Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
aa r X i v : . [ m a t h . A P ] A p r ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAREQUATIONS INVOLVING THE HALF-LAPLACIAN
XAVIER CABR´E AND ELEONORA CINTI
Abstract.
We establish sharp energy estimates for some solutions, such as global mini-mizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equa-tion ( − ∆) / u = f ( u ) in R n . Our energy estimates hold for every nonlinearity f and aresharp since they are optimal for one-dimensional solutions, that is, for solutions dependingonly on one Euclidian variable.As a consequence, in dimension n = 3, we deduce the one-dimensional symmetry ofevery global minimizer and of every monotone solution. This result is the analog of aconjecture of De Giorgi on one-dimensional symmetry for the classical equation − ∆ u = f ( u ) in R n . Introduction and results
In this paper we establish sharp energy estimates for some solutions of the fractionalnonlinear equation ( − ∆) / u = f ( u ) in R n , (1.1)where f : R → R is a C ,β function with 0 < β <
1. When f is a balanced bistablenonlinearity, for instance when f ( u ) = u − u , we call equation (1.1) of Allen-Cahn typeby the analogy with the corresponding equation involving the Laplacian instead of thehalf-Laplacian, − ∆ u = u − u in R n . (1.2)In 1978 De Giorgi conjectured that the level sets of every bounded solution of (1.2)which is monotone in one direction, must be hyperplanes, at least if n ≤
8. That is, suchsolutions depend only on one Euclidian variable. The conjecture has been proven to betrue in dimension n = 2 by Ghoussoub and Gui [12] and in dimension n = 3 by Ambrosioand the first author [2]. For 4 ≤ n ≤
8, if ∂ x n u >
0, and assuming the additional conditionlim x n →±∞ u ( x ′ , x n ) = ± x ′ ∈ R n − , it has been established by Savin [15]. Recently a counterexample to the conjecture for n ≥ Mathematics Subject Classification.
Primary: 35J60, 35R10; Secondary: 35B05, 35J20, 60J75.
Key words and phrases.
Half-Laplacian, energy estimates, symmetry properties, entire solutions.Both authors were supported by grants MTM2008-06349-C03-01 (Spain) and 2009SGR-345 (Catalunya).The second author was partially supported by University of Bologna (Italy), funds for selected researchtopics.
In this paper (see Theorem 1.4 below), we establish the one-dimensional symmetry ofbounded monotone solutions of (1.1) in dimension n = 3, that is, the analog of the conjec-ture of De Giorgi for the half-Laplacian in dimension 3. We recall that one-dimensional (or1-D) symmetry for bounded stable solutions of (1.1) in dimension n = 2 has been provenby the first author and Sol`a-Morales [6]. The same result in dimension n = 2 for the otherfractional powers of the Laplacian, i.e., for the equation( − ∆) s u = f ( u ) in R , with 0 < s < , has been established by the first author and Sire [4, 5] and by Sire and Valdinoci [16].A crucial ingredient in our proof of 1-D symmetry in R is a sharp energy estimate forglobal minimizers and for monotone solutions, that we state in Theorems 1.2 and 1.3 below.It is interesting to note that our method to prove the energy estimate also applies to thecase of saddle-shaped solutions in R m . These solutions are not global minimizers in general(this is indeed the case in dimensions 2 m ≤ R m . We treatthese solutions and their corresponding energy estimate at the end of this introduction.To study the nonlocal problem (1.1) we realize it as a local problem in R n +1+ with anonlinear Neumann condition on ∂ R n +1+ = R n . More precisely, if u = u ( x ) is a functiondefined on R n , we consider its harmonic extension v = v ( x, λ ) in R n +1+ = R n × (0 , + ∞ ). Itis well known (see [6, 9]) that u is a solution of (1.1) if and only if v satisfies ( ∆ v = 0 in R n +1+ , − ∂ λ v = f ( v ) on R n = ∂ R n +1+ . (1.3)Problem (1.3) allows to introduce the notions of energy and global minimality for asolution u of problem (1.1). Consider the cylinder C R = B R × (0 , R ) ⊂ R n +1+ , where B R is the ball of radius R centered at 0 in R n . We consider the energy functional E C R ( v ) = Z C R |∇ v | dxdλ + Z B R G ( v ) dx, (1.4)whose Euler-Lagrange equation is problem (1.3). The potential G , defined up to an additiveconstant, is given by G ( v ) = Z v f ( t ) dt. Using the energy functional (1.4), we introduce the notions of global minimizer and of layer solution of (1.1). We call layer solutions of (1.1) those bounded solutions that aremonotone increasing, say from − x -variables. After rotation, we cansuppose that the direction of monotonicity is the x n -direction, as in point c) of the followingdefinition. NERGY ESTIMATES FOR THE HALF-LAPLACIAN 3
Definition 1.1. a) We say that a bounded C ( R n +1+ ) function v is a global minimizer of(1.3) if, for all R > E C R ( v ) ≤ E C R ( w )for every C ( R n +1+ ) function w such that v ≡ w in R n +1+ \ C R .b) We say that a bounded C function u in R n is a global minimizer of (1.1) if itsharmonic extension v is a global minimizer of (1.3).c) We say that a bounded function u is a layer solution of (1.1) if ∂ x n u > R n andlim x n →±∞ u ( x ′ , x n ) = ± x ′ ∈ R n − . (1.5)Note that the functions w in point a) of Definition 1.1 need to agree with the solution v on the lateral boundary and on the top of the cylinder C R , but not on its bottom. Since itwill be useful in the sequel, we denote the lateral and top parts of the boundary of C R by ∂ + C R = ∂C R ∩ { λ > } . In some references, global minimizers are called “local minimizers”, where local standsfor the fact that the energy is computed in bounded domains.We recall that every layer solution is a global minimizer (see Theorem 1.4 in [6]).Our main result is the following energy estimate for global minimizers of problem (1.1).Given a bounded function u defined on R n , set c u = min { G ( s ) : inf R n u ≤ s ≤ sup R n u } . (1.6) Theorem 1.2.
Let f be any C ,β nonlinearity, with β ∈ (0 , , and u ∈ L ∞ ( R n ) be a globalminimizer of (1.1). Let v be the harmonic extension of u in R n +1+ .Then, for all R > , Z C R |∇ v | dxdλ + Z B R { G ( u ) − c u } dx ≤ CR n − log R, (1.7) where c u is defined by (1.6) and C is a constant depending only on n , || f || C ([inf u, sup u ]) , and || u || L ∞ ( R n ) . In particular, we have that Z C R |∇ v | dxdλ ≤ CR n − log R. (1.8)As a consequence, (1.7) and (1.8) also hold for layer solutions. We stress that this energyestimate is sharp because it is optimal for 1-D solutions, in the sense that for some explicit1-D solutions the energy is also bounded below by cR n − log R , for some constant c > R n (see Remark 2.2 below and section 2.1 of [6]).In dimensions n = 1 and n = 2 estimate (1.7) was established by the first author andSol`a-Morales in [6].In dimension n = 3, the energy estimate (1.7) holds also for monotone solutions whichdo not satisfy the limit assumption (1.5). These solutions are minimizers in some sense tobe explained later, but, in case that they exist, they are not known to be global minimizersas defined before. XAVIER CABR´E AND ELEONORA CINTI
Theorem 1.3.
Let n = 3 , f be any C ,β nonlinearity with β ∈ (0 , , and u be a boundedsolution of (1.1) such that ∂ e u > in R for some direction e ∈ R , | e | = 1 . Let v be itsharmonic extension in R .Then, for all R > , Z C R |∇ v | dxdλ + Z B R { G ( u ) − c u } dx ≤ CR log R, (1.9) where c u is defined by (1.6) and C is a constant depending only on || f || C ([inf u, sup u ]) and || u || L ∞ ( R ) . In dimension n = 3, Theorems 1.2 and 1.3 lead to the 1-D symmetry of global minimizersand of monotone solutions to problem (1.1). Theorem 1.4.
