American Mathematical Society

Entire solutions of semilinear elliptic equations in double-struck upper R cubed and a conjecture of De Giorgi

By Luigi Ambrosio and Xavier Cabré

Abstract

In 1978 De Giorgi formulated the following conjecture. Let u be a solution of normal upper Delta u equals u cubed minus u in all of double-struck upper R Superscript n such that StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 and partial-differential Subscript n Baseline u greater-than 0 in double-struck upper R Superscript n . Is it true that all level sets StartSet u equals lamda EndSet of u are hyperplanes, at least if n less-than-or-equal-to 8 ? Equivalently, does u depend only on one variable? When n equals 2 , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for n equals 3 . The question, however, remains open for n greater-than-or-equal-to 4 . The results for n equals 2 and 3 apply also to the equation normal upper Delta u equals upper F prime left-parenthesis u right-parenthesis for a large class of nonlinearities upper F .

1. Introduction

This paper is concerned with the study of bounded solutions of semilinear elliptic equations normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in the whole space double-struck upper R Superscript n , under the assumption that u is monotone in one direction, say, partial-differential Subscript n Baseline u greater-than 0 in double-struck upper R Superscript n . The goal is to establish the one-dimensional character or symmetry of u , namely, that u only depends on one variable or, equivalently, that the level sets of u 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 u element-of upper C squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis of

normal upper Delta u equals u cubed minus u

such that

StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 comma partial-differential Subscript n Baseline u greater-than 0

in the whole double-struck upper R Superscript n . Is it true that all level sets StartSet u equals lamda EndSet of u are hyperplanes, at least if n less-than-or-equal-to 8 ?

When n equals 2 , this conjecture was recently proved by Ghoussoub and Gui ReferenceGG. In the present paper we prove it for n equals 3 . The conjecture, however, remains open in all dimensions n greater-than-or-equal-to 4 . The proofs for n equals 2 and 3 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 epsilon -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 n less-than-or-equal-to 8 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,

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel limit Underscript x Subscript n Baseline right-arrow plus-or-minus normal infinity Endscripts u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis equals plus-or-minus 1 for all x prime element-of double-struck upper R Superscript n minus 1 Baseline period EndLayout

Here, the limits are not assumed to be uniform in x prime element-of double-struck upper R Superscript n minus 1 . Even in this simpler form, the conjecture was first proved in ReferenceGG for n equals 2 , in the present article for n equals 3 , and it remains open for n greater-than-or-equal-to 4 .

The positive answers to the conjecture for n equals 2 and 3 apply to more general nonlinearities than the scalar Ginzburg-Landau equation normal upper Delta u plus u minus u cubed equals 0 . Throughout the paper, we assume that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and that u is a bounded solution of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n satisfying partial-differential Subscript n Baseline u greater-than 0 in double-struck upper R Superscript n . Under these assumptions, Ghoussoub and Gui ReferenceGG have established that, when n equals 2 , u is a function of one variable only (see section 2 for the proof). Here, the only requirement on the nonlinearity is that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis .

The following are our results for n equals 3 . We start with the simpler case when the solution satisfies Equation1.1.

Theorem 1.1

Let u be a bounded solution of

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R cubed EndLayout

satisfying

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel partial-differential Subscript 3 Baseline u greater-than 0 in double-struck upper R cubed and limit Underscript x 3 right-arrow plus-or-minus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis equals plus-or-minus 1 for all x prime element-of double-struck upper R squared period EndLayout

Assume that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and that

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel upper F greater-than-or-equal-to min left-brace upper F left-parenthesis negative 1 right-parenthesis comma upper F left-parenthesis 1 right-parenthesis right-brace in left-parenthesis negative 1 comma 1 right-parenthesis period EndLayout

Then the level sets of u are planes, i.e., there exist a element-of double-struck upper R cubed and g element-of upper C squared left-parenthesis double-struck upper R right-parenthesis such that

u left-parenthesis x right-parenthesis equals g left-parenthesis a dot x right-parenthesis for all x element-of double-struck upper R cubed period

Note that the direction a of the variable on which u depends is not known apriori. Indeed, if u 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 x prime element-of double-struck upper R Superscript n minus 1 , then we are imposing an apriori choice of the direction a , namely, a dot x equals x Subscript n . In this respect, it has been established in ReferenceGG for n equals 3 , and more recently in ReferenceBBG, ReferenceBHM and ReferenceF2 for every dimension n , that if the limits in Equation1.1 are assumed to be uniform in x prime element-of double-struck upper R Superscript n minus 1 , then u only depends on the variable x Subscript n , that is, u equals u left-parenthesis x Subscript n Baseline right-parenthesis . This result applies to equation (Equation1.2) for various classes of nonlinearities upper F which always include the Ginzburg-Landau model.

Theorem 1.1 applies to upper F prime left-parenthesis u right-parenthesis equals u cubed minus u since upper F left-parenthesis u right-parenthesis equals left-parenthesis 1 minus u squared right-parenthesis squared slash 4 is a double-well potential with absolute minima at u equals plus-or-minus 1 . For this nonlinearity, the explicit one-dimensional solution (which is unique up to a translation of the independent variable) is given by hyperbolic tangent left-parenthesis s slash StartRoot 2 EndRoot right-parenthesis . Hence, in this case the conclusion of Theorem 1.1 is that

u left-parenthesis x right-parenthesis equals hyperbolic tangent left-parenthesis StartFraction a dot x minus c Over StartRoot 2 EndRoot EndFraction right-parenthesis in double-struck upper R cubed comma

for some c element-of double-struck upper R and a element-of double-struck upper R cubed with StartAbsoluteValue a EndAbsoluteValue equals 1 and a 3 greater-than 0 .

The hypothesis (Equation1.4) made on upper 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 one-dimensional solution. More precisely, if upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis satisfies upper F greater-than upper F left-parenthesis negative 1 right-parenthesis equals upper F left-parenthesis 1 right-parenthesis in left-parenthesis negative 1 comma 1 right-parenthesis and upper F prime left-parenthesis negative 1 right-parenthesis equals upper F prime left-parenthesis 1 right-parenthesis equals 0 , then h double-prime minus upper F prime left-parenthesis h right-parenthesis equals 0 has an increasing solution h left-parenthesis s right-parenthesis (which is unique up to a translation in s ) such that limit Underscript s right-arrow plus-or-minus normal infinity Endscripts h left-parenthesis s right-parenthesis equals plus-or-minus 1 ; see Lemma 3.2(ii).

The following result establishes for n equals 3 the conjecture of De Giorgi in the form stated in ReferenceDG. Namely, we do not assume that u right-arrow plus-or-minus 1 as x 3 right-arrow plus-or-minus normal infinity . The result applies to a class of nonlinearities which includes the model case upper F prime left-parenthesis u right-parenthesis equals u cubed minus u and also upper F prime left-parenthesis u right-parenthesis equals sine u , for instance.

Theorem 1.2

Let u be a bounded solution of

normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R cubed

satisfying

partial-differential Subscript 3 Baseline u greater-than 0 in double-struck upper R cubed period

Assume that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and that

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel upper F greater-than-or-equal-to min left-brace upper F left-parenthesis m right-parenthesis comma upper F left-parenthesis upper M right-parenthesis right-brace in left-parenthesis m comma upper M right-parenthesis EndLayout

for each pair of real numbers m less-than upper M satisfying upper F prime left-parenthesis m right-parenthesis equals upper F prime left-parenthesis upper M right-parenthesis equals 0 , upper F double-prime left-parenthesis m right-parenthesis greater-than-or-equal-to 0 and upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 . Then the level sets of u are planes, i.e., there exist a element-of double-struck upper R cubed and g element-of upper C squared left-parenthesis double-struck upper R right-parenthesis such that

u left-parenthesis x right-parenthesis equals g left-parenthesis a dot x right-parenthesis for all x element-of double-struck upper R cubed period

Our proof of Theorem 1.1 will only require upper F element-of upper C Superscript 1 comma 1 Baseline left-parenthesis double-struck upper R right-parenthesis , i.e., upper F prime Lipschitz. However, in Theorem 1.2 we need upper F prime of class upper C Superscript 1 .

