Entire solutions of semilinear elliptic equations in R<sup>3</sup> and a conjecture of De Giorgi
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ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS IN R 3 AND A CONJECTURE OF DE GIORGI Luigi Ambrosio and Xavier Cabr e 1. Intro duction This pap er is concerned with the study of b ounded 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 > 0 in R n . The goal is to establish the one-dimensional character or symmetry of u , namely, that u only dep ends on one variable or, equivalently, that the level sets of u are hyp erplanes. This typ e of symmetry question was raised by De Giorgi in 1978, who made the following conjecture { we quote literally (3), page 175 of DG]: Conjecture (DG]). Let us consider a solution 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 .Is it true that, for every 2 R ,the sets f u = g are hyperplanes, at least if n 8? When n =2, this conjecture was recently proved by Ghoussoub and Gui GG]. In the present pap er we proveitfor n =3. The conjecture, however, remains op en in all dimensions n 4. The pro ofs for n = 2 and 3 use some techniques in the linear theory develop ed by Berestycki, Caarelli and Nirenberg BCN] in one of their pap ers on qualitative prop erties of solutions of semilinear elliptic equations. The question of De Giorgi is also connected with the theories of minimal hyp ersurfaces and phase transitions. As we explain later in the intro duction, the conjecture is sometimes referred to as \the " -version of Bernstein problem for minimal graphs". This relation with Bernstein problem is probably the reason whyDe Giorgi states \at least if n 8" in the ab ove quotation. The authors would like to thank Mariano Giaquinta for several useful discussions. Most of this work was done while the second author was visiting the Universityof Pisa. He thanks the Department of Mathematics for its hospitality. 2000 Mathematics Subject Classication. Primary 35J60, 35B05, 35B40, 35B45. 1
2 LUIGI AMBROSIO AND XAVIER CABR E Most articles dealing with the question of De Giorgi have also considered the conjecture in a slightly simpler version. It consists of assuming that, in addition, (1.1) lim x n !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 . Even in this simpler form, the conjecture was rst proved in GG] for n = 2, in the present article for n =3, and it remains op en for n 4. The p ositive answers to the conjecture for n = 2 and 3 apply to more general 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 a bounded solution of u ; F 0 ( u )=0 in R n satisfying @ n u> 0in R n . Under these assumptions, Ghoussoub and Gui GG] have established that, when n = 2, u is a function of one variable only (see section 2 for the pro of ). Here, the only requirement on the nonlinearity is that F 2 C 2 ( R ). The following are our results for n =3. We start with the simpler case when the solution satises (1.1). Theorem 1.1. Let u be a bounded solution of (1.2) u ; F 0 ( u )= 0 in R 3 satisfying (1.3) @ 3 u> 0 in R 3 and lim x 3 !1 u ( x 0 x 3 )= 1 for al l 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., there exist a 2 R 3 and g 2 C 2 ( R ) such that u ( x )= g ( a x ) for al l x 2 R 3 : Note that the direction a of the variable on which u dep ends is not known apriori. Indeed, if u is a one-dimensional solution satisfying (1.3), we can \slightly" rotate 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 , then we are imp osing an apriori choice of the direction a , namely, a x = x n .In this resp ect, it has b een established in GG] for n = 3, and more recently in BBG], BHM] and F2] for every dimension n , that 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 , that is, u = u ( x n ). This result applies to equation (1.2) for various classes of nonlinearities F whichalways include the Ginzburg-Landau mo del. Theorem 1.1 applies to F 0 ( u ) = u 3 ; u since F ( u ) = (1 ; u 2 ) 2 = 4is a doublewell potential with absolute minima at u = 1. For this nonlinearity, the explicit
ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 3 one-dimensional solution (which is unique up to a translation of the indep endent variable) is given bytanh( 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 and a 3 > 0. The hyp othesis (1.4) made on F in Theorem 1.1 is a necessary condition for the existence of a one-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 a onedimensional solution. More precisely, if 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 ) = 0 has an increasing solution h ( s ) (which is unique up to a translation in s ) such that lim s !1 h ( s )= 1 see Lemma 3.2(ii). The following result establishes for n = 3 the conjecture of De Giorgi in the form stated in DG]. Namely, we do not assume that u ! 1 as x 3 !1 . The result applies to a class of nonlinearities which 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 be a bounded solution 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 numbers 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., there exist a 2 R 3 and g 2 C 2 ( R ) such that u ( x )= g ( a x ) for al l x 2 R 3 : Our pro of of Theorem 1.1 will only require F 2 C 1 1 ( R ), i.e., F 0 Lipschitz. However, in Theorem 1.2 we need F 0 of class C 1 . Question. Do Theorems 1.1 and 1.2 hold for every nonlinearity F 2 C 2 ?That is, can one remove hyp otheses (1.4) and (1.5) in these results? The rst partial result on the question of De Giorgi was found in 1980 by Mo dica and Mortola MM2]. They gave a p ositive answer to the conjecture for n = 2 under the additional assumption that the level sets of u are the graphs of an equiLipschitzian family of functions. Note that, since @ n u> 0, eachlevel set of u is the graph of a function of x 0 .