Let n = 3 and f be any C ,β nonlinearity with β ∈ (0 , . Let u be eithera bounded global minimizer of (1.1) , or a bounded solution of (1.1) monotone in somedirection e ∈ R , | e | = 1 .Then, u depends only on one variable, i.e., there exists a ∈ R and g : R → R , such that u ( x ) = g ( a · x ) for all x ∈ R . Equivalently, the level sets of u are planes. To prove 1-D symmetry, we use a standard Liouville type argument which requires anappropriate estimate for the kinetic energy. By a result of Moschini [14] (see Proposition6.1 in section 6 below), our energy estimate in R , Z C R |∇ v | dxdλ ≤ CR log R, allows to use such Liouville type result and deduce 1-D symmetry in R for global mini-mizers and for solutions monotone in one direction. Remark . As a consequence of Theorem 1.4, we obtain that for all
R > Z B R G ( v ( x, dx ≤ CR n − if 1 ≤ n ≤ , (1.10)if v is a bounded global minimizer or a bounded monotone solution of (1.3). This wasproven in [6] for n = 1 and n = 2. For n = 3, (1.10) follows from the n = 1 case after usingTheorem 1.4. In dimension n ≥ CR n − (instead of CR n − log R ) as in (1.10).In our next paper, using similar techniques, we establish sharp energy estimates for theother fractional powers of the Laplacian. More precisely, we prove that if u is a boundedglobal minimizer of ( − ∆) s u = f ( u ) in R n , with 0 < s < , (1.11)then the following energy estimate holds: E s,C R ( u ) ≤ CR n − s for 0 < s < , E s,C R ( u ) ≤ CR n − for 12 < s < . NERGY ESTIMATES FOR THE HALF-LAPLACIAN 5
Here the energy functional is defined using a local formulation in R n +1+ of problem (1.11),found by Caffarelli and Silvestre in [9]. If 1 / < s < E s,C R ( u ) ≤ CR n − ; in this casewe can deduce 1-D symmetry for global minimizers and monotone solutions in dimension n = 3.Back to the case s = 1 /
2, we have two different proofs of the energy estimate CR n − log R .The first one is very simple but applies only to Allen-Cahn type nonlinearities (suchas f ( u ) = u − u ) and to monotone solutions satisfying the limit assumption (1.5) or themore general (2.2) below. We present this very simple proof in section 2. It was found byAmbrosio and the first author [2] to prove the optimal energy estimate for − ∆ u = u − u in R n .Our second proof applies in more general situations and will lead to Theorems 1.2 and1.3. It is based on controlling the H (Ω)-norm of a function by its fractional Sobolev norm H / ( ∂ Ω) on the boundary.Let us recall the definition of the H / ( A ) norm, where A is either a Lipschitz open setof R n , or A = ∂ Ω and Ω is a Lipschitz open set of R n +1 . It is given by || w || H / ( A ) = || w || L ( A ) + Z A Z A | w ( z ) − w ( z ) | | z − z | n +1 dσ z dσ z . In our proof we will have A = ∂C R ⊂ R n +1 , the boundary of the cylinder Ω = C R .In the proof of Theorem 1.2 a crucial point will be the following well known result. If w is a function in H / ( ∂ Ω), where Ω is a bounded subset of R n +1 with Lipschitz boundary,then the harmonic extension w of w in Ω satisfies: Z Ω |∇ w | ≤ C (Ω) || w || H / ( ∂ Ω) . (1.12)For the sake of completeness (and since the proof will be important in our next paper [3]),we will recall a proof of this result in section 3 (see Proposition 3.1).To prove the sharp energy estimate for a global minimizer v in R n +1+ , we will bound itsenergy in the cylinder C R = B R × (0 , R ), using (1.12), by that of the harmonic extension w in C R of a well chosen function w defined on ∂C R . This function w must agree with v on ∂ + C R (the lateral and top boundaries of C R ), while it will be identically 1 on the portion B R − × { } of the bottom boundary. In this way, it will not pay potential energy in thisportion of the bottom boundary.By (1.12), we will need to control || w || H / ( ∂C R ) . After rescaling ∂C R to ∂C , we willcontrol the H / -norm of w using the following key result. It gives a bound on the H / -norm of functions on A which satisfy a certain gradient pointwise bound related with thedistance to a Lipschitz subset Γ of A . We will apply it in the sets A = ∂C and Γ = ∂B × { λ = 0 } , with a small parameter ε = 1 /R . Examples in which the following theorem applies are,among many others, A = B ⊂ R n the unit ball and Γ = B ∩ { x n = 0 } , and also A = B ⊂ R n and Γ = ∂B r for some r ∈ (0 , XAVIER CABR´E AND ELEONORA CINTI
Theorem 1.6.
Let A be either a bounded Lipschitz domain in R n or A = ∂ Ω , where Ω isa bounded open set of R n +1 with Lipschitz boundary. Let M ⊂ A be an open set (relativeto A ) with Lipschitz boundary (relative to A ) Γ ⊂ A . Let ε ∈ (0 , / .Let w : A → R be a Lipschitz function such that, for almost every x ∈ A , | w ( x ) | ≤ c (1.13) and | Dw ( x ) | ≤ c min (cid:26) ε , x, Γ) (cid:27) , (1.14) where D are all tangential derivatives to A , dist( x, Γ) is the distance from the point x tothe set Γ (either in R n or in R n +1 ), and c is a positive constant.Then, || w || H / ( A ) = || w || L ( A ) + Z A Z A | w ( z ) − w ( z ) | | z − z | n +1 dσ z dσ z ≤ c C | log ε | , (1.15) where C is a positive constant depending only on A and Γ . As we said, we will use this result with A = ∂C and Γ = ∂B × { λ = 0 } . Thus, in thiscase the constant C in (1.15) only depends on the dimension n . The gradient estimate(1.14), after rescaling ∂C R to ∂C and taking ε = 1 /R , will follow from the bound |∇ v ( x, λ ) | ≤ C λ for all x ∈ R n and λ ≥ , (1.16)satisfied by every bounded solution v of (1.3). Here the constant C depends only on n , || f || C , and || v || L ∞ ( R n +1+ ) . For λ ≥
1, (1.16) follows immediately from the fact that v is bounded and harmonic in B λ ( x, λ ) ⊂ R n +1+ . For λ ≤
1, estimate (1.16) for boundedsolutions of the nonlinear Neumann problem (1.3) was proven in Lemma 2.3 of [6].Our method to prove sharp energy estimates also applies to solutions which are minimiz-ers under perturbations vanishing on a suitable subset of R n , even if they are not in generalglobal minimizers as defined before. An important example of this are some saddle-shapedsolutions (or saddle solutions for short) of( − ∆) / u = f ( u ) in R m . The existence and qualitative properties of these solutions have been studied by the secondauthor in [10]. For equations of Allen-Cahn type involving the Laplacian, − ∆ u = f ( u ),saddle solutions have been studied in [7, 8].Saddle solutions are even with respect to the coordinate axes and odd with respect tothe Simons cone, which is defined as follows. For n = 2 m the Simons cone C is given by C = { x ∈ R m : x + ... + x m = x m +1 + ... + x m } . We define two new variables s = q x + · · · + x m and t = q x m +1 + · · · + x m , for which the Simons cone becomes C = { s = t } . NERGY ESTIMATES FOR THE HALF-LAPLACIAN 7
The existence of saddle solutions of (1.1) has been proven in [10] under the followinghypotheses on f : f is odd; (1.17) G ≥ G ( ±
1) in R , and G > − , f ′ is decreasing in (0 , . (1.19)Note that, if (1.17) and (1.18) hold, then f (0) = f ( ±
1) = 0 . Conversely, if f is odd in R , positive with f ′ decreasing in (0 ,
1) and negative in (1 , ∞ ) then f satisfies (1.17), (1.18)and (1.19). Hence, the nonlinearities f that we consider are of “balanced bistable type”,while the potentials G are of “double well type”. Our three assumptions (1.17), (1.18),(1.19) are satisfied by the scalar Allen-Cahn type equation( − ∆) / u = u − u . In this case we have that G ( u ) = (1 / − u ) . The three hypotheses also hold for theequation ( − ∆) / u = sin( πu ), for which G ( u ) = (1 /π )(1 + cos( πu )).The following result states the existence of at least one saddle solution for which oursharp energy estimate holds. Theorem 1.7.
Let f be a C ,β function for some < β < , satisfying (1.17) , (1.18) , and (1.19) . Then, there exists a saddle solution u of ( − ∆) / u = f ( u ) in R m , i.e., a boundedsolution u such that(a) u depends only on the variables s and t . We write u = u ( s, t ) ;(b) u > for s > t ;(c) u ( s, t ) = − u ( t, s ) .Moreover, | u | < in R m and for every R > , E C R ( v ) ≤ CR m − log R, where v is the harmonic extension of u in R m +1+ and C is a constant depending only on m and || f || C ([ − , . Observe that the saddle solution of the theorem satisfies the same optimal energy es-timate as global minimizers do, that is, CR n − log R = CR m − log R , even that in lowdimensions it is known [10] that saddle solutions are not global minimizers. Indeed saddlesolutions are not stable in dimension 2 (by a result of the first author and Sol`a-Morales [6])and in dimensions 4 and 6 (by a result of the second author [10]). As we will explain in thelast section, some saddle solutions are minimizers under perturbations vanishing on theSimons cone, and this will be enough to prove that they satisfy the sharp energy estimate.The paper is organized as follows: • In section 2 we prove the energy estimate for layer solutions of Allen-Cahn typeequations, using a simple argument found by Ambrosio and the first author [2]. • In section 3 we give the proof of the extension theorem and of the key Theorem1.6. • In section 4 we prove energy estimate (1.7) for global minimizers and for everynonlinearity f , that is, Theorem 1.2. XAVIER CABR´E AND ELEONORA CINTI • In section 5 we establish energy estimates for monotone solutions in R , Theorem1.3. • In section 6 we prove the 1-D symmetry result, that is, Theorem 1.4. • In section 7 we prove the energy estimate for saddle solutions, Theorem 1.7.2.
Energy estimate for monotone solutions of Allen-Cahn type equations
In this section we consider potentials G which satisfy hypothesis (1.18), that is, G ≥ G ( ±
1) in R and G > − , E C R defined by E C R ( v ) = Z C R |∇ v | dxdλ + Z B R G ( v ) dx. In general, it can be defined up to an additive constant c in the potential G ( v ) − c , but inthis case, by the assumption (1.18) on G , we take c = 0. Theorem 2.1.
