Minimal energy for the traveling waves of the Landau–Lifshitz equation

Minimal energy for the traveling waves of the Landau–Lifshitz equation

André de Laire

Laboratoire Paul Painlevé, Université Lille 1 59655 Villeneuve d’Ascq Cedex, France andre.de-laire@math.univ-lille1.fr

Abstract

We consider nontrivial finite energy traveling waves for the Landau–Lifshitz equation with easy- plane anisotropy. Our main result is the existence of a minimal energy for these traveling waves, in dimensions two, three and four. The proof relies on a priori estimates related with the theory of harmonic maps and the connection of the Landau–Lifshitz equation with the kernels appearing in the Gross–Pitaevskii equation.

2010Mathematics Subject Classification: 35J60, 35Q51, 37K40, 35B65, 58E20, 82D55

Keywords and phrases: Landau–Lifshitz equation, Traveling waves, Minimal energy, Harmonic maps, Asymptotic behavior

1 Introduction

In this work we consider the Landau–Lifshitz equation

∂tm+m×(∆m+λm3e3) = 0, m(t, x)∈S², t∈R, x∈R^N, (1.1) wheree3= (0,0,1),λ∈Randm= (m1, m2, m3). This equation was originally introduced by L. Landau and E. Lifshitz in [32] to describe the dynamics of magnetization in ferromagnetic materials. Here the parameter λ takes into account the anisotropy of such material. More precisely, the value λ = 0 corresponds to the isotropic case, meanwhileλ >0 andλ <0correspond to materials with an easy-axis and an easy-plane anisotropy, respectively (see [29, 25]).

The isotropic case λ= 0 recovers the Schödinger map equation, which has been intensively studied due to its applications in several areas of physics and mathematics (see [17, 38]). Forλ >0, the existence of solitary waves periodic in time have been established in [20, 41]. Moreover, Pu and Guo [42] showed that λ6= 0is a necessary condition to the existence of these types of solutions.

In this paper we are interested in the case of easy-plane anisotropyλ <0. By a scaling argument we can suppose from now on thatλ=−1. Then the energy of (1.1) is given by

E(m) = Z

R^N

e(m)dx≡1 2

Z

R^N

|∇m|²+m²₃ dx,

that it is formally conserved due to the Hamiltonian structure of (1.1). Ifmis smooth, by differentiating twice the condition|m(t, x)|² = 1we obtainm·∆m=−|∇m|², so that taking cross product ofmand (1.1), we can recast (1.1) as

m×∂tm= ∆m+|∇m|²m−(m3e3−m²₃m). (1.2) Using formal developments and numerical simulations, Papanicolaou and Spathis [40] found in dimensions N ∈ {2,3}nonconstant finite energy traveling waves of (1.2), propagating with speedc∈(0,1)along the x1-axis, i.e. of the form

mc(x, t) =u(x1−ct, x2, . . . , xN).

By substituting mc in (1.2), the profileusatisfies

−∆u=|∇u|²u+u²₃u−u3e3+cu×∂1u. (TWc) Notice that if usatisfies (TWc) with speed c, so does −uwith speed −c, therefore we can assume that c≥0. Also, we see that any constant in S¹× {0}satisfies (TWc), so that we refer to them as the trivial solutions. Since we are interested in finite energy solutions, the natural energy space to work in is

E(R^N) ={v∈L¹_loc(R^N;R³) :∇v∈L²(R^N), v3∈L²(R^N), |v|= 1a.e. onR^N}.

1.1 The minimal energy

Our main theorem is in the same spirit as the result proved by the author for the Gross–Pitaevskii equation in [9] (see also [2]). Precisely, we show the existence of a minimal value for the energy for the nontrivial traveling waves.

Theorem 1.1. Let N ∈ {2,3,4}. There exists a universal constant µ >0 such that if u ∈ E(R^N) is a nontrivial solution of (TWc) with c ∈(0,1], satisfying in addition that u is uniformly continuous if N ∈ {3,4}, then

E(u)≥µ. (1.3)

As noticed in [21] in dimension two, there is no smooth static solution of (TWc), i.e. with speed c= 0. More generally, we obtain the following result for static waves.