Question

Do Theorems 1.1 and 1.2 hold for every nonlinearity upper F element-of upper C squared ? 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 n equals 2 under the additional assumption that the level sets of u are the graphs of an equi-Lipschitzian family of functions. Note that, since partial-differential Subscript n Baseline u greater-than 0 , each level set of u is the graph of a function of x prime .

In 1985, Modica ReferenceM1 proved that if upper F greater-than-or-equal-to 0 in double-struck upper R , then every bounded solution u of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n satisfies the gradient bound

StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel one-half StartAbsoluteValue nabla u EndAbsoluteValue squared less-than-or-equal-to upper F left-parenthesis u right-parenthesis in double-struck upper R Superscript n Baseline period EndLayout

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 double-struck upper R Superscript n , 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 n equals 2 (see also ReferenceF3, where weaker assumptions than partial-differential Subscript 2 Baseline u greater-than 0 and more general elliptic operators are considered).

Under the additional assumption that u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis right-arrow plus-or-minus 1 as x Subscript n Baseline right-arrow plus-or-minus normal infinity uniformly in x prime element-of double-struck upper R Superscript n minus 1 , it is known that u only depends on the variable x Subscript n ; here, the hypothesis partial-differential Subscript n Baseline u greater-than 0 is not needed. This result was first proved in ReferenceGG for n equals 3 , and more recently in any dimension n by Barlow, Bass and Gui ReferenceBBG, Berestycki, Hamel and Monneau ReferenceBHM, and Farina ReferenceF2. Their results apply to various classes of nonlinearities upper F , which always include the Ginzburg-Landau model. These papers also contain related results where the assumption on the uniformity of the limits u right-arrow plus-or-minus 1 is replaced by various hypotheses on the level sets of u . 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 u equals u left-parenthesis x Subscript n Baseline right-parenthesis provided that u minimizes the energy functional in an infinite cylinder omega times double-struck upper R (with omega bounded) among the functions satisfying v left-parenthesis x prime comma x Subscript n Baseline right-parenthesis right-arrow plus-or-minus 1 as x Subscript n Baseline right-arrow plus-or-minus normal infinity uniformly in x prime element-of omega .

Our proof of the conjecture of De Giorgi in dimension 3 proceeds as the proof given in ReferenceBCN and ReferenceGG for n equals 2 . That is, for every coordinate x Subscript i , we consider the function sigma Subscript i Baseline equals partial-differential Subscript i Baseline u slash partial-differential Subscript n Baseline u . The goal is to show that sigma Subscript i 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 sigma Subscript i . The following energy estimate is the key result that will allow us to apply such a Liouville type theorem when n equals 3 . This energy estimate holds, however, in all dimensions and for arbitrary upper C squared left-parenthesis double-struck upper R right-parenthesis nonlinearities.

Theorem 1.3

Let u be a bounded solution of

normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n Baseline comma

where upper F is an arbitrary upper C squared left-parenthesis double-struck upper R right-parenthesis function. Assume that

partial-differential Subscript n Baseline u greater-than 0 in double-struck upper R Superscript n Baseline and limit Underscript x Subscript n Baseline right-arrow plus normal infinity Endscripts u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis equals 1 for all x prime element-of double-struck upper R Superscript n minus 1 Baseline period

For every upper R greater-than 1 , let upper B Subscript upper R Baseline equals StartSet StartAbsoluteValue x EndAbsoluteValue less-than upper R EndSet . Then,

integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u EndAbsoluteValue squared plus upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet d x less-than-or-equal-to upper C upper R Superscript n minus 1

for some constant upper C independent of upper R .

The energy functional in upper B Subscript upper R ,

upper E Subscript upper R Baseline left-parenthesis u right-parenthesis equals integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u EndAbsoluteValue squared plus upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet d x comma

has normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 as Euler-Lagrange equation. In 1989, Modica ReferenceM2 proved a monotonicity formula for the energy. It states that if

upper F greater-than-or-equal-to upper F left-parenthesis 1 right-parenthesis in double-struck upper R

and u is a bounded solution of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n , then the quantity

StartFraction upper E Subscript upper R Baseline left-parenthesis u right-parenthesis Over upper R Superscript n minus 1 Baseline EndFraction

is a nondecreasing function of upper R . 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 upper E Subscript upper R Baseline left-parenthesis u right-parenthesis slash upper R Superscript n minus 1 Baseline right-arrow 0 as upper R right-arrow normal infinity , then we would obtain that upper E Subscript upper R Baseline left-parenthesis u right-parenthesis equals 0 for any upper R greater-than 0 , and hence that u is constant in double-struck upper R Superscript n .

Note that the estimate of Theorem 1.3 is clearly true assuming that u is a one-dimensional solution; see (Equation3.7) 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 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 u 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 partial-differential Subscript n Baseline u greater-than 0 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 upper F left-parenthesis u right-parenthesis equals left-parenthesis 1 minus u squared right-parenthesis squared slash 4 . With u as in the conjecture, consider the blown-down sequence

u Subscript epsilon Baseline left-parenthesis y right-parenthesis equals u left-parenthesis y slash epsilon right-parenthesis for y element-of upper B 1 subset-of double-struck upper R Superscript n Baseline comma

and the penalized energy of u Subscript epsilon in upper B 1 :

upper H Subscript epsilon Baseline left-parenthesis u Subscript epsilon Baseline right-parenthesis equals integral Underscript upper B 1 Endscripts StartSet StartFraction epsilon Over 2 EndFraction StartAbsoluteValue nabla u Subscript epsilon Baseline EndAbsoluteValue squared plus StartFraction 1 Over epsilon EndFraction upper F left-parenthesis u Subscript epsilon Baseline right-parenthesis EndSet d y period

Note that upper H Subscript epsilon Baseline left-parenthesis u Subscript epsilon Baseline right-parenthesis is a bounded sequence, by Theorem 1.3. As epsilon right-arrow 0 , the functionals upper H Subscript epsilon normal upper Gamma -converge to a functional which is finite only for characteristic functions with values in StartSet negative 1 comma 1 EndSet and equal (up to the multiplicative constant 2 StartRoot 2 EndRoot slash 3 ) to the area of the hypersurface of discontinuity; see ReferenceMM1 and ReferenceLM. Heuristically, the sequence u Subscript epsilon is expected to converge to a characteristic function whose hypersurface of discontinuity upper S has minimal area or is at least stationary. The set upper S describes the behavior at infinity of the level sets of u , and upper S is expected to be the graph of a function defined on double-struck upper R Superscript n minus 1 (since the level sets of u are graphs due to hypothesis partial-differential Subscript n Baseline u greater-than 0 ). 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 double-struck upper R Superscript m Baseline equals double-struck upper R Superscript n minus 1 is known to be a hyperplane whenever m less-than-or-equal-to 7 , i.e., n less-than-or-equal-to 8 . 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 n equals 3 , the following Liouville type result for the equation nabla dot left-parenthesis phi squared nabla sigma right-parenthesis equals 0 , where phi equals partial-differential Subscript n Baseline u , sigma equals partial-differential Subscript i Baseline u slash partial-differential Subscript n Baseline u , and nabla dot denotes the divergence operator.