Let f be a C ,β function, for some < β < , satisfying (1.18) , where G ′ = − f . Let u be a bounded solution of problem (1.1) in R n , with | u | < in R n , and let v be the harmonic extension of u in R n +1+ . Assume that u x n > in R n (2.1) and lim x n → + ∞ u ( x ′ , x n ) = 1 for all x ′ ∈ R n − . (2.2) Then, for every
R > , Z C R |∇ v | dxdλ ≤ E C R ( v ) ≤ CR n − log R, for some constant C depending only on n and || f || C ([ − , .Remark . This energy estimate in dimension n = 1 has been proven by the first authorand Sol`a-Morales [6], using the gradient bound |∇ v ( x, λ ) | ≤ C | ( x, λ ) | for all x ∈ R and λ ≥ , (2.3)(see estimate (1.14) of [6]). Indeed, we next see that (2.3) leads to Z C R |∇ v | dxdλ ≤ C log R and also Z + ∞ dλ Z B R dx |∇ v | ≤ C log R. (2.4)That is, for n = 1, the energy estimate holds not only in the cylinder C R , but also inthe infinite cylinder B R × (0 , + ∞ ). Let us mention that for the explicit layer solutions insection 2.1 of [6], the upper bound C (1 + | ( x, λ ) | ) − for |∇ v | is also a lower bound for |∇ v | ,modulo a smaller multiplicative constant. As a consequence, the following computation NERGY ESTIMATES FOR THE HALF-LAPLACIAN 9 shows that the two previous upper bounds log R are also lower bounds for the Dirichletenergy after multiplying log R by a smaller constant.Estimate (2.4) holds, indeed: Z + ∞ dλ Z R − R dx |∇ v | ≤ C Z + ∞ dλ Z R − R dx
11 + x + λ ≤ C Z R − R dx Z + ∞ dλ x ) ·
11 + (cid:0) λ x (cid:1) ≤ C Z R − R (cid:20)
11 + x arctan λ x (cid:21) λ =+ ∞ λ =0 dx ≤ C Z R − R π x dx ≤ C log R. In higher dimensions, an analog of (2.3) is not available and therefore we need anothermethod to prove Theorem 2.1.
Proof of Theorem 2.1.
We follow an argument found by Ambrosio and the first author [2]to prove the energy estimate for layer solutions of the analog problem − ∆ u = f ( u ) in R n .It is based on sliding the function v , which is the harmonic extension of the solution u , inthe direction x n .Consider the function v t ( x, λ ) := v ( x ′ , x n + t, λ )defined for ( x, λ ) = ( x ′ , x n , λ ) ∈ R n × [0 , + ∞ ), where t ∈ R . For each t we have ( ∆ v t = 0 in R n +1+ , − ∂ λ v t = f ( v t ) on R n = ∂ R n +1+ . (2.5)Moreover, as stated in (1.16), the following bounds hold: | v t | ≤ |∇ v t | ≤ C λ . (2.6)Throughout the proof, C will denote different positive constants which depending only on n and || f || C ([ − , .A simple compactness argument implies thatlim t → + ∞ {| v t − | + |∇ v t |} = 0 (2.7)uniformly in compact sets of R n +1+ . Indeed, arguing by contradiction, assume that thereexist R > ε >
0, and a sequence t m → ∞ such that || v t m − || L ∞ ( C R ) + ||∇ v t m || L ∞ ( C R ) ≥ ε (2.8)for every m , where C R = B R × (0 , R ). Since v t m are all solutions of (1.3) in all the halfspace,the regularity results in [6] give C ( R n +1+ ) estimates for v t m uniform in m . Thus, thereexists a subsequence that converges in C ( R n +1+ ) to a bounded harmonic function v ∞ . By hypothesis (2.2), v ∞ ≡ ∂ R n +1+ , and thus by the maximum principle, v ∞ ≡ R n +1+ . This contradicts (2.8), by C convergence in compact sets of v t m towards v ∞ ≡ v t ( x, λ ) with respect to t by ∂ t v t ( x, λ ), we have ∂ t v t ( x, λ ) = v x n ( x ′ , x n + t, λ ) > x ∈ R n , λ ≥ . Note that v x n >
0, since v x n is the harmonic extension of the bounded function u x n > v t in the cylinder C R = B R × (0 , R ), E C R ( v t ) = Z C R |∇ v t | dxdλ + Z B R G ( v t ) dx. Note that, by (2.7), we have lim t → + ∞ E C R ( v t ) = 0 . (2.9)Next, we bound the derivative of E C R ( v t ) with respect to t . We use that v t is a solutionof problem (1.3), the bound (2.6) for | v t | and |∇ v t | , and the crucial fact that ∂ t v t >
0. Let ν denote the exterior normal to the lateral boundary ∂B R × (0 , R ) of the cylinder C R . Wehave ∂ t E C R ( v t ) = Z R dλ Z B R dx ∇ v t · ∇ ( ∂ t v t ) + Z B R dx G ′ ( v t ) ∂ t v t = Z R dλ Z ∂B R dσ ∂v t ∂ν ∂ t v t + Z B R ×{ λ = R } dx ∂v t ∂λ ∂ t v t ≥ − C Z R dλ λ Z ∂B R dσ ∂ t v t − CR Z B R ×{ λ = R } dx ∂ t v t . Hence, for every
T >
0, we have E C R ( v ) = E C R ( v T ) − Z T dt ∂ t E C R ( v t ) ≤ E C R ( v T ) + C Z T dt Z R dλ λ Z ∂B R dσ ∂ t v t + CR Z T dt Z B R ×{ λ = R } dx ∂ t v t = E C R ( v T ) + C Z R dλ λ Z ∂B R dσ Z T dt ∂ t v t + CR Z B R ×{ λ = R } dx Z T dt ∂ t v t = E C R ( v T ) + C Z R dλ λ Z ∂B R dσ ( v T − v ) + CR Z B R ×{ λ = R } dx ( v T − v ) ≤ E C R ( v T ) + CR n − log R + CR n − . Letting T → + ∞ and using (2.9), we obtain the desired estimate. (cid:3) H / estimate In this section we recall some definitions and properties about the spaces H / ( R n ) and H / ( ∂ Ω), where Ω is a bounded subset of R n +1 with Lipschitz boundary ∂ Ω (see [13]).
NERGY ESTIMATES FOR THE HALF-LAPLACIAN 11 H / ( R n ) is the space of functions u ∈ L ( R n ) such that Z R n Z R n | u ( x ) − u ( x ) | | x − x | n +1 dxdx < + ∞ , equipped with the norm || u || H / ( R n ) = (cid:18) || u || L ( R n ) + Z R n Z R n | u ( x ) − u ( x ) | | x − x | n +1 dxdx (cid:19) . Let now Ω be a bounded subset of R n +1 with Lipschitz boundary ∂ Ω. To define H / ( ∂ Ω),consider an atlas { ( O j , ϕ j ); j = 1 , ..., m } where { O j } is a family of open bounded setsin R n +1 such that { O j ∩ ∂ Ω; j = 1 , ..., m } cover ∂ Ω. The functions ϕ j are bilipschitzdiffeomorphisms such that • ϕ j : O j → U := { ( y, µ ) ∈ R n +1 : | y | < , − < µ < } , • ϕ j : O j ∩ Ω → U + := { ( y, µ ) ∈ R n +1 : | y | < , < µ < } , • ϕ j : O j ∩ ∂ Ω → { ( y, µ ) ∈ R n +1 : | y | < , µ = 0 } .Let { α j } be a partition of unity on ∂ Ω such that 0 ≤ α j ∈ C ∞ c ( O j ) , P mj =1 α j = 1 in ∂ Ω.If u is a function on ∂ Ω decompose u = P mj =1 uα j and define the function( uα j ) ◦ ϕ − j ( y,
0) := ( uα j )( ϕ − j ( y, , for every ( y, ∈ U ∩ { µ = 0 } . Since α j has compact support in O j ∩ ∂ Ω, the function ( uα j ) ◦ ϕ − j ( · ,
0) has compactsupport in U ∩ { µ = 0 } and therefore we may consider (( uα j ) ◦ ϕ − j )( · ,
0) to be defined in R n extending it by zero out of U ∩ { µ = 0 } . Now we define H / ( ∂ Ω) := { u : ( uα j ) ◦ ϕ − j ( · , ∈ H / ( R n ) , j = 1 , ..., m } equipped with the norm m X j =1 || ( uα j ) ◦ ϕ − j ( · , || H / ( R n ) ! . All these norms are independent of the choice of the system of local maps { O j , ϕ j } andof the partition of unity { α j } , and are all equivalent to || u || H / ( ∂ Ω) := (cid:18) || u || L ( ∂ Ω) + Z ∂ Ω Z ∂ Ω | u ( z ) − u ( z ) | | z − z | n +1 dσ z dσ z (cid:19) . We recall now the classical extension result that we will use in the proof of Theorem 1.2.
Proposition 3.1.
Let
Ω = R n +1+ or Ω be a bounded subset of R n +1 with Lipschitz boundary ∂ Ω , and let w belong to H / ( ∂ Ω) .Then, there exists a Lipschitz extension e w of w in Ω such that Z Ω |∇ e w | ≤ C || w || H / ( ∂ Ω) , (3.1) where C is a constant depending only on Ω . For the sake of completeness (and since the proof will be important in our next paper [3])we give the proof of this proposition.
