

R = ;



1R

'

R in

B

0 R

'

R =0 on

@B

0 R



normalized suc hthat

'

R (0) = 1. Note that



1R  0i sdecreasing in

R

and, hence, bo unded. Therefore, Harnac kinequalit yg ives that

'

R are bo unded, uniformly in

R

,

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 15 on ev ery compact set of R n ; 1 .By

W

2 p estimates, it follo ws that a subsequence of

'

R con verges in

W

2 p lo c to ap ositiv ef unction

'

2

W

2 p lo c ( R n ; 1 ), for ev ery

p<

1 , satisfying 

'

;

F

00 (

u

)

'

 0i n R n ; 1 (since



1R  0f or ev ery

R

). Finally ,assume that

n

=3 .F or eac h

i

2f 1



2 g ,w econsider the function



i =

@

i

u '

in R 2

:

Note that



i is well dened and weh av eenough regularit y to compute: r (

'

2 r



i )=

'



@

i

u

;

@

i

u



'

and hence



i r (

'

2 r



i )=

@

i

u



@

i

u

; (

@

i

u

) 2 (

'='

) =(

@

i

u

) 2

F

00 (

u

) ; (

@

i

u

) 2 (

'='

)  0



by (3.3). Next, we apply the Liouville prop ert yo fP rop osition 2.1 to this inequalit yi n R 2 . Since

'

i =

@

i

u

is bounded and the dimension is tw o, condition (2.2) holds. W e obtain that



i is constan t, that is, (3.11)

@

i

u

=

c

i

'

for some constan t

c

i .I f

c

1 =

c

2 = 0then

u

is equal to aconstan t

M

.I n this case, (3.10) ob viously implies that

F

00 (

M

)  0. If at least one

c

i is not zero, then

u

is constan ta long the direction (

c

2



;

c

1 ), by (3.11). Hence, taking

b

= j (

c

1

c

2 ) j ; 1 (

c

1

c

2 ), we nd that

u

(

x

0 )=

h

(

b



x

0 ) in R 2 for some function

h

. Using this relation and (3.11), we see that

c

i

'

=

c

i j (

c

1

c

2 ) j ; 1

h

0 (

b



x

0 ), and hence

h

0

>

0i n R . Next, wes ketc hthe pro of of Lemma 3.2, whic h is elemen tary . Pr oof of Lemma 3.2 .(i) Multiplying the equation by

h

0 and integrating, we nd that 2

F

(

h

) ; (

h

0 ) 2 =

c

is constan ti n R .Since

h

has nite limits as

s

! 1 we obtain (3.12) lim inf s !1

h

0 (

s

)=0



whence

c

is equal to both 2

F

(

m

1 )a nd 2

F

(

m

2 ). Since 2

F

(

h

)=

c

+(

h

0 ) 2

>c

and the image of

h

is (

m

1

m

2 ), we infer (3.6). The equalities

F

0 (

m

1 )=

F

0 (

m

2 )= 0follo w from the equation

h

00 ;

F

0 (

h

)= 0a nd from (3.12), using the mean value theorem. Finally ,t he integral in (3.7) is equal to R + 1 ;1 (

h

0 ) 2

ds

,w hic h can be estimated by (

m

2 ;

m

1 )sup

h

0  (

m

2 ;

m

1 ) p 2

D<

+ 1



16 LUIGI AMBR OSIO AND XA VIER CABR  E where

D

=s upt 2 ( m1 m2 )

F

(

t

) ;

F

(

m

1 ). (ii) Let

m

2 (

m

1

m

2 )a nd let



:(

m

1

m

2 ) ! R be the function



(

t

)= Z t _m 1 p 2

F

(

z

) ; 2

F

(

m

1 )

dz



well dened thanks to (3.6). By (3.5) and

F

(

m

1 )=

F

(

m

2 ), we infer that the image of



is the whole real line, and it is easy to chec kb yi ntegration that the unique increasing solution of

h

0 0 ;