Proposition 2.1

Let phi element-of upper L Subscript loc Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis be a positive function. Suppose that sigma element-of upper H Subscript loc Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis satisfies

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel sigma nabla dot left-parenthesis phi squared nabla sigma right-parenthesis greater-than-or-equal-to 0 in double-struck upper R Superscript n EndLayout

in the distributional sense. For every upper R greater-than 1 , let upper B Subscript upper R Baseline equals StartSet StartAbsoluteValue x EndAbsoluteValue less-than upper R EndSet and assume that

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel integral Underscript upper B Subscript upper R Baseline Endscripts left-parenthesis phi sigma right-parenthesis squared less-than-or-equal-to upper C upper R squared comma EndLayout

for some constant upper C independent of upper R . Then sigma 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 phi sigma . These hypotheses, which were more restrictive than (Equation2.2), could not be verified when trying to establish the conjecture of De Giorgi for n greater-than-or-equal-to 3 . We then realized that hypothesis (Equation2.2) could be verified when (and only when) n less-than-or-equal-to 3 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 u . We assume that u is a bounded solution of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in the distributional sense in double-struck upper R Superscript n . It follows that u is of class upper C Superscript 1 , and that nabla u is bounded in the whole double-struck upper R Superscript n , i.e.,

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel StartAbsoluteValue nabla u EndAbsoluteValue element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis period EndLayout

Indeed, applying interior upper W Superscript 2 comma p estimates, with p greater-than n , to the equation normal upper Delta u equals upper F prime left-parenthesis u right-parenthesis element-of upper L Superscript normal infinity in every ball upper B 2 left-parenthesis y right-parenthesis of radius 2 in double-struck upper R Superscript n , we find that

double-vertical-bar u double-vertical-bar Subscript upper W Sub Superscript 2 comma p Subscript left-parenthesis upper B 1 left-parenthesis y right-parenthesis right-parenthesis Baseline less-than-or-equal-to upper C StartSet double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper B 2 left-parenthesis y right-parenthesis right-parenthesis Baseline plus double-vertical-bar upper F prime left-parenthesis u right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper B 2 left-parenthesis y right-parenthesis right-parenthesis Baseline EndSet less-than-or-equal-to upper C

with upper C independent of y . Using the Sobolev embedding upper W Superscript 2 comma p Baseline left-parenthesis upper B 1 left-parenthesis y right-parenthesis right-parenthesis subset-of upper C Superscript 1 Baseline left-parenthesis ModifyingAbove upper B With bar Subscript 1 Baseline left-parenthesis y right-parenthesis right-parenthesis for p greater-than n , we conclude (Equation2.3) and that u element-of upper C Superscript 1 .

Next, we verify that

u element-of upper W Subscript loc Superscript 3 comma p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis for all 1 less-than-or-equal-to p less-than normal infinity semicolon

in particular, we have that

u element-of upper C Superscript 2 comma alpha Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis for all 0 less-than alpha less-than 1 period

Indeed, since upper F prime is upper C Superscript 1 , and u and nabla u are bounded, we have that upper F prime left-parenthesis u right-parenthesis element-of upper W Subscript loc Superscript 1 comma p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis , nabla upper F prime left-parenthesis u right-parenthesis equals upper F double-prime left-parenthesis u right-parenthesis nabla u , and

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel normal upper Delta partial-differential Subscript j Baseline u minus upper F double-prime left-parenthesis u right-parenthesis partial-differential Subscript j Baseline u equals 0 EndLayout

in the weak sense, for every index j . Since upper F double-prime left-parenthesis u right-parenthesis partial-differential Subscript j Baseline u element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis subset-of upper L Subscript loc Superscript p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis , we obtain partial-differential Subscript j Baseline u element-of upper W Subscript loc Superscript 2 comma p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis .

Proof of Theorem 1.1.

For each i element-of StartSet 1 comma 2 EndSet , we consider the functions

phi equals partial-differential Subscript 3 Baseline u and sigma Subscript i Baseline equals StartFraction partial-differential Subscript i Baseline u Over partial-differential Subscript 3 Baseline u EndFraction period

Note that sigma Subscript i is well defined since partial-differential Subscript 3 Baseline u greater-than 0 . We also have that sigma Subscript i is upper C Superscript 1 comma alpha (see the remarks made above about the regularity of u ) and that

phi squared nabla sigma Subscript i Baseline equals partial-differential Subscript 3 Baseline u nabla partial-differential Subscript i Baseline u minus partial-differential Subscript i Baseline u nabla partial-differential Subscript 3 Baseline u period

Since the right hand side of the last equality belongs to upper W Subscript loc Superscript 1 comma p Baseline left-parenthesis double-struck upper R cubed right-parenthesis , we can use that partial-differential Subscript i Baseline u and partial-differential Subscript 3 Baseline u satisfy the same linearized equation normal upper Delta w minus upper F double-prime left-parenthesis u right-parenthesis w equals 0 to conclude that

nabla dot left-parenthesis phi squared nabla sigma Subscript i Baseline right-parenthesis equals 0

in the weak sense in double-struck upper R cubed .

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

phi sigma Subscript i Baseline equals partial-differential Subscript i Baseline u comma

condition (Equation2.2) will be established if we show that, for each upper R greater-than 1 ,

StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel integral Underscript upper B Subscript upper R Baseline Endscripts StartAbsoluteValue nabla u EndAbsoluteValue squared less-than-or-equal-to upper C upper R squared EndLayout

for some constant upper C independent of upper R .

Recall that, by assumption, upper F greater-than-or-equal-to min left-brace upper F left-parenthesis negative 1 right-parenthesis comma upper F left-parenthesis 1 right-parenthesis right-brace in left-parenthesis negative 1 comma 1 right-parenthesis . Suppose first that min left-brace upper F left-parenthesis negative 1 right-parenthesis comma upper F left-parenthesis 1 right-parenthesis right-brace equals upper F left-parenthesis 1 right-parenthesis . In this case we have upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis 1 right-parenthesis greater-than-or-equal-to 0 in double-struck upper R cubed . Hence, applying Theorem 1.3 with n equals 3 (it is here and only here that we use n equals 3 ), we conclude that

one-half integral Underscript upper B Subscript upper R Baseline Endscripts StartAbsoluteValue nabla u EndAbsoluteValue squared less-than-or-equal-to integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u EndAbsoluteValue squared plus upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet less-than-or-equal-to upper C upper R squared period

This proves (Equation2.5). In case that min left-brace upper F left-parenthesis negative 1 right-parenthesis comma upper F left-parenthesis 1 right-parenthesis right-brace equals upper F left-parenthesis negative 1 right-parenthesis , we obtain the same conclusion by applying the previous argument with u left-parenthesis x prime comma x 3 right-parenthesis replaced by minus u left-parenthesis x prime comma minus x 3 right-parenthesis and with upper F left-parenthesis v right-parenthesis replaced by upper F left-parenthesis negative v right-parenthesis .

By Proposition 2.1, we have that sigma Subscript i is constant, that is,

partial-differential Subscript i Baseline u equals c Subscript i Baseline partial-differential Subscript 3 Baseline u

for some constant c Subscript i . Hence, u is constant along the directions left-parenthesis 1 comma 0 comma minus c 1 right-parenthesis and left-parenthesis 0 comma 1 comma minus c 2 right-parenthesis . We conclude that u is a function of the variable a dot x alone, where a equals left-parenthesis c 1 comma c 2 comma 1 right-parenthesis .

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 n equals 2 . The proof above shows that every bounded solution u of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R squared , with partial-differential Subscript 2 Baseline u greater-than 0 and upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis , is a function of one variable only. Here, no other assumption on upper F is required, since there is no need to apply Theorem 1.3. Indeed, when n equals 2 , (Equation2.5) is obviously satisfied since nabla u is bounded.

Remark 2.2

In ReferenceBCN, the authors raised the following question: Does Proposition 2.1 hold for n greater-than-or-equal-to 3 under the assumption phi sigma element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis – instead of (Equation2.2)? If the answer were yes, then the previous proof would establish the conjecture of De Giorgi in dimension n , since we have that phi sigma Subscript i Baseline equals partial-differential Subscript i Baseline u is bounded in double-struck upper R Superscript n . However, it has been established by Ghoussoub and Gui ReferenceGG for n greater-than-or-equal-to 7 , and later by Barlow ReferenceB for n greater-than-or-equal-to 3 , that the answer to the above question is negative.

We turn now to the

Proof of Theorem 1.3.