4 LUIGI AMBROSIO AND XAVIER CABR E In 1985, Mo dica M1] proved that if F 0 in R then every bounded solution u of u ; F 0 ( u )=0 in R n satises the gradient b ound (1.6) 1 2 jr u j 2 F ( u )in R n : In 1994, Caarelli, Garofalo and Segala CGS] generalized this bound to more general equations. They also showed that, if equality o ccurs in (1.6) at some point of R n then the conclusion of the conjecture of De Giorgi is true. More recently, Ghoussoub and Gui GG] have proved the conjecture in full generality when n =2 (see also F3], where weaker assumptions than @ 2 u > 0and more general elliptic op erators are considered). Under the additional assumption that u ( x 0 x n ) ! 1as x n !1 uniformly in x 0 2 R n ; 1 , it is known that u only dep ends on the variable x n here, the hyp othesis @ n u > 0 is not needed. This result was rst proved in GG] for n = 3, and more recently in any dimension n by Barlow, Bass and Gui BBG], Berestycki, Hamel and Monneau BHM], and Farina F2]. Their results apply to various classes of nonlinearities F , which always include the Ginzburg-Landau mo del. These pap ers also contain related results where the assumption on the uniformity of the limits u ! 1 is replaced by various hyp otheses on the level 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 moving planes metho d. Using a one-dimensional arrangement argument, Farina F1] proved the conclusion u = u ( x n ) provided that u minimizes the energy functional in an innite cylinder ! R (with ! b ounded) among the functions satisfying v ( x 0 x n ) ! 1as x n !1 uniformly in x 0 2 ! . Our pro of of the conjecture of De Giorgi in dimension 3 pro ceeds as the pro of given in BCN] and GG] for n = 2. That is, for every co ordinate x i , we consider the function i = @ i u=@ n u .The goal is to show that i is constant (then the conjecture follows immediately) and this will be achieved using a Liouville typ e result (Prop osition 2.1 b elow) for a degenerate elliptic equation satised by i . The following energy estimate is the key result that will allow us to apply such Liouville typ e theorem when n = 3. This energy estimate holds, however, in all dimensions and for arbitrary C 2 ( R ) nonlinearities. Theorem 1.3. Let u be a bounded solution of u ; F 0 ( u )=0 in R n where F is an arbitrary C 2 ( R ) function. Assume that @ n u> 0 in R n and lim x n ! + 1 u ( x 0 x n )= 1 for al l x 0 2 R n ; 1 : For every R> 1 , let B R = fj x j <R g . Then, Z B R 1 2 jr u j 2 + F ( u ) ; F (1) dx CR n ; 1
ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 5 for some constant C independent of R . The energy functional in B R , E R ( u )= Z B R 1 2 jr u j 2 + F ( u ) ; F (1) dx has u ; F 0 ( u ) = 0 as Euler-Lagrange equation. In 1989, Mo dica M2] proved a monotonicity formula for the energy. It states that if F F (1) in R and u is a b ounded solution of u ; F 0 ( u )=0 in R n , then the quantity E R ( u ) R n ; 1 is a nondecreasing function of R . Theorem 1.3 establishes that this quotient is, in addition, b ounded from ab ove. Moreover, the monotonicity formula shows that the upp er b ound in Theorem 1.3 is optimal: indeed, if E R ( u ) =R n ; 1 ! 0 as R ! 1 then we would obtain that E R ( u )= 0 for any R> 0, and hence that u is constant in R n . Note that the estimate of Theorem 1.3 is clearly true assuming that u is a onedimensional solution see (3.7) in Lemma 3.2(i). The estimate is also easy to prove for u as in Theorem 1.3 under the additional assumption that u is a lo cal minimizer of the energy see Remark 2.3. In this case, the estimate already app ears as a lemma in the work of Caarelli and Cordoba CC] on the convergence of intermediate level surfaces in phase transitions. The pro of of the estimate for u as in Theorem 1.3 involves a new idea. It originated from the pro of for lo cal minimizers and from a relation between the key hyp othesis @ n u > 0and the second variation of energy see section 2. Finally, we recall the heuristic argument that connects the conjecture of De Giorgi with Bernstein problem for minimal graphs. For simplicity let us supp ose that F ( u ) = (1 ; u 2 ) 2 = 4. With u as in the conjecture, consider the blown-down sequence u " ( y )= u ( y=" )for y 2 B 1 R n and the p enalized energy of u " in B 1 : H " ( u " )= Z B 1 " 2 jr u " j 2 + 1 " F ( u " ) dy : Note that H " ( u " )is a b ounded sequence, by Theorem 1.3. As " ! 