Proposition 1.2. Let N≥2. Assume thatu∈ E(R^N)is a solution of (TWc)withc= 0. Suppose also that uis uniformly continuous ifN≥3. Thenuis a trivial solution.

Theorem 1.1 shows that there are no small energy traveling wave solutions in dimensions two, three and four (assuming that they are uniformly continuous in dimensions three and four). This opens the door to have a scattering theory for equation (1.1) with λ=−1, similarly to the theory developed for the Gross–Pitaevskii equation by Gustafson, Nakanishi and Tsai [18, 19].

The one-dimensional case is different. IfN = 1, (TWc) is completely integrable and we can compute the solutions in E(R)explicitly. More precisely,

Proposition 1.3. Let N = 1,c≥0andu∈ E(R)be solution of (TWc).

(i) Ifc≥1, thenuis a trivial solution.

(ii) If0≤c <1 anduis nontrivial, then, up to invariances,uis given by u1=csech(p

1−c²x), u2= tanh(p

1−c²x), u3=p

1−c²sech(p

1−c²x).

Moreover, if0< c <1,

E(u) = 2p

1−c² and E(p(u)) = 2 sin(p(u)/2), wherep(u)denotes the momentum ofu.

We notice that equation (TWc) is invariant under translations and under the action ofS¹by a rotation around the e3-axis, that is ifu= (u1, u2, u3)is a solution of (TWc), so is

(u1cos(ϕ)−u2sin(ϕ), u1sin(ϕ) +u2cos(ϕ), u3),

for any ϕ ∈ R. Also, if u = (u1, u2, u3)(x) is a solution, so is u = (u1, u2,−u3)(−x). These are the invariances that we refer to in Proposition 1.3.

We provide the proof of Proposition 1.3 in Section 6, as well as the precise definition of momentum.

The relation E = 2 sin(p/2) is showed in Figure 1. In particular we note that there are solutions of small energy, but there is a maximum value for the energy and the momentum. We also remark that Proposition 1.3 provides a solution withc= 0, meanwhile Proposition 1.2 states that this is not possible in the caseN ≥2.

p E

E=p

π 2

Figure 1: Curve of energyE as a function of the momentumpin the one-dimensional case.

1.2 From Gross–Pitaevskii to Landau–Lifshitz

The results of this paper have been motivated by the numerical simulations in [40], where the authors determine a branch of nontrivial solutions of (TWc), axisymmetric around the x1-axis, for any speed c ∈ (0,1) in dimensions two and three. They also conjecture that there is no nontrivial finite energy solution of (TWc) forc≥1. Forcsmall, the existence of these traveling waves has been proved rigorously by Lin and Wei [33]. The branch of solutions is depicted in Figure 2. We see that the curve has a nonzero minimum, which represents the minimal energy in Theorem 1.1.

0 15 30 45 60

15 30

p E

c→0

c→1

E=p

c≈0.78

Figure 2: Curve of energyE as a function of the momentumpin the two-dimensional case.

Some properties of the solutions found in [40] are very similar to those of the traveling waves for the Gross–Pitaevskii equation obtained numerically by Jones, Putterman and Roberts [27, 26] and studied rigourously in [1, 2, 35]. In fact, if uis a solution of (TWc), the stereographic variable

ψ= u1+iu2

1 +u3

, satisfies

∆ψ+1− |ψ|²

1 +|ψ|²ψ−ic∂1ψ= 2 ¯ψ

1 +|ψ|²(∇ψ)², (1.4)

that seems like a perturbed equation for the traveling waves for the Gross–Pitaevskii equation, namely

∆Ψ + (1− |Ψ|²)Ψ−ic∂1Ψ = 0. (1.5)

However, other properties of the solutions are very different. For instance, the energy-momentum curve for the Gross–Pitaevskii equation tends to zero as the momentum goes to zero in the two-dimensional case, but there exists a minimal energy if N ≥3(see [2, 9]).