We consider the functions

u Superscript t Baseline left-parenthesis x right-parenthesis equals u left-parenthesis x prime comma x Subscript n Baseline plus t right-parenthesis comma

defined for x equals left-parenthesis x prime comma x Subscript n Baseline right-parenthesis element-of double-struck upper R Superscript n and t element-of double-struck upper R . For each t , we have

normal upper Delta u Superscript t Baseline minus upper F prime left-parenthesis u Superscript t Baseline right-parenthesis equals 0 in double-struck upper R Superscript n

and

StartAbsoluteValue u Superscript t Baseline EndAbsoluteValue plus StartAbsoluteValue nabla u Superscript t Baseline EndAbsoluteValue less-than-or-equal-to upper C in double-struck upper R Superscript n Baseline comma

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

limit Underscript t right-arrow plus normal infinity Endscripts u Superscript t Baseline left-parenthesis x right-parenthesis equals 1 for all x element-of double-struck upper R Superscript n Baseline period

Denoting the derivative of u Superscript t Baseline left-parenthesis x right-parenthesis with respect to t by partial-differential Subscript t Baseline u Superscript t Baseline left-parenthesis x right-parenthesis , we have

partial-differential Subscript t Baseline u Superscript t Baseline left-parenthesis x right-parenthesis equals partial-differential Subscript n Baseline u left-parenthesis x prime comma x Subscript n Baseline plus t right-parenthesis greater-than 0 for all x element-of double-struck upper R Superscript n Baseline period

We consider the energy of u Superscript t in the ball upper B Subscript upper R Baseline equals upper B Subscript upper R Baseline left-parenthesis 0 right-parenthesis defined by

upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis equals integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u Superscript t Baseline EndAbsoluteValue squared plus upper F left-parenthesis u Superscript t Baseline right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet d x period

Note that

StartLayout 1st Row with Label left-parenthesis 2.6 right-parenthesis EndLabel limit Underscript t right-arrow plus normal infinity Endscripts upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis equals 0 period EndLayout

Indeed, the term integral Underscript upper B Subscript upper R Endscripts StartSet upper F left-parenthesis u Superscript t Baseline right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet tends to zero as t right-arrow plus normal infinity by the Lebesgue dominated convergence theorem. To see that the term integral Underscript upper B Subscript upper R Endscripts left-parenthesis 1 slash 2 right-parenthesis StartAbsoluteValue nabla u Superscript t Baseline EndAbsoluteValue squared also tends to zero, we multiply normal upper Delta u Superscript t Baseline minus upper F prime left-parenthesis u Superscript t Baseline right-parenthesis equals 0 by u Superscript t Baseline minus 1 and we integrate by parts in upper B Subscript upper R . We obtain

integral Underscript upper B Subscript upper R Baseline Endscripts StartAbsoluteValue nabla u Superscript t Baseline EndAbsoluteValue squared equals integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts StartFraction partial-differential u Superscript t Baseline Over partial-differential nu EndFraction left-parenthesis u Superscript t Baseline minus 1 right-parenthesis minus integral Underscript upper B Subscript upper R Baseline Endscripts upper F prime left-parenthesis u Superscript t Baseline right-parenthesis left-parenthesis u Superscript t Baseline minus 1 right-parenthesis period

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

Next, we compute and bound the derivative of upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis with respect to t . We use the equation normal upper Delta u Superscript t Baseline minus upper F prime left-parenthesis u Superscript t Baseline right-parenthesis equals 0 , the upper L Superscript normal infinity bounds for u Superscript t and nabla u Superscript t , and the crucial fact partial-differential Subscript t Baseline u Superscript t Baseline greater-than 0 . We find that

StartLayout 1st Row 1st Column partial-differential Subscript t Baseline upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis 2nd Column equals integral Underscript upper B Subscript upper R Baseline Endscripts nabla u Superscript t Baseline nabla left-parenthesis partial-differential Subscript t Baseline u Superscript t Baseline right-parenthesis plus integral Underscript upper B Subscript upper R Baseline Endscripts upper F prime left-parenthesis u Superscript t Baseline right-parenthesis partial-differential Subscript t Baseline u Superscript t Baseline 2nd Row 1st Column Blank 2nd Column equals integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts StartFraction partial-differential u Superscript t Baseline Over partial-differential nu EndFraction partial-differential Subscript t Baseline u Superscript t Baseline 3rd Row with Label left-parenthesis 2.7 right-parenthesis EndLabel 1st Column Blank 2nd Column greater-than-or-equal-to minus upper C integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts partial-differential Subscript t Baseline u Superscript t Baseline period EndLayout

Hence, for each upper T greater-than 0 , we have

StartLayout 1st Row 1st Column upper E Subscript upper R Baseline left-parenthesis u right-parenthesis 2nd Column equals upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis minus integral Subscript 0 Superscript upper T Baseline d t partial-differential Subscript t Baseline upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis plus upper C integral Subscript 0 Superscript upper T Baseline d t integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts d sigma left-parenthesis x right-parenthesis partial-differential Subscript t Baseline u Superscript t Baseline left-parenthesis x right-parenthesis 3rd Row 1st Column Blank 2nd Column equals upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis plus upper C integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts d sigma left-parenthesis x right-parenthesis integral Subscript 0 Superscript upper T Baseline d t partial-differential Subscript t Baseline left-bracket u Superscript t Baseline left-parenthesis x right-parenthesis right-bracket 4th Row 1st Column Blank 2nd Column equals upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis plus upper C integral Underscript partial-differential upper B Subscript upper R Baseline Endscripts d sigma left-parenthesis x right-parenthesis left-parenthesis u Superscript upper T Baseline minus u right-parenthesis left-parenthesis x right-parenthesis 5th Row with Label left-parenthesis 2.8 right-parenthesis EndLabel 1st Column Blank 2nd Column less-than-or-equal-to upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis plus upper C StartAbsoluteValue partial-differential upper B Subscript upper R Baseline EndAbsoluteValue equals upper E Subscript upper R Baseline left-parenthesis u Superscript upper T Baseline right-parenthesis plus upper C upper R Superscript n minus 1 Baseline period EndLayout

Letting upper T right-arrow plus normal infinity 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 upper F prime Lipschitz – instead of upper F element-of upper C squared . The only delicate point to be checked is the linearized equation (Equation2.4), which is then used to derive the equation satisfied by sigma Subscript i in the weak sense. To verify (Equation2.4), we use that u element-of upper W Subscript loc Superscript 1 comma p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis intersection upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and that upper F prime is Lipschitz. It follows (see Theorem 2.1.11 of ReferenceZ) that upper F prime left-parenthesis u right-parenthesis element-of upper W Subscript loc Superscript 1 comma p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and nabla upper F prime left-parenthesis u right-parenthesis equals upper F double-prime left-parenthesis u right-parenthesis nabla u almost everywhere. Using this, we derive (Equation2.4).

Remark 2.3

Recall that u is said to be a local minimizer if, for every bounded domain normal upper Omega , u is an absolute minimizer of the energy in normal upper Omega on the class of functions agreeing with u on partial-differential normal upper Omega . It is easy to prove estimate upper E Subscript upper R Baseline left-parenthesis u right-parenthesis less-than-or-equal-to upper C upper R Superscript n minus 1 , for upper R greater-than 1 , whenever u element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis is a local minimizer. We just compare the energy of u with the energy of a function satisfying v identical-to 1 in upper B Subscript upper R minus 1 and v equals u on partial-differential upper B Subscript upper R . Take, for instance, v equals eta plus left-parenthesis 1 minus eta right-parenthesis u , where 0 less-than-or-equal-to eta less-than-or-equal-to 1 has compact support in upper B Subscript upper R and eta identical-to 1 in upper B Subscript upper R minus 1 . Then

StartLayout 1st Row 1st Column upper E Subscript upper R Baseline left-parenthesis u right-parenthesis 2nd Column less-than-or-equal-to upper E Subscript upper R Baseline left-parenthesis v right-parenthesis equals integral Underscript upper B Subscript upper R Baseline minus upper B Subscript upper R minus 1 Baseline Endscripts StartSet one-half StartAbsoluteValue nabla v EndAbsoluteValue squared plus upper F left-parenthesis v right-parenthesis minus upper F left-parenthesis 1 right-parenthesis EndSet d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C StartAbsoluteValue upper B Subscript upper R Baseline minus upper B Subscript upper R minus 1 Baseline EndAbsoluteValue less-than-or-equal-to upper C upper R Superscript n minus 1 Baseline comma EndLayout

with upper C independent of upper R .