Proof of Proposition 3.1. Case 1 . Ω = R n +1+ . Let ζ be a function belonging to H / ( R n ).We prove that there exists a Lipschitz extension e ζ of ζ in R n +1+ such that Z R n +1+ |∇ e ζ | dxdλ ≤ C Z R n Z R n | ζ ( x ) − ζ ( x ) | | x − x | n +1 dxdx. (3.2)Let K ( x ) be a nonnegative C ∞ function defined on R n with compact support in B andsuch that R R n K ( x ) dx = 1. Define e K ( x, λ ) on R n +1+ by e K ( x, λ ) := 1 λ n K (cid:16) xλ (cid:17) . Then, since Z R n e K ( x, λ ) dx = 1 for all λ > , (3.3)we obtain, differentiating with respect to x i and λ , Z R n ∂ x i e K ( x, λ ) dx = 0 and Z R n ∂ λ e K ( x, λ ) dx = 0 for all λ > . (3.4)In addition, for a constant C depending only on n , we have |∇ e K ( x, λ ) | ≤ Cλ n +1 for all ( x, λ ) ∈ R n +1+ . This holds, since the support of e K is contained in {| x | < λ } and, in this set, |∇ x e K ( x, λ ) | ≤ Cλ n +1 and | ∂ λ K ( x, λ ) | = (cid:12)(cid:12)(cid:12)(cid:12) − nλ n +1 K (cid:16) xλ (cid:17) − λ n ∇ K (cid:16) xλ (cid:17) · xλ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ n +1 . Now we define the extension e ζ as e ζ ( x, λ ) = Z R n e K ( x − x, λ ) ζ ( x ) dx, and we show that this function satisfies (3.2). Note also that, by (3.3), for every λ ≥ || e ζ ( · , λ ) || L ( R n ) ≤ || ζ || L ( R n ) . (3.5)To show (3.2), observe that, by (3.4), ∂ x i e ζ ( x, λ ) = Z R n ∂ x i e K ( x − x, λ ) ζ ( x ) dx = Z R n ∂ x i e K ( x − x, λ ) { ζ ( x ) − ζ ( x ) } dx, and thus | ∂ x i e ζ ( x, λ ) | ≤ C Z {| x − x | <λ } | ζ ( x ) − ζ ( x ) | λ n +1 dx. NERGY ESTIMATES FOR THE HALF-LAPLACIAN 13
In the same way | ∂ λ e ζ ( x, λ ) | ≤ C Z {| x − x | <λ } | ζ ( x ) − ζ ( x ) | λ n +1 dx. Hence, by Cauchy-Schwarz, |∇ e ζ ( x, λ ) | ≤ C Z {| x − x | <λ } | ζ ( x ) − ζ ( x ) | λ n +2 dx, and then Z R n +1+ |∇ e ζ | dxdλ ≤ C Z + ∞ dλ Z R n dx Z {| x − x | <λ } dx | ζ ( x ) − ζ ( x ) | λ n +2 ≤ C Z R n dx Z R n dx Z { λ> | x − x |} dλ | ζ ( x ) − ζ ( x ) | λ n +2 ≤ C Z R n Z R n | ζ ( x ) − ζ ( x ) | | x − x | n +1 dxdx. Case 2 . Consider now the general case of a function w belonging to H / ( ∂ Ω), where Ωis a bounded subset of R n +1 with Lipschitz boundary.Using the partition of unity { α j } introduced in the beginning of this section, we write w = P mj =1 wα j . Observe that, for every j = 1 , ..., m , Z ∂ Ω Z ∂ Ω | ( wα j )( z ) − ( wα j )( z ) | | z − z | n +1 dσ z dσ z ≤ C || w || H / ( ∂ Ω) , (3.6)where all constants C in the proof depend only on Ω. Indeed, Z ∂ Ω Z ∂ Ω | ( wα j )( z ) − ( wα j )( z ) | | z − z | n +1 dσ z dσ z = Z ∂ Ω Z ∂ Ω | ( wα j )( z ) − w ( z ) α j ( z ) + w ( z ) α j ( z ) − ( wα j )( z ) | | z − z | n +1 dσ z dσ z ≤ Z ∂ Ω Z ∂ Ω | α j ( z ) − α j ( z ) | | w ( z ) | | z − z | n +1 dσ z dσ z (3.7)+2 Z ∂ Ω Z ∂ Ω | w ( z ) − w ( z ) | | α j ( z ) | | z − z | n +1 dσ z dσ z . The integral in (3.7) is bounded by C || w || L ( ∂ Ω) . Indeed, using that α j is Lipschitz, we getthat the integral in (3.7) is controlled by C Z O j ∩ ∂ Ω | w ( z ) | dσ z Z O j ∩ ∂ Ω dσ z | z − z | n − ≤ C || w || L ( ∂ Ω) , where we have used spherical coordinates centered at z (after flattening the boundary) inthe last integral. From this, (3.6) follows. We flatten the boundary ∂ Ω using the local maps ϕ j introduced in the beginning of thissection, and consider the functions ζ j ( y,
0) := ( wα j )( ϕ − j ( y, , which are defined for ( y, ∈ U ∩ { µ = 0 } . Now ζ j ( · , U ∩ { µ =0 } , is defined in all of R n , and we are in the situation of case 1. We make the extension e ζ j of ζ j as in case 1. Since α j ∈ C ∞ c ( O j ), there exists a function β j ∈ C ∞ c ( O j ) such that β j ≡ α j . Thus β j ( e ζ j ◦ ϕ j ), extended by zero outside of O j , is welldefined as a function in Ω and agrees with wα j = β j wα j on ∂ Ω. We now define e w = m X j =1 β j ( e ζ j ◦ ϕ j ) in Ω , which agrees with w on ∂ Ω.Observe that, since ϕ j is a bilipschitz map and α j , β j ∈ C ∞ c ( O j ) for every j = 1 , ..., m ,we have |∇ e w | ≤ C m X j =1 n |∇ β j || e ζ j ◦ ϕ j | + | β j || ( ∇ e ζ j ) ◦ ϕ j | o , and thus Z Ω |∇ e w | ≤ C m X j =1 ( || e ζ j || L ( B × (0 , + Z R n +1+ |∇ e ζ j | ) . By (3.5) and (3.2) of case 1, we have for every j = 1 , ...m, || e ζ j || L ( B × (0 , + Z R n +1+ |∇ e ζ j | ≤ C (cid:26) || ζ j || L ( B ) + Z R n Z R n | ζ j ( y ) − ζ j ( y ) | | y − y | n +1 dydy (cid:27) ≤ C (cid:26) || w || L ( ∂ Ω) + Z R n Z R n | ζ j ( y ) − ζ j ( y ) | | y − y | n +1 dydy (cid:27) . Finally, using that ϕ j is a bilipschitz map for every j = 1 , ..., m , the definition of ζ j , and(3.6), we get Z R n Z R n | ζ j ( y ) − ζ j ( y ) | | y − y | n +1 dydy = Z B Z B | ( wα j )( ϕ − j ( y, − ( wα j )( ϕ − j ( y, || y − y | n +1 dydy ≤ C Z O j ∩ ∂ Ω Z O j ∩ ∂ Ω | ( wα j )( z ) − ( wα j )( z ) || ϕ j ( z ) − ϕ j ( z ) | n +1 dσ z dσ z ≤ C Z O j ∩ ∂ Ω Z O j ∩ ∂ Ω | ( wα j )( z ) − ( wα j )( z ) || z − z | n +1 dσ z dσ z ≤ C || w || H / ( ∂ Ω) . (cid:3) NERGY ESTIMATES FOR THE HALF-LAPLACIAN 15
Remark . Let w be the harmonic extension of w in Ω. Since w is the extension withminimal L (Ω)-norm of ∇ w , then we have that Z Ω |∇ w | dxdλ ≤ Z Ω |∇ e w | dxdλ ≤ C || w || H / ( ∂ Ω) . We give now the proof of the crucial Theorem 1.6.
Proof of Theorem 1.6.
The proof consists of two steps.