0, the functionals H " ;-converge to a functional whichis nite only for characteristic functions with values in f; 1 1 g and equal (up to the multiplicative constant 2 p 2 = 3) to the area of the hyp ersurface of discontinuity see MM1] and LM]. Heuristically, the sequence u " is exp ected to converge to a characteristic function whose hyp ersurface
6 LUIGI AMBROSIO AND XAVIER CABR E of discontinuity S has minimal area or is at least stationary. The set S describ es the behavior at innity of the level sets of u , and S is exp ected to be the graph of a function dened on R n ; 1 (since the level sets of u are graphs due to hyp othesis @ n u > 0). The conjecture of De Giorgi states that the level sets are hyp erplanes. The connection with the Bernstein problem (see chapter 7 of G] for a complete survey on this topic) is due to the fact that every minimal graph of a function dened on R m = R n ; 1 is known to be a hyp erplane whenever m 7, i.e., n 8. On the other hand, Bombieri, De Giorgi and Giusti gave in BDG] an example of minimal graph of a function of 8 variables dierent than a hyp erplane. In a forthcoming work AAC] with Alb erti, wewill use new variational metho ds to study the conjecture of De Giorgi in higher dimensions. In section 2 we prove Theorems 1.1 and 1.3. Section 3 is devoted to establish Theorem 1.2. 2. Pro of of Theorem 1.1 To prove the conjecture of De Giorgi in dimension 3, we will use the energy estimate of Theorem 1.3. It is this estimate that will allow us to apply, when n = 3, the following Liouville typ e result for the equation r ( ' 2 r )=0 (where ' = @ n u ) satised by = @ i u=@ n u . Prop osition 2.1. Let ' 2 L 1 lo c ( R n ) be a positive function. Suppose 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 ,let B R = fj x j <R g and assume that (2.2) Z B R ( ' ) 2 CR 2 for some constant C independent of R .Then is constant. The study of this typ e of Liouville prop erty, its connections with the sp ectrum of linear Schrodinger op erators, as well as its applications to symmetry prop erties of solutions of nonlinear elliptic equations, were develop ed by Berestycki, Caarelli and Nirenberg BCN]. In the pap ers BCN] and GG], this Liouville prop erty was shown to hold under various decay assumptions on ' . These hyp otheses, which were more restrictive than (2.2), could not b e veried when trying to establish the conjecture of De Giorgi for n 3. Wethen realized that hyp othesis (2.2) could be veried when (and only when) n 3 and that, at the same time, (2.2) was su cient to carry out the pro of of the Liouville prop erty given in BCN]. For convenience, we include b elow their pro of of Prop osition 2.1. See Remark 2.2 for another question regarding this Liouville prop erty. Before proving Theorem 1.3 and Prop osition 2.1, we use these results to give the detailed pro of of Theorem 1.1. First, we establish some simple b ounds and
ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 7 regularity results for the solution u . We assume that u is a b ounded solution of u ; F 0 ( u )= 0 in the distributional sense in R n .It follows that u is of class C 1 , and that r u is b ounded 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 , to the equation u = F 0 ( u ) 2 L 1 in every ball B 2 ( y )ofradius 2 in R n , we nd that 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 with C indep endent of y . Using the Sob olev embedding W 2 p ( B 1 ( y )) C 1 ( B 1 ( y )) for p>n , we conclude (2.3) and that u 2 C 1 . Next, we verify that u 2 W 3 p lo c ( R n ) for all 1 p< 1 in