This proof suggested we look for an appropriate path connecting u with the constant function 1 , in the general case of Theorem 1.3. We have seen that this is given by the solution u itself. Indeed, sliding u in the direction x Subscript n , we obtain the path u Superscript t Baseline left-parenthesis x right-parenthesis equals u left-parenthesis x prime comma x Subscript n Baseline plus t right-parenthesis connecting u for t equals 0 and the function 1 for t equals plus normal infinity in the ball upper B Subscript upper R Baseline equals upper B Subscript upper R Baseline left-parenthesis 0 right-parenthesis . 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 partial-differential Subscript n Baseline u greater-than 0 forces the second variation of energy in upper B Subscript upper R at u (and hence, also at each function u Superscript t in the path) to be nonnegative under perturbations vanishing on partial-differential upper B Subscript upper R . Indeed, partial-differential Subscript n Baseline u is a positive solution of the linearized equation normal upper Delta partial-differential Subscript n Baseline u minus upper F double-prime left-parenthesis u right-parenthesis partial-differential Subscript n Baseline u equals 0 . By a well-known result in the theory of the maximum principle, this implies that the first eigenvalue of the operator negative normal upper Delta plus upper F double-prime left-parenthesis u right-parenthesis in every ball upper B Subscript upper R is nonnegative (this result will be needed and established in the proof of Lemma 3.1). Therefore,

integral Underscript upper B Subscript upper R Baseline Endscripts StartAbsoluteValue nabla xi EndAbsoluteValue squared plus upper F double-prime left-parenthesis u right-parenthesis xi squared greater-than-or-equal-to 0 for all xi element-of upper C Subscript c Superscript normal infinity Baseline left-parenthesis upper B Subscript upper R Baseline right-parenthesis comma

which means that the second variation of energy is nonnegative under perturbations vanishing on partial-differential upper B Subscript upper R .

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

Proof of Proposition 2.1.

Let zeta be a upper C Superscript normal infinity function on double-struck upper R Superscript plus such that 0 less-than-or-equal-to zeta less-than-or-equal-to 1 and

zeta equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column if 0 less-than-or-equal-to t less-than-or-equal-to 1 comma 2nd Row 1st Column 0 2nd Column if t greater-than-or-equal-to 2 period EndLayout

For upper R greater-than 1 , let

zeta Subscript upper R Baseline left-parenthesis x right-parenthesis equals zeta left-parenthesis StartFraction StartAbsoluteValue x EndAbsoluteValue Over upper R EndFraction right-parenthesis for x element-of double-struck upper R Superscript n Baseline period

Multiplying (Equation2.1) by zeta Subscript upper R Superscript 2 and integrating by parts in double-struck upper R Superscript n , we obtain

StartLayout 1st Row 1st Column integral zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared 2nd Column less-than-or-equal-to minus 2 integral zeta Subscript upper R Baseline phi squared sigma nabla zeta Subscript upper R Baseline dot nabla sigma 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 2 left-bracket integral Underscript StartSet upper R less-than StartAbsoluteValue x EndAbsoluteValue less-than 2 upper R EndSet Endscripts zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared right-bracket Superscript 1 slash 2 Baseline left-bracket integral phi squared sigma squared StartAbsoluteValue nabla zeta Subscript upper R Baseline EndAbsoluteValue squared right-bracket Superscript 1 slash 2 Baseline 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C left-bracket integral Underscript StartSet upper R less-than StartAbsoluteValue x EndAbsoluteValue less-than 2 upper R EndSet Endscripts zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared right-bracket Superscript 1 slash 2 Baseline left-bracket StartFraction 1 Over upper R squared EndFraction integral Underscript upper B Subscript 2 upper R Baseline Endscripts left-parenthesis phi sigma right-parenthesis squared right-bracket Superscript 1 slash 2 Baseline comma EndLayout

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

StartLayout 1st Row with Label left-parenthesis 2.9 right-parenthesis EndLabel integral zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared less-than-or-equal-to upper C left-bracket integral Underscript StartSet upper R less-than StartAbsoluteValue x EndAbsoluteValue less-than 2 upper R EndSet Endscripts zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared right-bracket Superscript 1 slash 2 Baseline comma EndLayout

again with upper C independent of upper R . This implies that integral zeta Subscript upper R Superscript 2 Baseline phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared less-than-or-equal-to upper C and, letting upper R right-arrow normal infinity , we obtain

integral Underscript double-struck upper R Superscript n Baseline Endscripts phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared less-than-or-equal-to upper C period

It follows that the right hand side of (Equation2.9) tends to zero as upper R right-arrow normal infinity , and hence

integral Underscript double-struck upper R Superscript n Baseline Endscripts phi squared StartAbsoluteValue nabla sigma EndAbsoluteValue squared equals 0 period

We conclude that sigma 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 upper E Subscript upper R Baseline left-parenthesis u right-parenthesis less-than-or-equal-to upper C upper R squared . In the definition of upper E Subscript upper R Baseline left-parenthesis u right-parenthesis , we now replace the term upper F left-parenthesis 1 right-parenthesis of the previous section by upper F left-parenthesis sup u right-parenthesis . Looking at the proof of Theorem 1.3, we see that the difficulty arises when trying to show (Equation2.6), i.e., limit Underscript t right-arrow plus normal infinity Endscripts upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis equals 0 – since we no longer assume limit Underscript x 3 right-arrow plus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis equals sup u for all x prime . Hence, we consider the function

ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis equals limit Underscript x 3 right-arrow plus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis comma

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

limit sup upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis less-than-or-equal-to upper C upper R squared period

Proceeding exactly as in the proof of Theorem 1.3, this estimate will suffice to establish upper E Subscript upper R Baseline left-parenthesis u right-parenthesis less-than-or-equal-to upper C upper R squared and, under the assumptions made on upper F , 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 u overbar and its consequences.

Lemma 3.1

Let upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and let u be a bounded solution of normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n satisfying partial-differential Subscript n Baseline u greater-than 0 in double-struck upper R Superscript n . Then, the function

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel ModifyingAbove u With bar left-parenthesis x prime right-parenthesis equals limit Underscript x Subscript n Baseline right-arrow plus normal infinity Endscripts u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis EndLayout

is a bounded solution of

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel normal upper Delta u overbar minus upper F prime left-parenthesis u overbar right-parenthesis equals 0 in double-struck upper R Superscript n minus 1 EndLayout

and, in addition, there exists a positive function phi element-of upper W Subscript loc Superscript 2 comma p Baseline left-parenthesis double-struck upper R Superscript n minus 1 Baseline right-parenthesis for every p less-than normal infinity , such that

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel normal upper Delta phi minus upper F double-prime left-parenthesis u overbar right-parenthesis phi less-than-or-equal-to 0 in double-struck upper R Superscript n minus 1 Baseline period EndLayout

As a consequence, if n equals 3 , then u overbar is a function of one variable only. More precisely, either

(a) u overbar is equal to a constant upper M satisfying upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 , or

(b) there exist b element-of double-struck upper R squared , with StartAbsoluteValue b EndAbsoluteValue equals 1 , and a function h element-of upper C squared left-parenthesis double-struck upper R right-parenthesis such that h prime greater-than 0 in double-struck upper R and

ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis equals h left-parenthesis b dot x Superscript prime Baseline right-parenthesis for all x prime element-of double-struck upper R squared period

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

Lemma 3.2

Let upper F be a upper C squared left-parenthesis double-struck upper R right-parenthesis function.