From a mathematical point of view, (1.4) is a quasilinear Schrödinger equation meanwhile (1.5) is a semilinear Schrödinger equation. Therefore, it is not clear how to relate both equations. One of the purposes of this paper is to clarify this connection, to show how to exploit the similarities between (1.4) and (1.5), and how to deal with the extra difficulties of equation (TWc). In particular, we will discuss the regularity of the solutions of (TWc), some a priori bounds and their asymptotic behavior as|x| → ∞.

1.3 Sketch of the proof of Theorem 1.1

The starting point of our analysis is that for any solution u∈ E(R^N)of (TWc), there existsR≡R(u) such that we have the lifting

ˇ

u≡u1+iu2=̺e^iθ, onB(0, R)^c, (1.6) where ̺≡p

u²₁+u²₂ =p

1−u²₃ and ̺, θ ∈H˙¹(B(0, R)^c) (see Lemma 2.4). Let χ ∈C^∞(R^N) be such that |χ| ≤1, χ= 0 onB(0,2R)andχ= 1onB(0,3R)^c, ifR >0. In the case thatR= 0, we letχ= 1 onR^N. In this way, we can assume that the function χθand

G= (G1, . . . , GN)≡u1∇u2−u2∇u1− ∇(χθ), (1.7) is well-defined on R^N. Foru= (u1, u2, u3), equation (TWc) reads

−∆u1= 2e(u)u1+c(u2∂1u3−u3∂1u2), (1.8)

−∆u2= 2e(u)u2+c(u3∂1u1−u1∂1u3), (1.9)

−∆u3= 2e(u)u3−u3+c(u1∂1u2−u2∂1u1). (1.10) Then, using (1.8) and (1.9),

div(G) =u1∆u2−u2∆u1−∆(χθ)

=c(∂1u3−u3u·∂1u)−∆(χθ)

=c∂1u3−∆(χθ),

(1.11)

where we used the fact thatu·∂1u= 0. By combining with (1.10), we obtain

∆²u3−∆u3+c²∂₁₁² u3=−∆F+c∂1(divG), onR^N. (1.12) At this point we remark that the differential operator

∆²−∆ +c²∂₁₁²

is elliptic if and only if c≤1, which shows thatc= 1is a critical value for the equation (TWc).

Taking Fourier transform in (1.12), we get

|ξ|⁴+|ξ|²−c²ξ₁² b

u3(ξ) =|ξ|²Fb(ξ)−c XN j=1

ξ1ξjGbj(ξ), (1.13) and hence

b

u3(ξ) =Lc(ξ)



 bF(ξ)−c XN j=1

cξ1ξj

|ξ|²Gbj(ξ)



, (1.14)

where

Lc(ξ) = |ξ|²

|ξ|⁴+|ξ|²−c²ξ²₁. Equivalently, we can write (1.14) as the convolution equation

u3=Lc∗F−c XN j=1

Lc,j∗Gj, (1.15)

where Lbc=Lc and

Lbc,j= ξ1ξj

|ξ|⁴+|ξ|²−c²ξ²₁. (1.16) Similarly, from (1.11) and (1.14), forj ∈ {1, . . . , N},

∂j(χθ) =cL^c,j∗F−c² XN k=1

T^c,j,k∗Gk− XN k=1

R^j,k∗Gk, (1.17)

where

Tb^c,j,k= ξ1ξjξk

|ξ|²(|ξ|⁴+|ξ|²−c²ξ₁²) and Rb^j,k=ξjξk

|ξ|², for allj, k∈ {1, . . . , N}.

The kernelsL^c,L^c,j,R^j,kandT^c,j,kare the same as those appearing in the Gross–Pitaevskii equation (1.5). As a consequence, all the properties valid for the (1.5) depending only in the structure of these kernels, can be transfer to the Landau–Lifshitz equation. For instance, the asymptotic behavior theory developed in [14, 16, 4, 8] can be applied, after proving some algebraic decay of the solutions. We provide a precise statement in Theorem 1.6 at the end of this introduction.