F

0 (

h

)=0 in R satisfying

h

(0) =

m

is the inv erse function of



. _Finally ,w eg ivet he Pr oof of The orem 1.2 .Since

@

3

u>

0, the pro of of Theorem 1.1 sho ws that Theo-rem 1.2 will be established if wep rov e( 2.5) for ev ery

R>

1, i.e., Z BR jr

u

j 2 

CR

2 for some constan t

C

indep enden to f

R

. Let

m

= inf R 3

u

and

M

=s up R 3

u

and consider the functions

u

(

x

0 )= lim _x3 !;1

u

(

x

0

x

3 ) and

u

(

x

0 )= lim _x3 ! + 1

u

(

x

0

x

3 )

:

Note that

u<

u

in R 2 ,

m

= infR 2

u

,a nd

M

= sup R 2

u

.W eapply Lemma 3.1. If

u

is constan tt hen necessarily

u



M

,

F

0 (

M

)= 0b y( 3.2), and

F

00 (

M

)  0a sstated in Lemma 3.1. In case (b) of Lemma 3.1, we see that the function

h

satises (3.4). Hence, we can apply Lemma 3.2(i) with

m

1 =i nf

u<

m

2 =

M

=s up

u

,a nd we obtain again that

F

0 (

M

)= 0and, using (3.6), that

F

00 (

M

)  0. Hence, weh av e pro ved that wea lw ays ha ve 0

_F

(

M

)=0 and

F

00 (

M

)  0

:

In an analogous wa y, arguing with

u

(or simply replacing

u

(

x

0

x

3 )b y ;

u

(

x

0



;

x

3 ), and

F

(

v

)b y

F

( ;

v

)), we see that

F

0 (

m

)= 0 and

F

00 (

m

)  0

:

By the hyp othesis made on

F

,it follo ws that

F

 min f

F

(

m

)

F

(

M

) g in (

m

M

). Supp ose rst that min f

F

(

m

)

F

(

M

) g =

F

(

M

)( the other case reduces to this one,

ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUA TIONS 17 again by the same change of

u

and

F

as be fore). Then,

F

(

u

) ;

F

(

M

)  0i n R 3 . Hence, the theorem will be pro ved if we sho wt ha t Z BR  1 2jr

u

j 2 +

F

(

u

) ;

F

(

M

) 

dx



CR

2 for eac h

R>

1. To establish this, we pro ceed as in the pro of of Theorem 1.3. That is, we consider the functions

u

t (

x

)=

u

(

x

0

x

n +

t

)dened for

x

=(

x

0

x

n ) 2 R n and

t

2 R ,and the energy of

u

t in the ball

B

R =

B

R (0), dened no wb y

E

R (

u

t )= Z BR  1 2jr

u

t j 2 +

F

(

u

t );

F

(

M

) 

dx:

W e need to sho wt ha t

E

R (

u

)=

E

R (

u

0 ) 

CR

2 . The computations leading to inequalities (2.7) and (2.8) are still valid here {s ince the extra hyp othesis of The-orem 1.1, limx 3 !1

u

(

x

0

x

3 )=  1, was only used in the pro of of Theorem 1.3 to establish (2.6), i.e., limt ! + 1

E

R (

u

t )= 0. Using (2.8) we see that

E

R (

u

) 

CR

2 will hold if wev erify lim sup t ! + 1

E

R (

u

t )

CR

2

:

This inequalit y is an easy consequence of Lemmas 3.1 and 3.2(i). Indeed, using standard elliptic estimates and that

u

t (

x

)i ncreases in

B

R to

u

(

x

0 )a s

t

! + 1 ,w e ha ve lim _t! + 1

E

R (

u

t )= Z BR  1 2jr

u

(

x

0 ) j 2 +

F

(

u

(

x

0 )) ;

F

(

M

) 

dx



CR

Z B 0 R  1 2jr

u

(

x

0 ) j 2 +

F

(

u

(

x

0 )) ;

F

(

M

) 

dx

0



where

B

0 R = fj

x

0 j

<R

g R 2 .B ut the last integral R B 0 R f (1

=

2) jr

u

(

x

0 ) j 2 +

F

(

u

(

x

0 )) ;