Step 1 . Suppose that A = Q = { x ∈ R n : | x i | < i = 1 , ..., n } is a cube in R n , and that Γ = { x n = 0 } ∩ Q , where x = ( x ′ , x n ) ∈ R n − × R . We may assume c = 1 by replacing w by w/c . Byhypothesis we have that | w | ≤ A and that ( | Dw ( x ) | ≤ /ε for a.e. x ∈ Q with | x n | < ε | Dw ( x ) | ≤ / | x n | for a.e. x ∈ Q with | x n | > ε. (3.8)We need to estimate the H / -norm of w in Q , given by || w || H / ( Q ) = || w || L ( Q ) + Z Q Z Q | w ( x ) − w ( x ) | | x − x | n +1 dxdx. All constants C in step 1 depend only on n and differ from line to line. In this step, wetake 0 < ε ≤ / || w || L ( Q ) ≤ n . Let x ∈ Q +1 = { x ∈ Q : x n > } and let R x be aradius depending on the point x , defined by R x = ( ε if 0 < x n < εx n / ε < x n < . To bound || w || H / ( Q ) , we consider the two cases x ∈ B R x ( x ) and x / ∈ B R x ( x ), as follows: Z Q +1 dx Z Q dx | w ( x ) − w ( x ) | | x − x | n +1 == Z Q +1 dx Z Q ∩ B Rx ( x ) dx | w ( x ) − w ( x ) | | x − x | n +1 + Z Q +1 dx Z Q \ B Rx ( x ) dx | w ( x ) − w ( x ) | | x − x | n +1 := I + I . We use | w | ≤ I , and the gradient estimate (3.8) for w to bound I . In bothcases we use spherical coordinates, centered at x , calling r = | x − x | the radial coordinate. We have I ≤ Z Q +1 dx Z Q \ B Rx ( x ) dx | x − x | n +1 ≤ C Z Q +1 dx Z √ nR x dr r ≤ C Z Q +1 R x dx = C (cid:18)Z ε ε dx n + Z ε x n dx n (cid:19) ≤ C | log ε | . Next, we bound I . We have I = Z Q +1 dx Z Q ∩ B Rx ( x ) dx | w ( x ) − w ( x ) | | x − x | n +1 = Z Q +1 dx Z Q ∩ B Rx ( x ) dx | Dw ( y ( x, x )) | | x − x | n − , where y ( x, x ) ∈ Q ∩ B R x ( x ) is a point of the segment joining x and x .Now, (3.8) reads | Dw ( y ) | ≤ min { /ε, / | y n |} for a.e. y ∈ Q . We use the bound | Dw ( y ) | ≤ /ε when 0 < x n < ε . For ε < x n <
1, since y ( x, x ) ∈ B R x ( x ) = B x n / ( x ), wehave y n ( x, x ) ≥ x n − R x = x n /
2, and thus | Dw ( y ( x, x )) | ≤ /y n ( x, x ) ≤ /x n . Thus, usingspherical coordinates centered at x , I ≤ C Z ε dx n Z ε dr ε + C Z ε dx n Z x n / dr x n ≤ C + C Z ε x n dx n ≤ C | log ε | . Finally, for x ∈ Q − = { x ∈ Q : x n < } we proceed in the same way, and thus weconclude the proof of step 1. Step 2 . Suppose now the general situation of the theorem: A ⊂ R n is a bounded Lipschitzdomain, or A = ∂ Ω, where Ω is an open bounded subset of R n +1 with Lipschitz boundary.Recall that Γ ⊂ A is the boundary (relative to A ) of a Lipschitz open (relative to A ) subset M of A . From now on, we denote by B r ( p ) the ball in R n or in R n +1 indifferently, sincewe are considering together the cases A ⊂ R n and A = ∂ Ω with Ω ⊂ R n +1 . We define afinite open covering of A in the following way.First, for every p ∈ Γ, we choose a radius r p for which there exists a bilipschitzdiffeomorphism ϕ p : B r p ( p ) ∩ A → Q , where Q is the unit cube of R n , such that ϕ ( B r p ( p ) ∩ Γ) = { x ∈ Q : x n = 0 } .Let Γ be the closure of Γ in R n or R n +1 . Only in the case A ⊂ R n , it may happen thatΓ \ Γ = ∅ . In such case, for p ∈ Γ \ Γ, there exists a radius r p and a bilipschitz diffeomorphism ϕ p : B r p ( p ) → ( − , × ( − , n − such that ϕ p ( p ) = ( − , , ..., ϕ p ( B r p ( p ) ∩ A ) = Q =( − , n and ϕ p ( B r p ( p ) ∩ Γ) = Q ∩ { x n = 0 } . Thus, these last two properties hold for p ∈ Γ \ Γ, as for the points p ∈ Γ treated before.Since Γ is compact, we can cover it by a finite number m of open balls B r pi / ( p i ), i = 1 , ..., m , with half the radius r p i . We set A (1) r i / := B r pi / ( p i ) ∩ A and A (1) r i := B r pi ( p i ) ∩ A .Observe that the number m of balls and the Lipschitz constant of ϕ p i depend only on A and Γ, as all constants from now on. NERGY ESTIMATES FOR THE HALF-LAPLACIAN 17
Next, consider the compact set K := A \ S mi =1 A (1) r i / . For every q ∈ K , take a radius0 < s q ≤ (2 / q, Γ) for which there exists a bilipschitz diffeomorphism ϕ q : B s q ( q j ) ∩ A → Q . This is possible both if q ∈ A or if q ∈ ∂A . Cover K by l balls B s qj / ( q j ), j = 1 , ..., l , with center q j ∈ K and half of the radius s q j . Set A (2) s j / := B s qj / ( q j ) ∩ A and A (2) s j := B s qj ( q j ) ∩ A .Thus, { A (1) r i / , A (2) s j / } is a finite open covering of A . Set ε := min i,j { r i / , s j / , / } . If z and z are two points belonging to A such that | z − z | < ε , then there exists a set A (1) r i , or A (2) s j , such that both z and z belong to A (1) r i , or to A (2) s j . Hence we have { ( z, z ) ∈ A × A : | z − z | < ε } ⊂ m [ i =1 A (1) r i × A (1) r i ! ∪ l [ j =1 A (2) s j × A (2) s j ! . (3.9)Observe thatdist( y, Γ) ≥ dist( q j , Γ) − | y − q j | ≥ s q j − s q j = s j ≥ ε for every y ∈ A (2) s j . (3.10)Let L > ϕ p , ..., ϕ p m , ϕ − p , ..., ϕ − p m . Now, let w as in the statement of the theorem. Let us first treat the case 0 < ε ≤ / (2 L ) . Since Z A dσ z Z { z ∈ A : | z − z | >ε } dσ z | w ( z ) − w ( z ) | | z − z | n +1 ≤ c ε n +10 | A | = Cc , we only need to bound the double integral in { z ∈ A } × { z ∈ A : | z − z | < ε } . By (3.9),it suffices to bound the integrals in each A (1) r i × A (1) r i and in each A (2) s j × A (2) s j .Thus, for every i , consider Z A (1) ri Z A (1) ri | w ( z ) − w ( z ) | | z − z | n +1 dσ z dσ z . Recall that, by construction, there exists a bilipschitz map ϕ p i : A (1) r i → Q such that ϕ p i (Γ ∩ A (1) r i ) = { x ∈ Q : x n = 0 } . Thus, flattening the set A (1) r i using ϕ p i , we are in thesituation of step 1. More precisely, since ϕ p i is bilipschitz, we have that Z A (1) ri Z A (1) ri | w ( z ) − w ( z ) | | z − z | n +1 dσ z dσ z ≤ C Z Q Z Q | w i ( x ) − w i ( x ) | | x − x | n +1 dxdx, where we have set w i = w ◦ ϕ − p i .Given x ∈ Q , let y = ϕ − p i ( x ) ∈ A (1) r i . Recalling the definition of the Lipschitz constant L above, we have | x n | ≤ L dist( y, Γ) and hence | Dw i ( x ) | ≤ L | Dw ( y ) | ≤ Lc min (cid:26) ε , y, Γ) (cid:27) ≤ Lc min (cid:26) ε , L | x n | (cid:27) = L c min (cid:26) εL , | x n | (cid:27) . Thus we can apply the result proved in Step 1, with ε replaced by εL (note that we have εL ≤ /
2, as in Step 1), to the function w i / [(1 + L ) c ] . We obtain the desired bound Cc | log( εL ) | ≤ Cc | log ε | .Next, we consider the double integral in A (2) s j × A (2) s j , for any j ∈ { , ..., l } . Recall thatthere exists a bilipschitz diffeomorphism ϕ q j : A (2) s j → Q . Thus Z A (2) sj Z A (2) sj | w ( z ) − w ( z ) | | z − z | n +1 dσ z dσ z ≤ C Z Q Z Q | v j ( x ) − v j ( x ) | | x − x | n +1 dσ x dσ x , where now v j := w ◦ ϕ − q j . By (3.10) and (1.14), | Dw ( y ) | ≤ c /ε a.e. in A (2) s j , and | Dv j | ≤ C a.e. in Q . From this, the last double integral is bounded by C Z Q dx Z Q dx | x − x | n − ≤ C Z Q dx Z √ n dr ≤ C. This conclude the proof in case ε ≤ / (2 L ).Finally, given ε ∈ (0 , /
2) with ε > / (2 L ), since (1.14) holds with such ε , it also holdswith ε replaced by 1 / (2 L ). By the previous proof with ε taken to be 1 / (2 L ), the energy isbounded by C | log(1 / (2 L )) | ≤ C ≤ C | log ε | since ε < / (cid:3) Energy estimate for global minimizers
In this section we give the proof of Theorem 1.2. It is based on a comparison argument.The proof can be resumed in 3 steps. Let v be a global minimizer of (1.3).i) Construct a comparison function w , harmonic in C R , which takes the same valuesas v on ∂ + C R = ∂C R ∩ { λ > } and thus, by minimality of v , E C R ( v ) ≤ E C R ( w ) . ii) Use estimate (1.12): Z C R |∇ w | ≤ C || w || H / ( ∂C R ) . iii) Establish, using Theorem 1.6, the key estimate || w || H / ( ∂C R ) ≤ CR n − log R. Proof of Theorem 1.2.