Roughly speaking, the principle used to find a minimal energy for the traveling waves of (1.5) in [2, 9], written in the context of the equation (TWc), is that on one hand a convolution equation such as (1.15) should imply that

ku3kL^p≤C(kukW^k,q)E(u)^γ, (E1) for some q, k∈N, andγ >0. On the other hand, using (1.8), (1.9), (1.10) and integrating by parts one should get an a priori bound for the energy of the form:

E(u)≤C(kukW^ℓ,r)ku3k^δL^p, (E2) for some ℓ, r∈N, andδ >0. By putting together (E1) and (E2),

E(u)≤C(kukW^ℓ,r)C(kukW^k,p)^δE(u)^γδ.

Notice that we can assume thatE(u)>0because uis not constant. Ifγδ >1and if

C(kukW^ℓ,r)C(kukW^k,p)^δ≤M, (E3) for some constantM independent ofuandc, we can conclude that

1

M^1/(γδ−1) ≤E(u), so that we have the existence of a minimal energy.

In conclusion, we have reduced the proof of Theorem 1.1 to the proof of the estimates (E1), (E2) and (E3), for someγ, δ >0 such thatγδ >1.

1.3.1 Estimate(E3) and regularity of traveling waves Let us consider the quasilinear elliptic system,

∆u=f(x, u,∇u), in Ω,

where Ωis a smooth domain and f is a smooth function with quadratic growth

|f(x, z, p)| ≤A+B|p|².

We notice that the square-gradient term prevents us from invoking the usual elliptic regularity esti- mates. However, well-known regularity results imply that everycontinuous solution inH¹(Ω)belongs to H_loc^2,2(Ω)∩C_loc^0,α(Ω) (see [12, 31, 5, 28]), but in general we do not have nice a priori estimates such as in

theL^p-regularity theory because theH_loc^2,2(Ω)-norm depends on the modulus of continuity of the u. To exemplify this point, let us consider the harmonic map equation

−∆v=|∇v|²v, in Ω, v∈S². (1.18)

LetΩ =R^N,N ≥2, and assume that there existsC1, C2, α >0 such that

k∇vkL^∞(R^N)≤C1+C2k∇vk^αL²(R^N). (1.19) Note thatk∇vk²_L²_(R^N₎is the energy associated to (1.18). Since the function vλ(x) =v(λx), λ >0, also solves (1.18), we conclude thatvλ satisfies (1.19), but this implies

k∇vkL^∞(R^N)≤ 1 λ

C1+ C2

λ^α(N−2)/2k∇vk^αL²(R^N)

.

Then, lettingλ→ ∞, we deduce thatvis constant. Therefore an estimate such as (1.19) does not hold for (1.18) and probably neither for (TWc). This a big difference with the semilinear equation (1.5). Indeed, ifΨis a solution of (1.5), then (see [11, 2])

kΨkC^k(R^N)≤C(c, k, N).

In dimensionN = 2, Hélein [22, 23] proved that any finite energy solution of (1.18) is continuous and therefore smooth. In dimensionN ≥3, this result if false. In fact, ifN ≥3,v(x) =x/|x|is a discontinuous finite energy solution and Rivière [43] proved that (1.18) has almost everywhere discontinuous solutions with finite energy.

For these reasons, we need to treat differently the casesN = 2 andN ≥3. In any case, we establish in Section 2 a bound ofk∇ukL^∞(R^N)in terms of the energy, provided that the energy is small enough.

Proposition 1.4. Letc≥0andu∈ E(R²)be a solution of (TWc). Thenu∈C^∞(R²),u3∈L^p(R²)for all p∈[2,∞] and∇u∈W^k,p(R²) for all k∈Nand p∈[2,∞]. Moreover, there exist constantsε0 >0 andK >0, independent ofuandc, such that

ku3kL^∞(R²)≤K(1 +c)E(u)^1/2, (1.20) k∇ukL^∞(R²)≤K(1 +c)E(u)^1/4, (1.21) provided that E(u)≤ε0.