particular, we have that u 2 C 2 ( R n ) for all 0 << 1 : Indeed, since F 0 is C 1 , and u and r u are b ounded, wehavethat 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 every 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 ). Proof of Theorem 1.1 . For each i 2f 1 2 g ,we consider the functions ' = @ 3 u and i = @ i u @ 3 u : Note that i is well dened since @ 3 u > 0. We also have that i is C 1 (see the remarks made ab ove ab out the regularity of u ) and that ' 2 r i = @ 3 u r @ i u ; @ i u r @ 3 u: Note that the right hand side of the last equality belongs to W 1 p lo c ( R 3 ). Using that @ i u and @ 3 u satisfy the same linearized equation w ; F 00 ( u ) w = 0, we 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 erty of Prop osition 2.1. Since ' i = @ i u
8 LUIGI AMBROSIO AND XAVIER CABR E condition (2.2) will be established if we show that, for each R> 1, (2.5) Z B R jr u j 2 CR 2 for some constant C indep endent of 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 we have F ( u ) ; F (1) 0 in 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 B R jr u j 2 Z B R 1 2 jr u j 2 + F ( u ) ; F (1) CR 2 : This proves (2.5). In case that min f F ( ; 1) F (1) g = F ( ; 1), we obtain the same conclusion by applying the previous argument with u ( x 0 x 3 ) replaced by ; u ( x 0 ; x 3 ) and with F ( v ) replaced by F ( ; v ). By Prop osition 2.1, we have that i is constant, that is @ i u = c i @ 3 u for some constant c i . Hence, u is constant along the directions (1 0 ; c 1 ) and (0 1 ; c 2 ). We conclude that u is a function of the variable a x alone, where a =( c 1 c 2 1). When carried out in dimension 2, the previous pro of is essentially the one given in GG] to establish their extended version of the conjecture of De Giorgi for n =2. The pro of ab ove shows that every b ounded solution u of u ; F 0 ( u ) = 0 in R 2 , with @ 2 u > 0 and 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 obviously satised since r u is b ounded. Remark 2.2 . In BCN], the authors raised the following question: do es Prop osition 2.1 hold for n 3 under the assumption ' 2 L 1 ( R n ) { instead of (2.2)? If the answer were yes, then the previous pro of would establish the conjecture of De Giorgi in dimension n , since wehavethat ' i = @ i u is b ounded in R n . However, it has b een established by Ghoussoub and Gui GG] for n 7, and later by Barlow B] for n 3, that the answer to the ab ove question is negative. We turn now to the Proof of Theorem 1.3 . 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 . For each t , we have u t ; F 0 ( u t )=0 in R n
ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 9 and j u t j + jr u t j C in R n by (2.3) throughout the pro of, C will denote dierent p ositive constants indep endent of R and t . Note also that lim t ! + 1 u t ( x )=1 for all x 2 R n : Denoting the derivative of u t ( x ) with resp ect to t by @ t u t ( x ), we have @ t u t ( x )= @ n u ( x 0 x n + t ) > 0 for all x 2 R n : We consider the energy of u t in the ball B R = B R (0) dened by E R ( u t )= Z B R 1 2 jr 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 B R f F ( u t ) ; F (1) g tends to zero as t ! + 1 by the Leb esgue dominated convergence theorem. To see that the term R B R (1 = 2) jr u t j 2 also tends to zero, we multiply u t ; F 0 ( u t )=0 by u t ; 1and we integrate by parts in B R . We obtain Z B R jr u t j 2 = Z @B R @u t @ ( u t ; 1) ; Z B R F 0 ( u t )( u t ; 1) : Clearly, the last twointegrals converge to zero, again by the dominated convergence theorem. Next, we compute and b ound the derivativeof E R ( u t ) with resp ect to t . We use the equation u t ; F 0 ( u t ) = 0, the L 1 b ounds for u t and r u t , and the crucial fact @ t u t > 0. We nd that @ t E R ( u t )= Z B R r u t r ( @ t u t )+ Z B R F 0 ( u t ) @ t u t = Z @B R @u t @ @ t u t ; C Z @B R @ t u t : (2.7)