(i) Suppose that there exists a bounded function h element-of upper C squared left-parenthesis double-struck upper R right-parenthesis satisfying

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel h double-prime minus upper F prime left-parenthesis h right-parenthesis equals 0 and h prime greater-than 0 in double-struck upper R period EndLayout

Let m 1 equals inf Underscript double-struck upper R Endscripts h and m 2 equals sup Underscript double-struck upper R Endscripts h . Then, we have

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel upper F prime left-parenthesis m 1 right-parenthesis equals upper F prime left-parenthesis m 2 right-parenthesis equals 0 comma EndLayout

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel upper F greater-than upper F left-parenthesis m 1 right-parenthesis equals upper F left-parenthesis m 2 right-parenthesis in left-parenthesis m 1 comma m 2 right-parenthesis comma EndLayout and

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel integral Subscript negative normal infinity Superscript plus normal infinity Baseline StartSet one-half h prime left-parenthesis s right-parenthesis squared plus upper F left-parenthesis h left-parenthesis s right-parenthesis right-parenthesis minus upper F left-parenthesis m 2 right-parenthesis EndSet d s less-than plus normal infinity period EndLayout

(ii) Conversely, assume that m 1 less-than m 2 are two real numbers such that upper F satisfies left-parenthesis 3.5 right-parenthesis and left-parenthesis 3.6 right-parenthesis . Then there exists an increasing solution h of h double-prime minus upper F prime left-parenthesis h right-parenthesis equals 0 in double-struck upper R , with limit Underscript s right-arrow negative normal infinity Endscripts h left-parenthesis s right-parenthesis equals m 1 and limit Underscript s right-arrow plus normal infinity Endscripts h left-parenthesis s right-parenthesis equals m 2 . Such a solution is unique up to a translation of the independent variable s .

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 u overbar is a solution of normal upper Delta u overbar minus upper F prime left-parenthesis u overbar right-parenthesis equals 0 in double-struck upper R Superscript n minus 1 is easily verified viewing u overbar as a function of n variables, limit as t right-arrow plus normal infinity of the functions u Superscript t Baseline left-parenthesis x prime comma x Subscript n Baseline right-parenthesis equals u left-parenthesis x prime comma x Subscript n Baseline plus t right-parenthesis . By standard elliptic theory, u Superscript t Baseline right-arrow u overbar uniformly in the upper C Superscript 1 sense on compact sets of double-struck upper R Superscript n .

To check the existence of phi greater-than 0 satisfying (Equation3.3), we use that

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel partial-differential Subscript n Baseline u greater-than 0 and normal upper Delta partial-differential Subscript n Baseline u minus upper F double-prime left-parenthesis u right-parenthesis partial-differential Subscript n Baseline u equals 0 in double-struck upper R Superscript n Baseline period EndLayout

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

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel integral Underscript double-struck upper R Superscript n Baseline Endscripts StartAbsoluteValue nabla xi EndAbsoluteValue squared plus upper F double-prime left-parenthesis u right-parenthesis xi squared greater-than-or-equal-to 0 for all xi element-of upper C Subscript c Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis semicolon EndLayout

that is, the first eigenvalue (with Dirichlet boundary conditions) of negative normal upper Delta plus upper F double-prime left-parenthesis u right-parenthesis in every bounded domain is nonnegative. Indeed, (Equation3.9) can be easily proved by multiplying the equation in (Equation3.8) by xi squared slash partial-differential Subscript n Baseline u – recall that partial-differential Subscript n Baseline u element-of upper C Superscript 1 comma alpha – and integrating by parts to obtain

integral StartFraction 2 xi Over partial-differential Subscript n Baseline u EndFraction nabla partial-differential Subscript n Baseline u dot nabla xi plus upper F double-prime left-parenthesis u right-parenthesis xi squared equals integral StartFraction xi squared Over left-parenthesis partial-differential Subscript n Baseline u right-parenthesis squared EndFraction StartAbsoluteValue nabla partial-differential Subscript n Baseline u EndAbsoluteValue squared period

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

Next, we claim that

StartLayout 1st Row with Label left-parenthesis 3.10 right-parenthesis EndLabel integral Underscript double-struck upper R Superscript n minus 1 Baseline Endscripts StartAbsoluteValue nabla eta EndAbsoluteValue squared plus upper F double-prime left-parenthesis u overbar right-parenthesis eta squared greater-than-or-equal-to 0 for all eta element-of upper C Subscript c Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n minus 1 Baseline right-parenthesis period EndLayout

To show this, we take rho greater-than 0 and psi Subscript rho Baseline element-of upper C Superscript normal infinity Baseline left-parenthesis double-struck upper R right-parenthesis with 0 less-than-or-equal-to psi Subscript rho Baseline less-than-or-equal-to 1 , 0 less-than-or-equal-to psi prime Subscript rho Baseline less-than-or-equal-to 2 , psi Subscript rho Baseline equals 0 in left-parenthesis negative normal infinity comma rho right-parenthesis union left-parenthesis 2 rho plus 2 comma plus normal infinity right-parenthesis , and psi Subscript rho Baseline equals 1 in left-parenthesis rho plus 1 comma 2 rho plus 1 right-parenthesis , and we apply (Equation3.9) with xi left-parenthesis x right-parenthesis equals eta left-parenthesis x prime right-parenthesis psi Subscript rho Baseline left-parenthesis x Subscript n Baseline right-parenthesis . We obtain, after dividing the expression by alpha Subscript rho Baseline equals integral psi Subscript rho Superscript 2 Baseline , that

integral Underscript double-struck upper R Superscript n minus 1 Endscripts StartAbsoluteValue nabla eta left-parenthesis x prime right-parenthesis EndAbsoluteValue squared plus integral Underscript double-struck upper R Superscript n minus 1 Endscripts eta squared left-parenthesis x prime right-parenthesis integral Underscript double-struck upper R Endscripts StartFraction left-parenthesis psi prime Subscript rho right-parenthesis squared left-parenthesis x Subscript n Baseline right-parenthesis Over alpha Subscript rho Baseline EndFraction plus integral Underscript double-struck upper R Superscript n minus 1 Endscripts eta squared left-parenthesis x prime right-parenthesis integral Underscript double-struck upper R Endscripts upper F double-prime left-parenthesis u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis right-parenthesis StartFraction psi Subscript rho Superscript 2 Baseline left-parenthesis x Subscript n Baseline right-parenthesis Over alpha Subscript rho Baseline EndFraction

is nonnegative. Passing to the limit as rho right-arrow plus normal infinity , and using upper F element-of upper C squared and that u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis converges to ModifyingAbove u With bar left-parenthesis x prime right-parenthesis as x Subscript n Baseline right-arrow plus normal infinity uniformly in compact sets of double-struck upper R Superscript n minus 1 , we conclude (Equation3.10). This is the crucial point where we need upper F element-of upper C squared , and not only upper F element-of upper C Superscript 1 comma 1 .

Now, (Equation3.10) implies that the first eigenvalue lamda Subscript 1 comma upper R of negative normal upper Delta plus upper F double-prime left-parenthesis u overbar right-parenthesis in the ball upper B prime Subscript upper R Baseline equals StartSet x prime element-of double-struck upper R Superscript n minus 1 Baseline colon StartAbsoluteValue x Superscript prime Baseline EndAbsoluteValue less-than upper R EndSet is nonnegative for every upper R greater-than 1 . Let phi Subscript upper R Baseline greater-than 0 be the corresponding first eigenfunction in upper B prime Subscript upper R :

StartLayout Enlarged left-brace 1st Row 1st Column normal upper Delta phi Subscript upper R Baseline minus upper F double-prime left-parenthesis u overbar right-parenthesis phi Subscript upper R Baseline equals minus lamda Subscript 1 comma upper R Baseline phi Subscript upper R Baseline 2nd Column in upper B prime Subscript upper R comma 2nd Row 1st Column phi Subscript upper R Baseline equals 0 2nd Column on partial-differential upper B prime Subscript upper R comma EndLayout

normalized such that phi Subscript upper R Baseline left-parenthesis 0 right-parenthesis equals 1 . Note that lamda Subscript 1 comma upper R Baseline greater-than-or-equal-to 0 is decreasing in upper R and, hence, bounded. Therefore, the Harnack inequality gives that phi Subscript upper R are bounded, uniformly in upper R , on every compact set of double-struck upper R Superscript n minus 1 . By upper W Superscript 2 comma p estimates, it follows that a subsequence of phi Subscript upper R converges in upper W Subscript loc Superscript 2 comma p to a positive function phi element-of upper W Subscript loc Superscript 2 comma p Baseline left-parenthesis double-struck upper R Superscript n minus 1 Baseline right-parenthesis , for every p less-than normal infinity , satisfying normal upper Delta phi minus upper F double-prime left-parenthesis u overbar right-parenthesis phi less-than-or-equal-to 0 in double-struck upper R Superscript n minus 1 (since lamda Subscript 1 comma upper R Baseline greater-than-or-equal-to 0 for every upper R ).