Denoting byU C(R^N)the set of uniformly continuous functions, in the higher dimensional case, we have

Proposition 1.5. Let N ≥ 3, c ≥ 0 and u ∈ E(R^N)∩U C(R^N) be a solution of (TWc). Then u ∈ C^∞(R^N)and∇u∈W^k,p(R^N)for all k∈N andp∈[2,∞]. Moreover, ifN ∈ {3,4}andc∈[0,1], there exist ε0, K, α >0, independent ofuandc, such that

ku3kL^∞(R^N)≤KE(u)^α, (1.22) k∇ukL^∞(R^N)≤KE(u)^α, (1.23) provided that E(u)≤ε0.

As we will show, Propositions 1.4 and 1.5 will be enough to get the universal constantM in estimate (E3). The proof of Proposition 1.5 is the only point of the paper where the condition N ≤4 is used. It is straightforward to verify that if the estimates (1.22) and (1.23) are satisfy for some dimensionN, then Theorem 1.1 holds for this N.

1.3.2 Estimates(E1) and (E2)

In order to prove Theorem 1.1, we can assume thatE(u)is small. Then, by (1.20) and (1.22), we only need to prove (E1) and (E2) for traveling waves such thatku3k ≤1/2.

In Section 3, we will prove that estimate (E1) holds withp= 4,γ= 1 ifN = 2, p= 2, γ= _2(N^2N+3₋₁₎ if N ≥3, and k = 1, q =∞in both cases. The main element in the proof is the study of the Fourier multiplier L^c done by the author in [9] ifN ≥3and by Chiron and Maris [7] if N= 2.

Sections 4 and 5 are devoted to establish some Pohozaev identities and a priori bounds that allow us to obtain estimate (E2). More precisely, under the conditionku3k ≤1/2, we show that

E(u)≤Kku3k^δL^p, withp= 4,δ= 4ifN= 2, andp= 2,δ= 2 ifN ≥3.

1.4 Asymptotic behavior at infinity

As remarked before, the arguments given by Gravejat in [14, 16] apply to (1.15) and (1.17), since they rely mainly on the structure of the kernels. This allows to establish the precise limit at infinity of the finite energy solutions of (TWc).

Theorem 1.6. Let N ≥2 and c ∈ (0,1). Assume that u∈ E(R^N) is a solution of (TWc). Suppose further thatu∈U C(R^N)ifN≥3. Then there exist a constantλ_∞∈Cof modulus one and two functions ˇ

u_∞, u3,∞∈C(S^N−1;R)such that

|x|^N−1(ˇu(x)−λ_∞)−iλ_∞uˇ_∞ x

|x|

→0, (1.24)

|x|^Nu3(x)−u3,∞

x

|x|

→0, (1.25)

uniformly as |x| → ∞. Moreover, assuming without loss of generality thatλ_∞= 1, we have

ˇ

u_∞(σ) = ασ1

(1−c²+c²σ₁²)^N2 + XN j=2

βjσj

(1−c²+c²σ₁²)^N2 , (1.26)

u3,∞(σ) =αc 1

(1−c²+c²σ₁²)^N2 − N σ²₁ (1−c²+c²σ₁²)^N+22

!

− XN j=2

βj

N cσ1σj

(1−c²+c²σ₁²)^N+22 , (1.27) where σ= (σ1, . . . , σN)∈S^N−1,

α= Γ ^N₂

2π^N2 (1−c²)^N−32

2c Z

R^N

e(u)u3dx−(1−c²) Z

R^N

G1(x)dx

and

βj =−Γ ^N₂

2π^N2 (1−c²)^N−12 Z

R^N

Gj(x)dx.

In particular, since the solutions found in [33] are uniformly continuous, Theorem 1.6 applies to those solutions. For the sake of completeness we sketch the proof of Theorem 1.6 in Section 7.

Remark 1.7. The analogous constantsαandβj found in [16] for the asymptotic behavior at infinity of the nontrivial finite energy traveling waves to the Gross–Pitaevskii equation (1.5) can be written in terms the energy and (vectorial) momentum. Later, Wei and Yao [47] have proved that all the coordinates of the momentum of these solutions are zero, except the first one. As a consequence the traveling waves are asymptotically axisymmetric (see [47] for details). It is not clear if the arguments in [47] can be generalized in the context of the Landau–Lifshitz equation. However, we think that the same conclusion holds in our case, that isβj= 0, for all2≤j≤N, which would imply the asymptotically axisymmetry of the solutions of (TWc).