16 LUIGI AMBROSIO AND XAVIER CABR E for some constant C indep endent of R . Let m =inf R 3 u and M =sup R 3 u and consider the functions 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 ) : Note that u < u in R 2 , m =inf R 2 u , and M =sup R 2 u . We apply Lemma 3.1. If u is constant then necessarily u M , F 0 ( M )= 0 by (3.2), and F 00 ( M ) 0 as stated 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 = inf u<m 2 = M =sup u ,and we obtain again that F 0 ( M ) = 0 and, using (3.6), that F 00 ( M ) 0. Hence, we have proved that we always have F 0 ( M )= 0 and F 00 ( M ) 0 : In an analogous way, arguing with u (or simply replacing u ( x 0 x 3 )by ; u ( x 0 ; x 3 ), and F ( v )by F ( ; v )), wesee that F 0 ( m )= 0 and F 00 ( m ) 0 : By the hyp othesis made on F , it follows 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, again by the same change of u and F as b efore). Then, F ( u ) ; F ( M ) 0in R 3 . Hence, the theorem will be proved if we show that Z B R 1 2 jr u j 2 + F ( u ) ; F ( M ) dx CR 2 for each 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 now by E R ( u t )= Z B R 1 2 jr u t j 2 + F ( u t ) ; F ( M ) dx: We need to show that E R ( u ) = E R ( u 0 ) CR 2 .The computations leading to inequalities (2.7) and (2.8) are still valid here { since the extra hyp othesis of Theorem 1.1, lim x 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., lim t ! + 1 E R ( u t ) = 0. Using (2.8) we see that E R ( u ) CR 2 will hold if we verify lim sup t ! + 1 E R ( u t ) CR 2 :
ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 17 This inequality is an easy consequence of Lemmas 3.1 and 3.2(i). Indeed, using standard elliptic estimates and that u t ( x ) increases in B R to u ( x 0 )as t ! + 1 ,we have lim t ! + 1 E R ( u t )= Z B R 1 2 jr u ( x 0 ) j 2 + F ( u ( x 0 )) ; F ( M ) dx CR Z B 0 R 1 2 jr 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 .But 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 ,whichis computed in a two-dimensional ball, is bounded by CR , since u is a function of one variable only (by Lemma 3.1), and in this variable the energy is integrable on all the real line, by (3.7). The pro of is now complete. References AAC] G. Alb erti, L. Ambrosio and X. Cabre, forthcoming. B] M. T. Barlow, On the Liouvil le property for divergenceform operators , Canad. J. Math. 50 (1998), 487{496. BBG] M. T. Barlow, R. F. Bass and C. Gui, The Liouvil le property and a conjecture of De Giorgi , preprint. BCN] H. Berestycki, L. Caarelli and L. Nirenberg, Further qualitative properties for el liptic equations in unbounded domains ,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 69{94. BHM] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some el liptic equations , preprint. BDG] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem , Invent. Math. 7 (1969), 243{268. CC] L. Caarelli and A. Cordoba, Uniform convergence of a singular perturbation problem , Comm. Pure Appl. Math. 48 (1995), 1{12. CGS] L. Caarelli, N. Garofalo and F. Segala, A gradient bound for entire solutions of quasilinear equations and its consequences , Comm. Pure Appl. Math. 47 (1994), 1457{1473. DG] E. De Giorgi, Convergenceproblems for functionals and operators , Pro c. Int. Meeting on Recent Metho ds in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna (1979), 131{188. F1] A. Farina, Some remarks on a conjecture of De Giorgi , Calc. Var. Partial Dierential Equations 8 (1999), 233{245. F2] A. Farina, Symmetry for solutions of semilinear el liptic equations in R N and related conjectures , Ricerche di Matematica XLVI I I (1999), 129{154. F3] A. Farina, forthcoming. GG] N. Ghoussoub and C. Gui, On a conjecture ofDeGiorgi and some relatedproblems , Math. Ann. 311 (1998), 481{491. G] E. Giusti, Minimal Surfaces and Functions of Bounded Variation , Birkhauser Verlag, Basel-Boston (1984). LM] S. Luckhaus and L. Mo dica, The Gibbs{Thompson relation within the gradient theory of phase transitions ,Arch. Rational Mech. Anal. 107 (1989), 71{83.
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