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

## 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 ReferenceDG:

### Conjecture (ReferenceDG).

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 ,ReferenceGG. 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 ReferenceBCN 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 .ReferenceGG 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 .ReferenceGG 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 .Equation1.1.

### 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 (Equation1.3), we can “slightly” rotate coordinates to obtain a new solution still satisfying (Equation1.3). Instead, if we further assume that the limits in Equation1.1 are uniform in then we are imposing an apriori choice of the direction , namely, , In this respect, it has been established in .ReferenceGG for and more recently in ,ReferenceBBG, ReferenceBHM and ReferenceF2 for every dimension that if the limits in ,Equation1.1 are assumed to be uniform in then , only depends on the variable that is, , This result applies to equation ( .Equation1.2) 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 (Equation1.4) 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 ReferenceDG. 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 ( ?Equation1.4) and (Equation1.5) in these results?

The first partial result on the question of De Giorgi was found in 1980 by Modica and Mortola ReferenceMM2. 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 ReferenceM1 proved that if in then every bounded solution , of in satisfies the gradient bound

In 1994, Caffarelli, Garofalo and Segala ReferenceCGS generalized this bound to more general equations. They also showed that, if equality occurs in (Equation1.6) at some point of then the conclusion of the conjecture of De Giorgi is true. More recently, Ghoussoub and Gui ,ReferenceGG have proved the conjecture in full generality when (see also ReferenceF3, 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 ReferenceGG for and more recently in any dimension , by Barlow, Bass and Gui ReferenceBBG, Berestycki, Hamel and Monneau ReferenceBHM, and Farina ReferenceF2. 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 .ReferenceBBG uses probabilistic methods, ReferenceBHM uses the sliding method, and ReferenceGG and ReferenceF2 are based on the moving planes method.

Using a one-dimensional arrangement argument, Farina ReferenceF1 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 ReferenceBCN and ReferenceGG 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 ReferenceM2 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 (Equation3.7) 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 ReferenceCC 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 )ReferenceMM1 and ReferenceLM. 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 ).ReferenceG 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 .ReferenceBDG established the existence of a smooth and entire minimal graph of a function of eight variables different than a hyperplane.

In a forthcoming work ReferenceAAC 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 ReferenceBCN. In the papers ReferenceBCN and ReferenceGG, this Liouville property was shown to hold under various decay assumptions on These hypotheses, which were more restrictive than ( .Equation2.2), could not be verified when trying to establish the conjecture of De Giorgi for We then realized that hypothesis ( .Equation2.2) could be verified when (and only when) and that, at the same time, (Equation2.2) was sufficient to carry out the proof of the Liouville property given in ReferenceBCN. 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 ( ,Equation2.3) 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 (Equation2.2) 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 (Equation2.5). 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 ReferenceGG 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 ( ,Equation2.5) is obviously satisfied since is bounded.

### Remark 2.2

In ReferenceBCN, the authors raised the following question: Does Proposition 2.1 hold for under the assumption – instead of (Equation2.2)? 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 .ReferenceGG for and later by Barlow ,ReferenceB 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 ,

and

by (Equation2.3); throughout the proof, will denote different positive constants independent of and Note also that .

Denoting the derivative of with respect to by we have ,

We consider the energy of in the ball defined by

Note that

Indeed, the term tends to zero as by the Lebesgue dominated convergence theorem. To see that the term also tends to zero, we multiply by and we integrate by parts in We obtain .

Clearly, the last two integrals converge to zero, again by the dominated convergence theorem.

Next, we compute and bound the derivative of with respect to We use the equation . the , bounds for and and the crucial fact , We find that .

Hence, for each we have ,

Letting and using (Equation2.6), we obtain the desired estimate.