Finally, assume that n equals 3 . For each i element-of StartSet 1 comma 2 EndSet , we consider the function

sigma Subscript i Baseline equals StartFraction partial-differential Subscript i Baseline u overbar Over phi EndFraction in double-struck upper R squared period

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

nabla dot left-parenthesis phi squared nabla sigma Subscript i Baseline right-parenthesis equals phi normal upper Delta partial-differential Subscript i Baseline u overbar minus partial-differential Subscript i Baseline u overbar normal upper Delta phi comma

and hence

StartLayout 1st Row 1st Column sigma Subscript i Baseline nabla dot left-parenthesis phi squared nabla sigma Subscript i Baseline right-parenthesis 2nd Column equals partial-differential Subscript i Baseline u overbar normal upper Delta partial-differential Subscript i Baseline u overbar minus left-parenthesis partial-differential Subscript i Baseline u overbar right-parenthesis squared left-parenthesis normal upper Delta phi slash phi right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis partial-differential Subscript i Baseline u overbar right-parenthesis squared upper F double-prime left-parenthesis u overbar right-parenthesis minus left-parenthesis partial-differential Subscript i Baseline u overbar right-parenthesis squared left-parenthesis normal upper Delta phi slash phi right-parenthesis 3rd Row 1st Column Blank 2nd Column greater-than-or-equal-to 0 comma EndLayout

by (Equation3.3).

Next, we apply the Liouville property of Proposition 2.1 to this inequality in double-struck upper R squared . Since phi sigma Subscript i Baseline equals partial-differential Subscript i Baseline u overbar is bounded and the dimension is two, condition (Equation2.2) holds. We obtain that sigma Subscript i is constant, that is,

StartLayout 1st Row with Label left-parenthesis 3.11 right-parenthesis EndLabel partial-differential Subscript i Baseline u overbar equals c Subscript i Baseline phi EndLayout

for some constant c Subscript i . If c 1 equals c 2 equals 0 , then u overbar is equal to a constant upper M . In this case, (Equation3.10) obviously implies that upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 .

If at least one c Subscript i is not zero, then u is constant along the direction left-parenthesis c 2 comma minus c 1 right-parenthesis , by (Equation3.11). Hence, taking b equals StartAbsoluteValue left-parenthesis c 1 comma c 2 right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline left-parenthesis c 1 comma c 2 right-parenthesis , we find that ModifyingAbove u With bar left-parenthesis x prime right-parenthesis equals h left-parenthesis b dot x prime right-parenthesis in double-struck upper R squared for some function h . Using this relation and (Equation3.11), we see that c Subscript i Baseline phi equals c Subscript i Baseline StartAbsoluteValue left-parenthesis c 1 comma c 2 right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline h prime left-parenthesis b dot x prime right-parenthesis , and hence h prime greater-than 0 in double-struck upper R .

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

Proof of Lemma 3.2.

(i) Multiplying the equation by h prime and integrating, we find that 2 upper F left-parenthesis h right-parenthesis minus left-parenthesis h prime right-parenthesis squared equals c is constant in double-struck upper R . Since h has finite limits as s right-arrow plus-or-minus normal infinity we obtain

StartLayout 1st Row with Label left-parenthesis 3.12 right-parenthesis EndLabel limit inf h prime left-parenthesis s right-parenthesis equals 0 comma EndLayout

whence c is equal to both 2 upper F left-parenthesis m 1 right-parenthesis and 2 upper F left-parenthesis m 2 right-parenthesis . Since 2 upper F left-parenthesis h right-parenthesis equals c plus left-parenthesis h prime right-parenthesis squared greater-than c and the image of h is left-parenthesis m 1 comma m 2 right-parenthesis , we infer (Equation3.6). The equalities upper F prime left-parenthesis m 1 right-parenthesis equals upper F prime left-parenthesis m 2 right-parenthesis equals 0 follow from the equation h double-prime minus upper F prime left-parenthesis h right-parenthesis equals 0 and from (Equation3.12), using the mean value theorem. Finally, the integral in (Equation3.7) is equal to integral Subscript negative normal infinity Superscript plus normal infinity Baseline left-parenthesis h prime right-parenthesis squared d s , which can be estimated by

left-parenthesis m 2 minus m 1 right-parenthesis sup h Superscript prime Baseline less-than-or-equal-to left-parenthesis m 2 minus m 1 right-parenthesis StartRoot 2 upper D EndRoot less-than plus normal infinity comma

where upper D equals sup Underscript t element-of left-parenthesis m 1 comma m 2 right-parenthesis Endscripts upper F left-parenthesis t right-parenthesis minus upper F left-parenthesis m 1 right-parenthesis .

(ii) Let m element-of left-parenthesis m 1 comma m 2 right-parenthesis and let phi colon left-parenthesis m 1 comma m 2 right-parenthesis right-arrow double-struck upper R be the function

phi left-parenthesis t right-parenthesis equals integral Subscript m Superscript t Baseline StartFraction 1 Over StartRoot 2 upper F left-parenthesis z right-parenthesis minus 2 upper F left-parenthesis m 1 right-parenthesis EndRoot EndFraction d z comma

well defined thanks to (Equation3.6). By (Equation3.5) and upper F left-parenthesis m 1 right-parenthesis equals upper F left-parenthesis m 2 right-parenthesis , we infer that the image of phi is the whole real line, and it is easy to check by integration that the unique increasing solution of h double-prime minus upper F prime left-parenthesis h right-parenthesis equals 0 in double-struck upper R satisfying h left-parenthesis 0 right-parenthesis equals m is the inverse function of phi .

Finally, we give the

Proof of Theorem 1.2.

Since partial-differential Subscript 3 Baseline u greater-than 0 , the proof of Theorem 1.1 shows that Theorem 1.2 will be established if we prove (Equation2.5) for every upper R greater-than 1 , i.e.,

integral Underscript upper B Subscript upper R Baseline Endscripts StartAbsoluteValue nabla u EndAbsoluteValue squared less-than-or-equal-to upper C upper R squared

for some constant upper C independent of upper R .

Let

m equals inf Underscript double-struck upper R cubed Endscripts u and upper M equals sup Underscript double-struck upper R cubed Endscripts u comma

and consider the functions

ModifyingBelow u With bar left-parenthesis x Superscript prime Baseline right-parenthesis equals limit Underscript x 3 right-arrow negative normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis and ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis equals limit Underscript x 3 right-arrow plus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis period

Note that u underbar less-than u overbar in double-struck upper R squared , m equals inf Underscript double-struck upper R squared Endscripts u underbar , and upper M equals sup Underscript double-struck upper R squared Endscripts u overbar . We apply Lemma 3.1. If u overbar is constant, then necessarily u overbar identical-to upper M , upper F prime left-parenthesis upper M right-parenthesis equals 0 by (Equation3.2), and upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 as stated in Lemma 3.1. In case (b) of Lemma 3.1, we see that the function h satisfies (Equation3.4). Hence, we can apply Lemma 3.2(i) with m 1 equals inf u overbar less-than m 2 equals upper M equals sup u overbar , and we obtain again that upper F prime left-parenthesis upper M right-parenthesis equals 0 and, using (Equation3.6), that upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 . Hence, we have proved that we always have

upper F prime left-parenthesis upper M right-parenthesis equals 0 and upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 period

