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

By Luigi Ambrosio and Xavier Cabré

Abstract

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 .

1. Introduction

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:

Conjecture (Reference DG).

Let us consider a solution of

such that

in the whole . Is it true that all level sets of are hyperplanes, at least if ?

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 -version of the Bernstein problem for minimal graphs”. This relation with the Bernstein problem is probably the reason why De Giorgi states “at least if 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.

Theorem 1.1.

Let be a bounded solution of

satisfying

Assume that and that

Then the level sets of are planes, i.e., there exist and such that

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.

Theorem 1.2.

Let be a bounded solution of

satisfying

Assume that and that

for each pair of real numbers satisfying , and . Then the level sets of are planes, i.e., there exist and such that

Our proof of Theorem 1.1 will only require , i.e., Lipschitz. However, in Theorem 1.2 we need of class .

Question.

Do Theorems 1.1 and 1.2 hold for every nonlinearity ? That is, can one remove hypotheses (Equation 1.4) and (Equation 1.5) in these results?

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.

Theorem 1.3.

Let be a bounded solution of

where is an arbitrary function. Assume that

For every , let . Then,

for some constant independent of .

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 -converge to a functional which is finite only for characteristic functions with values in 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.

In section 2 we prove Theorems 1.1 and 1.3. Section 3 is devoted to establishing Theorem 1.2.

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.

Proposition 2.1.

Let be a positive function. Suppose that satisfies

in the distributional sense. For every , let and assume that

for some constant independent of . Then is constant.

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 .

Proof of Theorem 1.1.

For each , we consider the functions

Note that is well defined since . We also have that is (see the remarks made above about the regularity of ) and that

Since the right hand side of the last equality belongs to , we can use that and satisfy the same linearized equation to conclude that

in the weak sense in .

Our goal is to apply to this equation the Liouville property of Proposition 2.1. Since

condition (Equation 2.2) will be established if we show that, for each ,

for some constant independent of .

Recall that, by assumption, in . Suppose first that . In this case we have in . Hence, applying Theorem 1.3 with (it is here and only here that we use ), we conclude that

This proves (Equation 2.5). In case that , we obtain the same conclusion by applying the previous argument with replaced by and with replaced by .

By Proposition 2.1, we have that is constant, that is,

for some constant . Hence, is constant along the directions and . We conclude that is a function of the variable alone, where .

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.

Remark 2.2.

In Reference BCN, the authors raised the following question: Does Proposition 2.1 hold for under the assumption – instead of (Equation 2.2)? If the answer were yes, then the previous proof would establish the conjecture of De Giorgi in dimension , since we have that is bounded in . However, it has been established by Ghoussoub and Gui Reference GG for , and later by Barlow Reference B for , that the answer to the above question is negative.

We turn now to the

Proof of Theorem 1.3.

We consider the functions

defined for and . For each , we have