Notations. We use the standard notations “·” and “×” for the inner and cross product, respectively.

Fory∈R^N andr≥0,B(y, r)orBr(y)denote the open ball of centery and radiusr(which is empty forr= 0). In the case that there is no confusion, we simply putBr.

Givenx= (x1, x2),f :R²→R²,f = (f1, f2), we setx^⊥ = (−x2, x1), andcurl(f) =∂1f2−∂2f1. We also use the skew gradient∇^⊥ = (−∂2, ∂1).

For a functiong:R^ℓ→R³,g= (g1, g2, g3), we define ˇgas the complex-valued functiongˇ=g1+ig2. We identify∇gwith the matrix inR^ℓ,3whose columns are ∇g1,∇g2and∇g3.

LetA= [A1|A2|A3]andA˜= [ ˜A1|A˜2|A˜3]be two matrices inR^ℓ,3, then A: ˜A=

X3 j=1

Aj·A˜j,

and for any vectorb∈R³, A.b∈R³. denotes the standard matrix-vector product.

EitherF(f)or fbstands for the Fourier transform off, namely F(f)(ξ) =fb(ξ) =

Z

R^N

f(x)e^−ix·ξdx.

We also adopt the standard notation K(·,·, . . .) to represent a generic constant that depends only on each of its arguments.

2 Estimates for k u

k

and k∇ u k

In this section we use some of the elements developed to the study of the harmonic map equation. In particular, the next lemma is a consequence of the Wente lemma [48, 6, 46] and Hélein’s trick [22, 23].

Lemma 2.1. Let Ω ⊂R² be a smooth bounded domain and g ∈ L²(Ω). Assume that u ∈ H¹(Ω,S²) satisfies

−∆u=|∇u|²u+g, inΩ. (2.1)

Let r >0 andx∈Ωsuch that B(x, r)⊆Ω. Then for anyi∈ {1,2,3} we have

B(x,r/2)osc ui≤K min

k∇ukL²(B(x,r)),kuikL^∞(∂Br) +k∇uk²L²(B(x,r))

+rkgkL²(B(x,r)) 1 +k∇ukL²(B(x,r))

,

(2.2)

for some universal constantK >0. In particularu∈C(Ω). Moreover, if the trace ofuon∂Ωbelongs to C(∂Ω), thenu∈C( ¯Ω)and

kuikL^∞(Ω)≤ kuikL^∞(∂Ω)+K(Ω)(k∇uk²L²(Ω)+kgkL²(Ω)(1 +k∇ukL²(Ω))), (2.3) for some constant K(Ω)depending only on Ω.

Proof. As for the standard harmonic maps, we recast (2.1) as

−∆ui= X3 j=1

vi,j· ∇uj+gi, inΩ, i= 1,2,3, (2.4) where vi,j=ui∇uj−uj∇ui. Then

div(vi,j) =ujgi−uigj and kdiv(vi,j)kL²(Ω)≤2kgkL²(Ω). (2.5) Let us considerhi,j∈H²(Ω)the solution of

(∆hi,j= div(vi,j), in Ω,

hi,j= 0, on∂Ω. (2.6)

Thus

k∇hi,jkL²(Ω)≤ kvi,jkL²(Ω)≤2k∇ukL²(Ω). (2.7) Sincediv(vi,j− ∇hi,j) = 0, withvi,j− ∇hi,j∈L²(Ω), there existswi,j∈H¹(Ω)(see e.g. [13, Thm 2.9]) such that

vi,j=∇hi,j+∇^⊥wi,j, inΩ. (2.8)

Now we decomposeuas ui=φi+ϕi+ψi, where φi, ϕi, ψi are the solutions of the equations (−∆φi= 0, inU,

φi =ui, on∂U, (2.9)

(−∆ϕi=∇hi:∇u+gi, inU,

ϕi = 0, on∂U, (2.10)