F

(

M

) g

dx

0 ,w hic hi sc omputed in at wo-dimensional ball, is bo unded by

CR

,since

u

is af unction of one variable only (b yL emma 3.1), and in this variable the energy is integrable on all the real line, by (3.7). The pro of is no w complete. References AA C] G. Alb erti, L. Am brosio and X. Cabr e, forthcoming. B] M. T. Barlo w, On the Liouvil le pr op erty for diver genc ef orm op erators ,C anad. J. Math. 50 (1998), 487{496. BBG] M. T. Barlo w, R. F. Bass and C. Gui, The Liouvil le pr op erty and a conje ctur eo fD e Gior gi ,preprin t. BCN] H. Berest ycki, L. Caarelli and L. Niren be rg, Further qualitative pr op erties for ell iptic equations in unb ounde dd om ain s ,A nn. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 69{94.

18 LUIGI AMBR OSIO AND XA VIER CABR  E BHM] H. Berest ycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounde de ntir e solutions of some ell iptic equations ,p rep rin t. BDG] E. Bom bieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein pr oblem , Inv en t. Math. 7 (1969), 243{268. CC] L. Caarelli and A. Cordoba, Uniform conver genc eo fa singular perturb ation pr oblem , Comm. Pure Appl. Math. 48 (1995), 1{12. CGS] L. Caarelli, N. Garofalo and F. Segala, Ag radient bound for entir esolutions of quasi-line ar equations and its conse quenc es ,C omm. Pure Appl. Math. 47 (1994), 1457{1473. DG] E. De Giorgi, Conver genc ep roblems for functionals and op erators ,P roc .I nt. Meeting on Recen tM etho ds in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna (1979), 131{188. F1] A. Farina, Some remarks on a conje ctur eo fD e Gi or gi ,Calc. Var. Partial Dieren tial Equations 8 (1999), 233{245. F2] A. Farina, Symmetry for solutions of semiline ar elliptic equations in R N and relate d conje ctur es ,Ricerc he di Matematica XL V III (1999), 129{154. F3] A. Farina, forthcoming. GG] N. Ghoussoub and C. Gui, On ac onje ctur eo fD eG ior gi and some relate dp roblems ,M ath. Ann. 311 (1998), 481{491. G] E. Giusti, Minimal Surfac es and Functions of Bounde dV ariation ,B irkh auser Verlag, Basel-Boston (1984). LM] S. Luc khaus and L. Mo dica, The Gibbs{Thompson relation within the gradient the ory of phase transitions ,A rch. Rational Mec h. Anal. 107 (1989), 71{83. M1] L. Mo dica, Ag radient bound and a Liouvil le the or em for nonline ar Poisson equations , Comm. Pure Appl. Math. 38 (1985), 679{684. M2] L. Mo dica, Monotonicity of the ener gy for entir es olutions of semiline ar elliptic equations , Partial dieren tial equations and the calculus of variations, Vol. II., Progr. Nonlinear Dieren tial Equations Appl. 2, Birkh auser, Boston (1989), 843{850. MM1] L. Mo dica and S. Mortola, Un esempio di ; ; -conver genza ,B oll. Un. Mat. Ital. B (5) 14 (1977), 285{299. MM2] L. Mo dica and S. Mortola, Some entir esolutions in the plane of nonline ar Poisson equa-tions ,B oll. Un. Mat. Ital. B (5) 17 (1980), 614{622. Z] W. P. Ziemer, We akly Dier entiable Functions ,S pringer Verlag, New York (1989). L. Ambr osio. Scuola Normale Superiore di Pisa. Piazza dei Ca valieri, 7. 56126 Pisa. It al y. E-mail addr ess : luigi@ambrosio .sn s.i t X. Cabr e. Dep ar tament de Ma tem atica A plicad a 1, U niversit atP ol it ecnica de Ca taluny a. Dia gonal, 647. 08028 Bar celona. Sp ain. E-mail addr ess : cabre@ma1.upc. es