Let v be a bounded global minimizer of (1.3). Let u be its trace on ∂ R n +1+ . Recall the definition (1.6) of the constant c u . Let s ∈ [inf u, sup u ] be such that G ( s ) = c u .Through the proof, C denotes positive constants depending only on n , || f || C ([inf u, sup u ]) and || u || L ∞ ( R n ) . As explained in (1.16), v satisfies the following bounds: | v | ≤ C and |∇ v ( x, λ ) | ≤ C λ for every x ∈ R n , λ ≥ . (4.1)We estimate the energy E C R ( v ) of v using a comparison argument. We define a function w = w ( x, λ ) on C R in the following way. First we define w ( x,
0) on the base of the
NERGY ESTIMATES FOR THE HALF-LAPLACIAN 19 cylinder to be equal to a smooth function g ( x ) which is identically equal to s in B R − and g ( x ) = v ( x,
0) for | x | = R . The function g is defined as follows: g = sη R + (1 − η R ) v ( · , , (4.2)where η R is a smooth function depending only on r = | x | such that η ≡ B R − and η ≡ B R . Thus, g satisfies g ∈ [inf u, sup u ] and |∇ g | ≤ C in B R . (4.3)Then we define w ( x, λ ) as the unique solution of the Dirichlet problem ∆ w = 0 in C R w ( x,
0) = g ( x ) on B R × { λ = 0 } w ( x, λ ) = v ( x, λ ) on ∂C R ∩ { λ > } . (4.4)Since v is a global minimizer of E C R and w = v on ∂C R ∩ { λ > } , then Z C R |∇ v | dxdλ + Z B R { G ( u ) − c u } dx ≤ Z C R |∇ w | dxdλ + Z B R { G ( w ( x, − c u } dx. We prove next that Z C R |∇ w | dxdλ + Z B R { G ( w ( x, − c u } dx ≤ CR n − log R. Observe that the potential energy is bounded by CR n − . Indeed, by definition w ( x,
0) = s in B R − , and hence Z B R { G ( w ( x, − c u } dx = Z B R \ B R − { G ( g ( x )) − c u } dx ≤ C | B R \ B R − | ≤ CR n − . Thus, we only need to bound the Dirichlet energy. First of all, rescaling, we set w ( x, λ ) = w ( Rx, Rλ ) , for ( x, λ ) ∈ C = B × (0 , ε = 1 /R. Observe that Z C R |∇ w | = CR n − Z C |∇ w | . Thus, we need to prove that Z C |∇ w | ≤ C log R = C | log ε | . (4.5) Since w is harmonic in C , Proposition 3.1 gives that Z C |∇ w | dxdλ ≤ C || w || H / ( ∂C ) . To control || w || H / ( ∂C ) , we apply Theorem 1.6 to w | ∂C in A = ∂C , taking Γ = ∂B × { λ = 0 } .Since | w | ≤ C , we only need to check (1.14) in ∂C . In the bottom boundary, B × { } ,this is simple. Indeed w ≡ s in B − ε , and thus we need only to control |∇ w ( x, | = ε − |∇ g ( Rx ) | ≤ Cε − for | x | > − ε , by (4.3). Here dist( x, ∂B ) < ε , and thus (1.14) holdshere.Next, to verify (1.14) in ∂C ∩ { λ > } we use that w = v here and that we know |∇ v ( x, λ ) | ≤ C λ for every ( x, λ ) ∈ C R , as stated in (4.1). Thus, the tangential derivatives of w in ∂C ∩ { λ > } satisfy |∇ w ( x, λ ) | ≤ CR Rλ = Cε + λ ≤ C min (cid:26) ε , λ (cid:27) . Since dist(( x, λ ) , Γ) ≥ λ on ∂C ∩ { λ > } , w | ∂C satisfies the hypotheses of Theorem 1.6.We conclude that (4.5) holds. (cid:3) Energy estimate for monotone solutions in R The following lemma will play a key role in this section to establish the energy estimatefor monotone solutions in dimension n = 3. Lemma 5.1.
Let f be a C ,β function, for some < β < , and u a bounded solution ofequation (1.1) in R , such that u x > . Let v be the harmonic extension of u in R . Set v ( x , x , λ ) := lim x →−∞ v ( x, λ ) and v ( x , x , λ ) := lim x → + ∞ v ( x, λ ) . Then, v and v are solutions of (1.3) in R , and each of them is either constant or itdepends only on λ and one Euclidian variable in the ( x , x ) -plane. As a consequence, each u = v ( · , and u = v ( · , is either constant or 1-D.Moreover, set m = inf u ≤ e m = sup u and f M = inf u ≤ M = sup u . Then, G >G ( m ) = G ( e m ) in ( m, e m ) , G ′ ( m ) = G ′ ( e m ) = 0 and G > G ( f M ) = G ( M ) in ( f M , M ) , G ′ ( f M ) = G ′ ( M ) = 0 . Proof.
The function v ( x ′ , λ ) = lim x → + ∞ v ( x ′ , x , λ ) is the harmonic extension of u . Thekey point of the proof is to verify that v is a stable solution of problem (1.3) in R andthen apply Theorem 1.5 of [6] on 1-D symmetry in R .The fact that v is a solution of problem (1.3) in R is easily verified viewing v as afunction of 4 variables, limit as t → + ∞ of the solutions v t ( x ′ , x , λ ) = v ( x ′ , x + t, λ ). Bystandard elliptic theory, v t → v uniformly in the C sense on compact sets of R . NERGY ESTIMATES FOR THE HALF-LAPLACIAN 21
Now we prove that v ( x ′ , λ ) is a stable solution of problem (1.3) in R . By Lemma 4.1 of[6], the stability of v is equivalent to the existence of a function ϕ > R which satisfies ( ∆ ϕ = 0 in R − ∂ϕ∂λ = f ′ ( v ) ϕ on ∂ R . (5.1)To check the existence of ϕ > v x > R and that satisfiesthe problem ( ∆ v x = 0 in R − ∂v x ∂λ = f ′ ( v ) v x on ∂ R . This gives that v is stable in R , i.e. Z R |∇ ξ | dxdλ − Z R f ′ ( v ) ξ dx ≥ , for every ξ ∈ C ∞ c ( R ) . (5.2)Next, we claim that Z R |∇ η | dx ′ dλ − Z R f ′ ( v ) η dx ′ ≥ , for every η ∈ C ∞ c ( R ) . (5.3)To show this, we take ρ > ψ ρ ∈ C ∞ ( R ) with 0 ≤ ψ ρ ≤ , ≤ ψ ′ ρ ≤ , ψ ρ = 0 in( −∞ , ρ ) ∪ (2 ρ + 2 , + ∞ ), and ψ ρ = 1 in ( ρ + 1 , ρ + 1), and we apply (5.2) with ξ ( x, λ ) = η ( x ′ , λ ) ψ ρ ( x ) . We obtain after dividing the expression by α ρ = R ψ ρ , that Z R dx ′ dλ |∇ η ( x ′ , λ ) | + Z R dx ′ dλ η ( x ′ , λ ) Z R dx ( ψ ′ ρ ) ( x ) α ρ − Z R dx ′ dλ η ( x ′ , λ ) Z R dx f ′ ( v ( x ′ , x , ψ ρ ( x ) α ρ ≥ ρ → + ∞ , and using f ∈ C and that v ( x ′ , x , λ ) → v ( x ′ , λ ) as x → + ∞ uniformly in compact sets of R , we obtain (5.3).Since v ( x ′ , λ ) is a stable solution of problem (1.3) in R , by Theorem 1.5 (point b) in[6] we deduce that v is constant or v depends only on λ and one Euclidian variable in the x ′ -plane. Now note that the function 2( v − f M ) / ( M − f M ) − f forwhich there exists a layer solutions for problem (1.3) in dimension n = 1, and restatingthe conclusion for v , we get G ′ ( f M ) = G ′ ( M ) = 0 and G > G ( f M ) = G ( M ) in ( f M , M ).In the same way, we prove that the conclusion holds for v and that G ′ ( e m ) = G ′ ( m ) = 0and G > G ( e m ) = G ( m ) in ( m, e m ). (cid:3) Remark . We claim that in the case of Allen-Cahn type equations, we could prove theenergy estimate (1.9) for monotone solutions in dimension n = 3 using the same argumentas in the proof of Theorem 2.1. The only difficulty is that in this section we do not assumelim x → + ∞ v = 1, and then we do not know if lim T →∞ E C R ( v T ) = 0 (see (2.9) in the proof ofTheorem 2.1). Using Lemma 5.1, we have that u ( x , x ) = lim x → + ∞ u ( x , x , x ) is either a constant or it depends only on one variable. Then, applying Theorem 1.6 of [6], whichgives the energy bounds for 1-D solutions, we deduce that lim T → + ∞ E C R ( v T ) ≤ CR log R ,and this is enough to carry out the proof of Theorem 2.1 in the present setting.Before giving the proof of Theorem 1.3, we need the following proposition. It is theanalog of Theorem 4.4 of [1] and asserts that the monotonicity of a solution implies itsminimality among a suitable family of functions. Proposition 5.3.
Let f be any C ,β nonlinearity, with β ∈ (0 , . Let u be a boundedsolution of (1.1) in R n such that u x n > , and let v be its harmonic extension in R n +1+ .Then, Z C R |∇ v ( x, λ ) | dxdλ + Z B R G ( v ( x, dx ≤ Z C R |∇ w ( x, λ ) | dxdλ + Z B R G ( w ( x, dx, for every w ∈ C ( R n +1+ ) such that w = v on ∂ + C R = ∂C R ∩ { λ > } and v ≤ w ≤ v in C R , where v and v are defined by v ( x ′ , λ ) := lim x n →−∞ v ( x ′ , x n , λ ) and v ( x ′ , λ ) := lim x n → + ∞ v ( x ′ , x n , λ ) . Proof.