Now that Theorems 1.3 and 1.1 are proved, we can verify that these results only require Lipschitz – instead of The only delicate point to be checked is the linearized equation ( .Equation2.4), which is then used to derive the equation satisfied by in the weak sense. To verify (Equation2.4), we use that and that is Lipschitz. It follows (see Theorem 2.1.11 of ReferenceZ) that and almost everywhere. Using this, we derive (Equation2.4).

### Remark 2.3

Recall that is said to be a local minimizer if, for every bounded domain , is an absolute minimizer of the energy in on the class of functions agreeing with on It is easy to prove estimate . for , whenever , is a local minimizer. We just compare the energy of with the energy of a function satisfying in and on Take, for instance, . where , has compact support in and in Then .

with independent of .

This proof suggested we look for an appropriate path connecting with the constant function in the general case of Theorem ,1.3. We have seen that this is given by the solution itself. Indeed, *sliding* in the direction we obtain the path , connecting for and the function 1 for in the ball Moreover, this path is made by functions which are all solutions of the same Euler-Lagrange equation. .

At the same time, it is interesting to observe that the condition forces the second variation of energy in at (and hence, also at each function in the path) to be nonnegative under perturbations vanishing on Indeed, . is a positive solution of the linearized equation By a well-known result in the theory of the maximum principle, this implies that the first eigenvalue of the operator . in every ball is nonnegative (this result will be needed and established in the proof of Lemma 3.1). Therefore,

which means that the second variation of energy is nonnegative under perturbations vanishing on .

Finally, we present the proof of the Liouville property exactly as given in ReferenceBCN.

### Proof of Proposition 2.1.

Let be a function on such that and

For let ,

Multiplying (Equation2.1) by and integrating by parts in we obtain ,

for some constant independent of Using hypothesis ( .Equation2.2), we infer that

again with independent of This implies that . and, letting we obtain ,

It follows that the right hand side of (Equation2.9) tends to zero as and hence ,

We conclude that is constant.

## 3. Proof of Theorem 1.2

To prove Theorem 1.2, we proceed as in the previous section. We need to establish the energy estimate In the definition of . we now replace the term , of the previous section by Looking at the proof of Theorem .1.3, we see that the difficulty arises when trying to show (Equation2.6), i.e., – since we no longer assume for all Hence, we consider the function .

which is a solution of the same semilinear equation, but now in Using a method developed by Berestycki, Caffarelli and Nirenberg .ReferenceBCN to study symmetry of solutions in half spaces, we establish a stability property for which will imply that is actually a solution depending on one variable only. As a consequence, we will obtain that the energy of in a two-dimensional ball of radius is bounded by and, hence, that

Proceeding exactly as in the proof of Theorem 1.3, this estimate will suffice to establish and, under the assumptions made on the conjecture. The rest of this section is devoted to giving the precise proof of Theorem ,1.2.

We start with a lemma that states the stability property of and its consequences.

### Lemma 3.1

Let and let be a bounded solution of in satisfying in Then, the function .

is a bounded solution of

and, in addition, there exists a positive function for every such that ,

As a consequence, if then , is a function of one variable only. More precisely, either

(a) is equal to a constant satisfying or ,

(b) there exist with , and a function , such that in and

The following lemma, which is elementary, is concerned with one-dimensional solutions. We will use its first part.

### Lemma 3.2

Let be a function.

(i) Suppose that there exists a bounded function satisfying

Let and Then, we have .

and

(ii) Conversely, assume that are two real numbers such that satisfies and Then there exists an increasing solution . of in with , and Such a solution is unique up to a translation of the independent variable . .

We start with the proof of Lemma 3.1. Here, we employ several ideas taken from section 3 of ReferenceBCN.

### Proof of Lemma 3.1.

The fact that is a solution of in is easily verified viewing as a function of variables, limit as of the functions By standard elliptic theory, . uniformly in the sense on compact sets of .