(−∆ψi=∇^⊥wi:∇u, in U,

ψi= 0, on∂U, (2.11)

where∇hi= [∇hi,1| ∇hi,2| ∇hi,3],∇^⊥wi= [∇^⊥wi,1| ∇^⊥wi,2| ∇^⊥wi,3], andU is an open smooth domain such that Br⊆U ⊆Ω. We now prove (2.2) for r= 1, supposing thatB1⊆U, since then (2.2) follows from a scaling argument. First, invoking Theorem A.1 we have that

Bosc1/2

φi ≤Kminn

k∇φikL²(B_2/3),kφikL²(B_2/3)

o. (2.12)

Also, some standard computations and the maximum principle yield

k∇φikL²(U)≤ k∇uikL²(U) and kφikL^∞(U)≤ kuikL^∞(∂U). (2.13) Thus from (2.12) and (2.13) we conclude that

Bosc1/2

φi≤Kmin

k∇uikL²(U),kuikL^∞(∂U) . (2.14) Forϕi, Theorem A.2 gives

oscB1

ϕi≤K(k∇h· ∇ukL^3/2(B1)+kgkL²(B1)). (2.15) To estimate the first term in the r.h.s. of (2.15), we use the Hölder inequality

k∇hi:∇ukL^3/2(B1)≤ k∇hkL⁶(B1)k∇ukL²(B1), (2.16) and the Sobolev embedding theorem

k∇hkL⁶(B1)≤K(k∇hkL²(B1)+kD²hkL²(B1)). (2.17) By using (2.5), (2.15), (2.16), (2.17) and L²-regularity estimates for (2.6), we are led to

oscB1

ϕi≤KkgkL²(B1)(1 +k∇ukL²(B1)). (2.18) Similarly, sinceW^2,p(U)֒→C( ¯U), for allp >1, we also have

kϕikC( ¯U)≤K(U)kgkL²(U)(1 +k∇ukL²(U))). (2.19) To estimateψi we invoke the Wente estimate (see [46], [23]), so that

kψikC( ¯U)+ osc

U ψi≤Kk∇wkL²(U)k∇ukL²(U)≤Kk∇uk²L²(U), (2.20) where we have used (2.7) and (2.8) for the last inequality.

Therefore, takingU =B1and putting together (2.14), (2.18) and (2.20), we conclude (2.2) withr= 1.

If the trace of u on ∂Ω belongs to C(∂Ω), we take Ω = U and then from (2.9) we deduce that φi∈C²(Ω)∩C( ¯Ω). Sinceϕi, ψi∈C( ¯Ω), we conclude thatui∈C( ¯Ω)and (2.3) follows from (2.13), (2.19) and (2.20).

Lemma 2.2. Let y∈R²,r >0 andBr≡B(y, r). Assume thatu∈H¹(Br,S²) satisfies

−∆u=|∇u|²u+f(x, u(x),∇u(x)), inBr, (2.21) where f is a continuous function such that |f(x, z, p)| ≤ C1+C2|p|, for some constants C1, C2≥0, for a.e. x∈Br,z∈R³,p∈R^3×3. Suppose that

A≡A(u, r)≡oscBru(1 +r²(C1+C₂²)) 1−3 oscBru ≤ 1

32. (2.22)

Then

kD²ukL²(Br/2)+k∇uk²L⁴(Br/2)≤Kr⁻¹ k∇ukL²(Br)+kgkL²(Br)

, (2.23)

whereg(x) =f(x, u(x),∇u(x)). Assume further thatf(x, z, p) = ˜f(x) +Rf(x, z, p), for some continuous functions f˜, Rf, such that |Rf(x, z, p)| ≤ C3|p|, for some constant C3 ≥0, for a.e. x∈ Br, z ∈R³, p∈R^3×3. Then,

k∇ukL^∞(B_r/4)≤Kr⁻¹k∇ukL²(Br)+Kr^−2/3

k∇uk²L²(Br)(r⁻²+r^−4/3) +kgk²L²(Br)

+kf˜kL³(Br)+C3r^−1/3k∇uk^1/3_L²_(Br) k∇uk^1/3_L²_(Br)+kgk^1/3_L²_(Br) ,

(2.24)

whereK is some universal constant.

