In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and in Is it true that all level sets . of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for , The question, however, remains open for . The results for . and 3 apply also to the equation for a large class of nonlinearities .
This paper is concerned with the study of bounded solutions of semilinear elliptic equations in the whole space under the assumption that , is monotone in one direction, say, in The goal is to establish the one-dimensional character or symmetry of . namely, that , only depends on one variable or, equivalently, that the level sets of are hyperplanes. This type of symmetry question was raised by De Giorgi in 1978, who made the following conjecture – we quote (3), page 175 of Reference DG:
When this conjecture was recently proved by Ghoussoub and Gui ,Reference GG. In the present paper we prove it for The conjecture, however, remains open in all dimensions . The proofs for . and use some techniques in the linear theory developed by Berestycki, Caffarelli and Nirenberg Reference BCN in one of their papers on qualitative properties of solutions of semilinear elliptic equations.
The question of De Giorgi is also connected with the theories of minimal hypersurfaces and phase transitions. As we explain later in the introduction, the conjecture is sometimes referred to as “the of the Bernstein problem for minimal graphs”. This relation with the Bernstein problem is probably the reason why De Giorgi states “at least if -version in the above quotation. ”
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,
Here, the limits are not assumed to be uniform in Even in this simpler form, the conjecture was first proved in .Reference GG for in the present article for , and it remains open for ,.
The positive answers to the conjecture for and apply to more general nonlinearities than the scalar Ginzburg-Landau equation Throughout the paper, we assume that . and that is a bounded solution of in satisfying in Under these assumptions, Ghoussoub and Gui .Reference GG have established that, when , is a function of one variable only (see section 2 for the proof). Here, the only requirement on the nonlinearity is that .
The following are our results for We start with the simpler case when the solution satisfies .Equation 1.1.
Note that the direction of the variable on which depends is not known apriori. Indeed, if is a one-dimensional solution satisfying (Equation 1.3), we can “slightly” rotate coordinates to obtain a new solution still satisfying (Equation 1.3). Instead, if we further assume that the limits in Equation 1.1 are uniform in then we are imposing an apriori choice of the direction , namely, , In this respect, it has been established in .Reference GG for and more recently in ,Reference BBG, Reference BHM and Reference F2 for every dimension that if the limits in ,Equation 1.1 are assumed to be uniform in then , only depends on the variable that is, , This result applies to equation ( .Equation 1.2) for various classes of nonlinearities which always include the Ginzburg-Landau model.
Theorem 1.1 applies to since is a double-well potential with absolute minima at For this nonlinearity, the explicit one-dimensional solution (which is unique up to a translation of the independent variable) is given by . Hence, in this case the conclusion of Theorem .1.1 is that
for some and with and .
The hypothesis (Equation 1.4) made on 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 one-dimensional solution. More precisely, if satisfies in and then , has an increasing solution (which is unique up to a translation in such that ) see Lemma ;3.2(ii).
The following result establishes for the conjecture of De Giorgi in the form stated in Reference DG. Namely, we do not assume that as The result applies to a class of nonlinearities which includes the model case . and also for instance. ,
The first partial result on the question of De Giorgi was found in 1980 by Modica and Mortola Reference MM2. They gave a positive answer to the conjecture for under the additional assumption that the level sets of are the graphs of an equi-Lipschitzian family of functions. Note that, since each level set of , is the graph of a function of .
In 1985, Modica Reference M1 proved that if in then every bounded solution , of in satisfies the gradient bound
In 1994, Caffarelli, Garofalo and Segala Reference CGS generalized this bound to more general equations. They also showed that, if equality occurs in (Equation 1.6) at some point of then the conclusion of the conjecture of De Giorgi is true. More recently, Ghoussoub and Gui ,Reference GG have proved the conjecture in full generality when (see also Reference F3, where weaker assumptions than and more general elliptic operators are considered).
Under the additional assumption that as uniformly in it is known that , only depends on the variable here, the hypothesis ; is not needed. This result was first proved in Reference GG for and more recently in any dimension , by Barlow, Bass and Gui Reference BBG, Berestycki, Hamel and Monneau Reference BHM, and Farina Reference F2. Their results apply to various classes of nonlinearities which always include the Ginzburg-Landau model. These papers also contain related results where the assumption on the uniformity of the limits , is replaced by various hypotheses on the level sets of The paper .Reference BBG uses probabilistic methods, Reference BHM uses the sliding method, and Reference GG and Reference F2 are based on the moving planes method.
Using a one-dimensional arrangement argument, Farina Reference F1 proved the conclusion provided that minimizes the energy functional in an infinite cylinder (with bounded) among the functions satisfying as uniformly in .