In an analogous way, arguing with u underbar (or simply replacing u left-parenthesis x prime comma x 3 right-parenthesis by minus u left-parenthesis x prime comma minus x 3 right-parenthesis , and upper F left-parenthesis v right-parenthesis by upper F left-parenthesis negative v right-parenthesis ), we see that

upper F prime left-parenthesis m right-parenthesis equals 0 and upper F double-prime left-parenthesis m right-parenthesis greater-than-or-equal-to 0 period

By the hypothesis made on upper F , it follows that upper F greater-than-or-equal-to min left-brace upper F left-parenthesis m right-parenthesis comma upper F left-parenthesis upper M right-parenthesis right-brace in left-parenthesis m comma upper M right-parenthesis . Suppose first that min left-brace upper F left-parenthesis m right-parenthesis comma upper F left-parenthesis upper M right-parenthesis right-brace equals upper F left-parenthesis upper M right-parenthesis (the other case reduces to this one, again by the same change of u and upper F as before). Then, upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 in double-struck upper R cubed . Hence, the theorem will be proved if we show that

integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u EndAbsoluteValue squared plus upper F left-parenthesis u right-parenthesis minus upper F left-parenthesis upper M right-parenthesis EndSet d x less-than-or-equal-to upper C upper R squared

for each upper R greater-than 1 .

To establish this, we proceed as in the proof of Theorem 1.3. That is, we consider the functions u Superscript t Baseline left-parenthesis x right-parenthesis equals u left-parenthesis x prime comma x Subscript n Baseline plus t right-parenthesis defined for x equals left-parenthesis x prime comma x Subscript n Baseline right-parenthesis element-of double-struck upper R Superscript n and t element-of double-struck upper R , and the energy of u Superscript t in the ball upper B Subscript upper R Baseline equals upper B Subscript upper R Baseline left-parenthesis 0 right-parenthesis , defined now by

upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis equals integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla u Superscript t Baseline EndAbsoluteValue squared plus upper F left-parenthesis u Superscript t Baseline right-parenthesis minus upper F left-parenthesis upper M right-parenthesis EndSet d x period

We need to show that upper E Subscript upper R Baseline left-parenthesis u right-parenthesis equals upper E Subscript upper R Baseline left-parenthesis u Superscript 0 Baseline right-parenthesis less-than-or-equal-to upper C upper R squared . The computations leading to inequalities (Equation2.7) and (Equation2.8) are still valid here – since the extra hypothesis of Theorem 1.1, limit Underscript x 3 right-arrow plus-or-minus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis equals plus-or-minus 1 , was only used in the proof of Theorem 1.3 to establish (Equation2.6), i.e., limit Underscript t right-arrow plus normal infinity Endscripts upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis equals 0 . Using (Equation2.8) we see that upper E Subscript upper R Baseline left-parenthesis u right-parenthesis less-than-or-equal-to upper C upper R squared will hold if we verify

limit sup upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis less-than-or-equal-to upper C upper R squared period

This inequality is an easy consequence of Lemmas 3.1 and 3.2(i). Indeed, using standard elliptic estimates and that u Superscript t Baseline left-parenthesis x right-parenthesis increases in upper B Subscript upper R to ModifyingAbove u With bar left-parenthesis x prime right-parenthesis as t right-arrow plus normal infinity , we have

StartLayout 1st Row 1st Column limit Underscript t right-arrow plus normal infinity Endscripts upper E Subscript upper R Baseline left-parenthesis u Superscript t Baseline right-parenthesis 2nd Column equals integral Underscript upper B Subscript upper R Baseline Endscripts StartSet one-half StartAbsoluteValue nabla ModifyingAbove u With bar left-parenthesis x prime right-parenthesis EndAbsoluteValue squared plus upper F left-parenthesis ModifyingAbove u With bar left-parenthesis x prime right-parenthesis right-parenthesis minus upper F left-parenthesis upper M right-parenthesis EndSet d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C upper R integral Underscript upper B Subscript upper R Superscript prime Baseline Endscripts StartSet one-half StartAbsoluteValue nabla ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis EndAbsoluteValue squared plus upper F left-parenthesis ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis right-parenthesis minus upper F left-parenthesis upper M right-parenthesis EndSet d x Superscript prime Baseline comma EndLayout

where upper B prime Subscript upper R Baseline equals StartSet StartAbsoluteValue x prime EndAbsoluteValue less-than upper R EndSet subset-of double-struck upper R squared . But the last integral

integral Underscript upper B prime Subscript upper R Endscripts StartSet left-parenthesis 1 slash 2 right-parenthesis StartAbsoluteValue nabla ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis EndAbsoluteValue squared plus upper F left-parenthesis ModifyingAbove u With bar left-parenthesis x Superscript prime Baseline right-parenthesis right-parenthesis minus upper F left-parenthesis upper M right-parenthesis EndSet d x Superscript prime Baseline comma

which is computed in a two-dimensional ball, is bounded by upper C upper R , since u overbar 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 upper F element-of upper C squared . That is, the additional hypothesis Equation1.5 on upper F is not needed in this theorem.

Mathematical Fragments

Equation (1.1)
StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel limit Underscript x Subscript n Baseline right-arrow plus-or-minus normal infinity Endscripts u left-parenthesis x prime comma x Subscript n Baseline right-parenthesis equals plus-or-minus 1 for all x prime element-of double-struck upper R Superscript n minus 1 Baseline period EndLayout
Theorem 1.1

Let u be a bounded solution of

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R cubed EndLayout

satisfying

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel partial-differential Subscript 3 Baseline u greater-than 0 in double-struck upper R cubed and limit Underscript x 3 right-arrow plus-or-minus normal infinity Endscripts u left-parenthesis x prime comma x 3 right-parenthesis equals plus-or-minus 1 for all x prime element-of double-struck upper R squared period EndLayout

Assume that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and that

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel upper F greater-than-or-equal-to min left-brace upper F left-parenthesis negative 1 right-parenthesis comma upper F left-parenthesis 1 right-parenthesis right-brace in left-parenthesis negative 1 comma 1 right-parenthesis period EndLayout

Then the level sets of u are planes, i.e., there exist a element-of double-struck upper R cubed and g element-of upper C squared left-parenthesis double-struck upper R right-parenthesis such that

u left-parenthesis x right-parenthesis equals g left-parenthesis a dot x right-parenthesis for all x element-of double-struck upper R cubed period
Theorem 1.2

Let u be a bounded solution of

normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R cubed

satisfying

partial-differential Subscript 3 Baseline u greater-than 0 in double-struck upper R cubed period

Assume that upper F element-of upper C squared left-parenthesis double-struck upper R right-parenthesis and that

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel upper F greater-than-or-equal-to min left-brace upper F left-parenthesis m right-parenthesis comma upper F left-parenthesis upper M right-parenthesis right-brace in left-parenthesis m comma upper M right-parenthesis EndLayout

for each pair of real numbers m less-than upper M satisfying upper F prime left-parenthesis m right-parenthesis equals upper F prime left-parenthesis upper M right-parenthesis equals 0 , upper F double-prime left-parenthesis m right-parenthesis greater-than-or-equal-to 0 and upper F double-prime left-parenthesis upper M right-parenthesis greater-than-or-equal-to 0 . Then the level sets of u are planes, i.e., there exist a element-of double-struck upper R cubed and g element-of upper C squared left-parenthesis double-struck upper R right-parenthesis such that

u left-parenthesis x right-parenthesis equals g left-parenthesis a dot x right-parenthesis for all x element-of double-struck upper R cubed period
Equation (1.6)
StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel one-half StartAbsoluteValue nabla u EndAbsoluteValue squared less-than-or-equal-to upper F left-parenthesis u right-parenthesis in double-struck upper R Superscript n Baseline period EndLayout
Theorem 1.3

Let u be a bounded solution of

normal upper Delta u minus upper F prime left-parenthesis u right-parenthesis equals 0 in double-struck upper R Superscript n Baseline comma

where upper F is an arbitrary