To check the existence of satisfying (Equation3.3), we use that

It is well known in the theory of the maximum principle that (Equation3.8) leads to

that is, the first eigenvalue (with Dirichlet boundary conditions) of in every bounded domain is nonnegative. Indeed, (Equation3.9) can be easily proved by multiplying the equation in (Equation3.8) by – recall that – and integrating by parts to obtain

Then, (Equation3.9) follows by the Cauchy-Schwarz inequality.

Next, we claim that

To show this, we take and with , , in and , in and we apply ( ,Equation3.9) with We obtain, after dividing the expression by . that ,

is nonnegative. Passing to the limit as and using , and that converges to as uniformly in compact sets of we conclude ( ,Equation3.10). This is the crucial point where we need and not only , .

Now, (Equation3.10) implies that the first eigenvalue of in the ball is nonnegative for every Let . be the corresponding first eigenfunction in :

normalized such that Note that . is decreasing in and, hence, bounded. Therefore, the Harnack inequality gives that are bounded, uniformly in on every compact set of , By . estimates, it follows that a subsequence of converges in to a positive function for every , satisfying , in (since for every ).

Finally, assume that For each . we consider the function ,

Note that is well defined and we have enough regularity to compute:

and hence

by (Equation3.3).

Next, we apply the Liouville property of Proposition 2.1 to this inequality in Since . is bounded and the dimension is two, condition (Equation2.2) holds. We obtain that is constant, that is,

for some constant If . then , is equal to a constant In this case, ( .Equation3.10) obviously implies that .

If at least one is not zero, then is constant along the direction by ( ,Equation3.11). Hence, taking we find that , in for some function Using this relation and ( .Equation3.11), we see that and hence , in .

Next, we sketch the proof of Lemma 3.2, which is elementary.

### Proof of Lemma 3.2.

(i) Multiplying the equation by and integrating, we find that is constant in Since . has finite limits as we obtain

whence is equal to both and Since . and the image of is we infer ( ,Equation3.6). The equalities follow from the equation and from (Equation3.12), using the mean value theorem. Finally, the integral in (Equation3.7) is equal to which can be estimated by ,

where .

(ii) Let and let be the function

well defined thanks to (Equation3.6). By (Equation3.5) and we infer that the image of , is the whole real line, and it is easy to check by integration that the unique increasing solution of in satisfying is the inverse function of .

Finally, we give the

### Proof of Theorem 1.2.

Since the proof of Theorem ,1.1 shows that Theorem 1.2 will be established if we prove (Equation2.5) for every i.e., ,

for some constant independent of .

Let

and consider the functions

Note that in , and , We apply Lemma .3.1. If is constant, then necessarily , by (Equation3.2), and as stated in Lemma 3.1. In case (b) of Lemma 3.1, we see that the function satisfies (Equation3.4). Hence, we can apply Lemma 3.2(i) with and we obtain again that , and, using (Equation3.6), that Hence, we have proved that we always have .

In an analogous way, arguing with (or simply replacing by and , by we see that ),

By the hypothesis made on it follows that , in Suppose first that . (the other case reduces to this one, again by the same change of and as before). Then, in Hence, the theorem will be proved if we show that .

for each .

To establish this, we proceed as in the proof of Theorem 1.3. That is, we consider the functions defined for and and the energy of , in the ball defined now by ,

We need to show that The computations leading to inequalities ( .Equation2.7) and (Equation2.8) are still valid here – since the extra hypothesis of Theorem 1.1, was only used in the proof of Theorem ,1.3 to establish (Equation2.6), i.e., Using ( .Equation2.8) we see that will hold if we verify

This inequality is an easy consequence of Lemmas 3.1 and 3.2(i). Indeed, using standard elliptic estimates and that increases in to as we have ,

where But the last integral .

which is computed in a two-dimensional ball, is bounded by since , 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 (Equation3.7). The proof is now complete.

## Added in proof

In the forthcoming article ReferenceAAC with Alberti, we have proved that Theorem 1.2 holds for every nonlinearity That is, the additional hypothesis .Equation1.5 on is not needed in this theorem.