Our proof of the conjecture of De Giorgi in dimension proceeds as the proof given in Reference BCN and Reference GG for That is, for every coordinate . we consider the function , The goal is to show that . is constant (then the conjecture follows immediately) and this will be achieved using a Liouville type result (Proposition 2.1 below) for a nonuniformly elliptic equation satisfied by The following energy estimate is the key result that will allow us to apply such a Liouville type theorem when . This energy estimate holds, however, in all dimensions and for arbitrary . nonlinearities.
The energy functional in ,
has as Euler-Lagrange equation. In 1989, Modica Reference M2 proved a monotonicity formula for the energy. It states that if
and is a bounded solution of in then the quantity ,
is a nondecreasing function of Theorem .1.3 establishes that this quotient is, in addition, bounded from above. Moreover, the monotonicity formula shows that the upper bound in Theorem 1.3 is optimal: indeed, if as then we would obtain that , for any and hence that , is constant in .
Note that the estimate of Theorem 1.3 is clearly true assuming that is a one-dimensional solution; see (Equation 3.7) in Lemma 3.2(i). The estimate is also easy to prove for as in Theorem 1.3 under the additional assumption that is a local minimizer of the energy; see Remark 2.3. In this case, the estimate already appears as a lemma in the work of Caffarelli and Córdoba Reference CC on the convergence of intermediate level surfaces in phase transitions. The proof of the estimate for as in Theorem 1.3 involves a new idea. It originated from the proof for local minimizers and from a relation between the key hypothesis and the second variation of energy; see section 2.
Finally, we recall the heuristic argument that connects the conjecture of De Giorgi with the Bernstein problem for minimal graphs. For simplicity let us suppose that With . as in the conjecture, consider the blown-down sequence
and the penalized energy of in :
Note that is a bounded sequence, by Theorem 1.3. As the functionals , to a functional which is finite only for characteristic functions with values in -converge and equal (up to the multiplicative constant to the area of the hypersurface of discontinuity; see )Reference MM1 and Reference LM. Heuristically, the sequence is expected to converge to a characteristic function whose hypersurface of discontinuity has minimal area or is at least stationary. The set describes the behavior at infinity of the level sets of and , is expected to be the graph of a function defined on (since the level sets of are graphs due to hypothesis The conjecture of De Giorgi states that the level sets are hyperplanes. The connection with the Bernstein problem (see Chapter 17 of ).Reference G for a complete survey on this topic) is due to the fact that every minimal graph of a function defined on is known to be a hyperplane whenever i.e., , On the other hand, Bombieri, De Giorgi and Giusti .Reference BDG established the existence of a smooth and entire minimal graph of a function of eight variables different than a hyperplane.
In a forthcoming work Reference AAC with Alberti, we will use new variational methods to study the conjecture of De Giorgi in higher dimensions.
2. Proof 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 the following Liouville type result for the equation , where , , and , denotes the divergence operator.
The study of this type of Liouville property, its connections with the spectrum of linear Schrödinger operators, as well as its applications to symmetry properties of solutions of nonlinear elliptic equations, were developed by Berestycki, Caffarelli and Nirenberg Reference BCN. In the papers Reference BCN and Reference GG, this Liouville property was shown to hold under various decay assumptions on These hypotheses, which were more restrictive than ( .Equation 2.2), could not be verified when trying to establish the conjecture of De Giorgi for We then realized that hypothesis ( .Equation 2.2) could be verified when (and only when) and that, at the same time, (Equation 2.2) was sufficient to carry out the proof of the Liouville property given in Reference BCN. For convenience, we include below their proof of Proposition 2.1. See Remark 2.2 for another question regarding this Liouville property.
Before proving Theorem 1.3 and Proposition 2.1, we use these results to give the detailed proof of Theorem 1.1. First, we establish some simple bounds and regularity results for the solution We assume that . is a bounded solution of in the distributional sense in It follows that . is of class and that , is bounded in the whole i.e., ,
Indeed, applying interior estimates, with to the equation , in every ball of radius in we find that ,
with independent of Using the Sobolev embedding . for we conclude ( ,Equation 2.3) and that .
Next, we verify that
in particular, we have that
Indeed, since is and , and are bounded, we have that , and ,
in the weak sense, for every index Since . we obtain ,.
When carried out in dimension 2, the previous proof is essentially the one given in Reference GG to establish their extended version of the conjecture of De Giorgi for The proof above shows that every bounded solution . of in with , and is a function of one variable only. Here, no other assumption on , is required, since there is no need to apply Theorem 1.3. Indeed, when ( ,Equation 2.5) is obviously satisfied since is bounded.
We turn now to the