This property of minimality of monotone solutions among functions w such that v ≤ w ≤ v follows from the following two results:i) Uniqueness of solution to the problem ∆ w = 0 in C R ,w = v on ∂ + C R = ∂C R × { λ > } , − ∂ λ w = f ( w ) on ∂ C R = B R × { λ = 0 } ,v ≤ w ≤ v in C R . (5.4)Thus, the solution must be w ≡ v . This is the analog of Lemma 3.1 of [6], and below wecomment on its proof.ii) Existence of an absolute minimizer for E C R in the set C v = { w ∈ H ( C R ) : w ≡ v on ∂ + C R , v ≤ w ≤ v in C R } . This is the analog of Lemma 2.10 of [6].The statement of the proposition follows from the fact that by i) and ii), the monotonesolution v , by uniqueness, must agree with the absolute minimizer in C R .To prove points i) and ii), we proceed exactly as in [6], with the difference that herewe do not assume lim x n →±∞ v = ±
1. We have only to substitute − v and v , respectively, in the proofs of Lemma 3.1 and Lemma 2.10 in [6]. For this, it is impor-tant that v and v are, respectively, a strict subsolution and a strict supersolution of theDirichlet–Neumann mixed problem (5.4). We make a short comment about these proofs. NERGY ESTIMATES FOR THE HALF-LAPLACIAN 23
The proof of uniqueness is based, as in Lemma 3.1 of [6], on sliding the function v ( x, λ )in the direction x n . We set v t ( x , ..., x n , λ ) = v ( x , ..., x n + t, λ ) for every ( x, λ ) ∈ C R . Since v t → v as t → + ∞ uniformly in C R and v < w < v , then w < v t in C R , for t largeenough. We want to prove that w < v t in C R for every t >
0. Suppose that s > t > w < v t in C R . Then by applying the maximum principleand Hopf’s lemma we get a contradiction, since one would have w ≤ v s in C R and w = v s at some point in C R \ ∂ + C R .To prove the existence of an absolute minimizer for E C R in the convex set C v , we proceedexactly as in the proof of Lemma 2.10 of [6], substituting − v and v , respectively. (cid:3) We give now the proof of the energy estimate in dimension 3 for monotone solutionswithout the limit assumptions.
Proof of Theorem 1.3.
We follow the proof of Theorem 5.2 of [1]. We need to prove thatthe comparison function w , used in the proof of Theorem 1.2, satisfies v ≤ w ≤ v . Thenwe can apply Proposition 5.3 to make the comparison argument with the function w (asfor global minimizers). We recall that w is the solution of problem (4.4), ∆ w = 0 in C R w ( x,
0) = g ( x ) on B R × { λ = 0 } w ( x, λ ) = v ( x, λ ) on ∂C R ∩ { λ > } , (5.5)where g = sη R + (1 − η R ) v ( · , v ≤ s ≤ inf v , then v ≤ g ≤ v and hence v and v are respectively, subsolution and supersolutions of (5.5). It follows that v ≤ w ≤ v , as desired.To show that sup v ≤ s ≤ inf v , let m = inf u = inf u and M = sup u = sup u , where u and u are defined in Lemma 5.1. Set e m = sup u and f M = inf u , obviously e m and f M belong to [ m, M ]. By Lemma 5.1, u and u are either constant or monotone 1-D solutions;moreover, G > G ( m ) = G ( e m ) in ( m, e m ) (5.6)in case m < e m (i.e. u not constant), and G > G ( M ) = G ( f M ) in ( f M , M ) (5.7)in case f M < M (i.e. u not constant).In all four possible cases (that is, each u and u is constant or one-dimensional), we deducefrom (5.6) and (5.7) that e m ≤ f M and that there exists s ∈ [ e m, f M ] such that G ( s ) = c u (recall that c u is the infimum of G in the range of u ). We conclude thatsup u = sup v ≤ e m ≤ s ≤ f M ≤ inf v = inf u. Hence, we can apply Proposition 5.3 to make comparison argument with the function w and obtain the desired energy estimate. (cid:3) R In this section we present the Liouville result due to Moschini [14] that we will use inthe proof of 1-D symmetry in dimension n = 3. Set F = (cid:26) F : R + → R + , F is nondecreasing and Z + ∞ rF ( r ) = + ∞ (cid:27) . Note that F includes the function F ( r ) = log r . Proposition 6.1 ([14]) . Let ϕ ∈ L ∞ loc ( R n +1+ ) be a positive function. Suppose that σ ∈ H ( R n +1+ ) satisfies ( − σ div( ϕ ∇ σ ) ≤ in R n +1+ − σ∂ λ σ ≤ on ∂ R n +1+ (6.1) in the weak sense. Let the following condition hold: lim sup R → + ∞ R F ( R ) Z C R ( ϕσ ) dx < ∞ (6.2) for some F ∈ F .Then, σ is constant.In particular, this statement holds with F ( R ) = log R .Remark . In [14], the author proves the previous result under the assumption + ∞ X j =0 F (2 j +1 ) = + ∞ (6.3)on F . This is equivalent to R + ∞ ( rF ( r )) − dr = + ∞ . Indeed, since the function j F (2 j +1 )is nondecreasing, we have that + ∞ X j =3 F (2 j +1 ) ≤ Z + ∞ dsF (2 s +1 ) = 1log 2 Z ∞ drrF ( r ) ≤ + ∞ X j =2 F (2 j +1 ) . Thus, (6.3) holds if and only if F ∈ F . Proof of Proposition 6.1.
We present the proof following that of Theorem 5.1 of [14], herein C R instead of B R . Set ∂ + C R := ∂C R ∩ { λ > } . Since σ satisfies (6.1), we havediv( σϕ ∇ σ ) ≥ ϕ |∇ σ | . (6.4)On the other hand Z ∂ + C R σϕ ∂σ∂ν ds ≤ (cid:18)Z ∂ + C R ϕ |∇ σ | ds (cid:19) (cid:18)Z ∂ + C R ( ϕσ ) ds (cid:19) , (6.5)where ν denotes the outer normal vector on ∂ + C R . Now, set, as in [14], D ( R ) = Z C R ϕ |∇ σ | dx. NERGY ESTIMATES FOR THE HALF-LAPLACIAN 25
Integrating (6.4) over C R , using that − σ∂ λ σ ≤ ∂ C R = ∂C R ∩{ λ = 0 } , and using (6.5), we get D ( R ) ≤ D ′ ( R ) (cid:18)Z ∂ + C R ( ϕσ ) ds (cid:19) , (6.6)which is the analog of (5.5) in [14] on ∂ + C R instead of ∂B R .Assume that σ is not constant. Then, there exists R > D ( R ) > R > R . Integrating (6.6) and using Schwarz inequality, we get that, for every r > r > R ,( r − r ) Z C r \ C r ( ϕσ ) dx ! − = ( r − r ) (cid:18)Z r r dR Z ∂ + C R ds ( ϕσ ) (cid:19) − ≤ Z r r dR (cid:18)Z ∂ + C R ds ( ϕσ ) (cid:19) − ≤ D ( r ) − D ( r ) . (6.7)Next, choose r = 2 j +1 r ∗ and r = 2 j r ∗ , for some r ∗ > R , for every j = 0 , ..., N −
1. Using(6.2), (6.7) and summing over j , we find that1 D ( r ∗ ) ≥ C N − X j =0 F (2 j +1 r ∗ ) . (6.8)If j is such that r ∗ ≤ j , then, by hypothesis on F , F (2 j +1 r ∗ ) ≤ F (2 j + j +1 ). Thus, by(6.3), the sum in (6.8) diverges as N → ∞ and hence D ( r ∗ ) = 0 for every r ∗ > R , whichis a contradiction. (cid:3) We can give now the proof of the 1-D symmetry result.
Proof of Theorem 1.4.
Without loss of generality we can suppose e = (0 , , ϕ ∈ C ( R ) ∩ C ( R ) suchthat ϕ > R and ( ∆ ϕ = 0 in R − ∂ϕ∂λ = f ′ ( v ) ϕ on ∂ R . Note that, if u is a monotone solution in the direction x , then we can choose ϕ = v x ,where v is the harmonic extension of u in the half space. For i = 1 , , σ i = v x i ϕ . We prove that σ i is constant in R , using the Liouville result of Proposition 6.1 and ourenergy estimate.Since ϕ ∇ σ i = ϕ ∇ v x i − v x i ∇ ϕ, we have that div( ϕ ∇ σ i ) = 0 in R . Moreover, the normal derivative − ∂ λ σ i is zero on ∂ R . Indeed, ϕ ∂ λ σ i = ϕv λx i − v x i ϕ λ = 0since both v x i and ϕ satisfy the same boundary condition − ∂ λ v x i − f ′ ( v ) v x i = 0 , − ∂ λ ϕ − f ′ ( v ) ϕ = 0 . Now, using our energy estimates (1.7) or (1.9), we have for n = 3, Z C R ( ϕσ i ) ≤ Z C R |∇ v | ≤ CR log R, for every R > . Thus, using Proposition 6.1, we deduce that σ i is constant for every i = 1 , ,
3, i.e., v x i = c i ϕ for some constant c i , with i = 1 , , . We conclude the proof observing that if c = c = c = 0 then v is constant. Otherwisewe have c i v x j − c j v x i = 0 for every i = j, and we deduce that v depends only on λ and on the variable parallel to the vector ( c , c , c ).Thus, u ( x ) = v ( x,
0) is 1-D. (cid:3) Energy estimate for saddle-shaped solutions
In this section we prove that the energy estimate (1.7) holds also for some saddle solutions(which are known [10] not to be global minimizers in dimensions 2 m ≤
6) of the problem( − ∆) / u = f ( u ) in R m . Here, we suppose that f is balanced and bistable, that is f satisfies hypotheses (1.17),(1.18), and (1.19).We recall that saddle solutions are even with respect to the coordinate axes and oddwith respect to the Simons cone, which is defined as follows: C = { x ∈ R m : x + ... + x m = x m +1 + ... + x m } . If we set s = q x + · · · + x m and t = q x m +1 + · · · + x m , then the Simons cone becomes C = { s = t } . We say that a solution u of problem (1.1) isa saddle solution if it satisfies the following properties:a) u depends only on the variables s and t . We write u = u ( s, t );b) u > s > t ;c) u ( s, t ) = − u ( t, s ) . In [10], the second author proves the existence of a saddle solution u = u ( x ) to problem(1.1), by proving the existence of a solution v = v ( x, λ ) to problem (1.3) with the followingproperties:a) v depends only on the variables s, t and λ . We write v = v ( s, t, λ ); NERGY ESTIMATES FOR THE HALF-LAPLACIAN 27 b) v > s > t ;c) v ( s, t, λ ) = − v ( t, s, λ ).The proof of the existence of such function v is simple and it uses a non-sharp energyestimate. Next, we sketch the proof.We use the following notations: O := { x ∈ R m : s > t } ⊂ R m , e O := { ( x, λ ) ∈ R m +1+ : x ∈ O} ⊂ R m +1+ . Note that ∂ O = C . Let B R be the open ball in R m centered at the origin and of radius R . We will considerthe open bounded sets O R := O ∩ B R = { s > t, | x | = s + t < R } ⊂ R m , e O R := O R × (0 , R ) , and e O R,L := O R × (0 , L ) . Note that ∂ O R = ( C ∩ B R ) ∪ ( ∂B R ∩ O ) . Moreover we define the set e H ( e O R,L ) = { v ∈ H ( e O R,L ) : v ≡ ∂ + e O R,L , v = v ( s, t, λ ) a.e. } . Proof of Theorem 1.7.
The proof of existence of the saddle solution v in R m +1+ can beresumed in three steps. Step a ). For every
R > L > v R,L of the energy functional E e O R,L ( v ) = Z e O R,L |∇ v | + Z O R G ( v )among all functions belonging to the space e H ( e O R,L ). The existence of such minimizer,that may be taken to satisfy | v R,L | ≤ v R,L is a solution of the equation (1.3) written inthe ( s, t, λ ) variables and we can assume that v R,L ≥ e O R,L . Step b ). Extend v R,L to B R × (0 , L ) by odd reflection with respect to C × (0 , L ), that is, v R,L ( s, t, λ ) = − v R,L ( t, s, λ ). Then, v R,L is a solution in B R × (0 , L ). Step c ). Define v as the limit of the sequence v R,L as R → + ∞ , taking L = R γ → + ∞ with 1 / ≤ γ <
1. With the aid of a non-sharp energy estimate, verify that v v is a saddle solution. This step could also be carried out using thesharp energy estimate that we prove next.Here, it is important to observe that the solution v constructed in this way is not aglobal minimizer in R m +1+ (indeed it is not stable in dimensions 2 m = 4 , e O , or in other words, it is a minimizer under perturbationsvanishing on the Simons cone. Next, we use this fact to prove the energy estimate E e O R ( v ) ≤ CR m − log R in the set e O R = O R × (0 , R ), using a comparison argument as for globalminimizers. As before, we want to construct a comparison function w in e O R which agrees with v on ∂ + e O R and such that E e O R ( w ) = Z e O R |∇ w | + Z O R G ( w ) ≤ CR m − log R. (7.1)We define the function w = w ( x, λ ) = w ( s, t, λ ) in e O R in the following way.First we define w ( x,
0) on the base O R of e O R to be equal to a smooth function g ( x )which is identically equal to 1 in O R − ∩ { ( s − t ) / √ > } and g ( x ) = v ( x,
0) on ∂ O R . Thefunction g is defined as follows: g = η R min (cid:26) , s − t √ (cid:27) + (1 − η R ) v ( · , , (7.2)where η R is a smooth function depending only on r = | x | = ( s + t ) / such that η R ≡ O R − and η R ≡ O R . Let w = w ( x, λ ) = w ( s, t, λ ) be any Lipschitz function inthe closure of e O R (the precise function w will be chosen later) such that ( w ( x,
0) = g ( x ) on O R × { λ = 0 } w ( x, λ ) = v ( x, λ ) on ∂ e O R ∩ { λ > } . (7.3)Since v is a global minimizer of E e O R and w = v on ∂ e O R ∩ { λ > } , then Z e O R |∇ v | dxdλ + Z O R G ( u ) dx ≤ Z e O R |∇ w | dxdλ + Z O R G ( w ( x, dx. We establish now the bound (7.1) for the energy E e O R ( w ) of w .Observe that the potential energy of w is bounded by CR m − , indeed Z O R G ( w ( x, dx ≤ C (cid:12)(cid:12)(cid:12)(cid:12) O R − ∩ (cid:26) s − t √ < (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + C |O R \ O R − |≤ C Z R − { ( t + √ m − t m } t m − dt + CR m − ≤ CR m − . Next, we bound the Dirichlet energy of w . First of all, as in the proof of the energyestimate for global minimizers, we rescale and set w ( x, λ ) = w ( Rx, Rλ ) for every ( x, λ ) ∈ e O . Thus, the Dirichlet energy of w in e O R , satisfies Z e O R |∇ w | = CR m − Z e O |∇ w | . NERGY ESTIMATES FOR THE HALF-LAPLACIAN 29
Setting ε = 1 /R , we need to prove that Z e O |∇ w | ≤ C | log ε | . (7.4)Set s = | ( x , ..., x m ) | and t = | ( x m +1 , ..., x m ) | , for every x = ( x , ..., x m ) ∈ O . We observethat Z e O |∇ w | dxdλ == C Z dλ Z { s + t < ,s>t ≥ } (cid:8) ( ∂ s w ) + ( ∂ t w ) + ( ∂ λ w ) (cid:9) s m − t m − dsdt ≤ C Z dλ Z { s + t < ,s>t ≥ } (cid:8) ( ∂ s w ) + ( ∂ t w ) + ( ∂ λ w ) (cid:9) dsdt. We can see the last integral as an integral in the set { ( s, t, λ ) ∈ R : s + t < , s > t ≥ , < λ < } ⊂ R . We consider now w the even reflection of w with respect to { t = 0 } . We set ( s = z t = | z | , and we define w ( z, λ ) = w ( z , z , λ ) := w ( s, t, λ ) in the Lipschitz setΩ = { ( z , z , λ ) : z + z < , z > | z | , < λ < } ⊂ R . We have that Z dλ Z { s + t < ,s>t> } (cid:8) ( ∂ s w ) + ( ∂ t w ) + ( ∂ λ w ) (cid:9) dsdt ≤ Z dλ Z { z + z < ,z > | z |} |∇ w | dz dz . Next we apply Proposition 3.1 and Theorem 1.6 to the function w in Ω. Observe thatΩ is Lipschitz as a subset of R , but it is not Lipschitz at the origin if seen as a subset of R m +1 . We now take w to be the harmonic extension in Ω ⊂ R of the boundary valuesgiven by (7.3), after rescaling by R and doing even reflection with respect to t = | z | = 0.Since w is harmonic in Ω ⊂ R , Proposition 3.1 gives that Z Ω |∇ w | dz dz dλ ≤ C || w || H / ( ∂ Ω) . To bound the quantity || w || H / ( ∂ Ω) , we apply Theorem 1.6 with A = ∂ Ω andΓ = (cid:0)(cid:8) z + z < , z = | z | (cid:9) × { λ = 0 } (cid:1) ∪ (cid:0)(cid:8) z + z = 1 , z > | z | (cid:9) × { λ = 0 } (cid:1) . Since | w | ≤
1, we need only to check (1.14) in ∂ Ω. By the definition of w , we have that w ( z, ≡ z, Γ) > ε , while for dist( z, Γ) < ε , |∇ w ( z , z , | = |∇ w ( s, t, | = ε − |∇ g ( Rx, | ≤ Cε − = C min { ε − , (dist( z, Γ)) − } . Moreover, as in the proof of Theorem 1.2, to verify (1.14) in ∂ Ω ∩ { λ > } we use that w ≡ v here and the gradient bound (1.16) for v . Thus, |∇ w ( z , z , λ ) | ≤ CR Rλ = Cε + λ ≤ C min (cid:26) ε , λ (cid:27) . Hence, w satisfies the hypotheses of Theorem 1.6 and we conclude that (7.4) holds. (cid:3) References [1] G. Alberti, L. Ambrosio and X. Cabr´e,
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ICREA and Universitat Polit`ecnica de Catalunya, Diagonal 647, Departament de Mate-m`atica Aplicada 1, 08028 Barcelona (Spain)
E-mail address : [email protected] Universit`a di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5,40126 Bologna (Italy) and Universitat Polit`ecnica de Catalunya, Diagonal 647, Departa-ment de Matem`atica Aplicada 1, 08028 Barcelona (Spain)
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