Proof. As mentioned before, Lemma 2.1 and the quadratic growth of the r.h.s. of (2.21) imply that u ∈ H_loc^2,2(Ω). In fact, this could be seen by repeating the following arguments with finite differences instead of weak derivatives. As standard in the analysis of this type of equations, we let ρ∈(0, r)and χ∈C₀^∞(Br), withχ(x) = 1if|x| ≤ρ,

|χ| ≤1 and |∇χ| ≤K/(r−ρ), onBr. (2.25) Then setting η =χ|∇u|, taking inner product in (2.1) with (u−u(x0))η² and integrating by parts we obtain

Z

Br

|∇u|²η²+ 2 Z

Br

(∇u.(u−u(x0)))·(η∇η) = Z

Br

|∇u|²u·(u−u(x0))η²+ Z

Br

η²g·(u−u(x0)). (2.26) Then, using the elementary inequality2ab≤a²+b²,

|η²g·(u−u(x0))| ≤η²(C1+C2|∇u|) osc

Br

(u)≤C1η²osc

Br

(u)+

η²|∇u|²osc

Br

(u) +1

4C₂²η²osc

Br

(u).

In a similar fashion, we estimate the remaining terms in (2.26). Then, using the Poincaré inequality kηkL²(Br)≤ r

j0k∇ηkL²(Br),

where j0≈2.4048is the first zero of the Bessel function, and that|u|= 1, we conclude that Z

Br

|∇u|²η²≤ oscBru(1 +r²(C1+C₂²)) 1−3 oscBru

Z

Br

|∇η|², where we bounded 1/j0and1/(4j0)by 1 to simplify the estimate. Thus,

Z

Br

|∇u|⁴χ²≤A Z

Br

(|D²u|²χ²+|∇u|²|∇χ|²)). (2.27) On the other hand, taking inner product in (2.21) with∂k(χ²∂ku), integrating by parts and summing overk= 1,2, we have

− Z

Br

χ²|D²u|²−2 X

i∈{1,2} j,k∈{1,2,3}

Z

Br

∂jkuiχ∂jχ∂kui= X

i∈{1,2}

Z

Br

(|∇u|²ui+gi)(2χ∇χ∇ui+χ²∆ui).

(2.28) Using again the inequalities2ab≤εa²+b²/εand ab≤εa²+b²/4ε, we are led to

Z

Br

χ²|D²u|²≤ 1 1−3ε

Z

Br

(2 +ε⁻¹)|∇u|²|∇χ|²+ (1 + 4ε⁻¹)|∇u|⁴χ²+ (1 + 4ε⁻¹)χ²|g²|)

. (2.29) Then, minimizing with respect toε, it follows that

Z

Br

χ²|D²u|²≤16 Z

Br

(|∇u|²|∇χ|²+|∇u|⁴χ²+χ²|g²|). (2.30)

By combining (2.25), (2.27) and (2.30), we infer that Z

Bρ

|∇u|⁴≤ KA 1−16A

1 (r−ρ)²

Z

Br

|∇u|²+ Z

Br

|g|²

, (2.31)

Z

Bρ

|D²u|²≤ K 1−16A

1 +A (r−ρ)²

Z

Br

|∇u|²+ Z

Br

|g|²

. (2.32)

Takingρ=r/2 and using thatA≤1/32, (2.23) follows.

Now we decomposeuas ui=φi+ψi, where

(−∆φi = 0, inBr/2,

φi=ui, on∂Br/2, (2.33)

(−∆ψi=|∇u|²ui+ ˜fi+ (Rf(x, u,∇u))i, in Br/2,

ψi= 0, on∂Br/2, (2.34)

Sinceφi is a harmonic function,

k∇φikL^∞(Br/4)≤Kr⁻¹k∇φikL²(Br/2), so that using also (2.13), we obtain the estimate

k∇φikL^∞(Br/4)≤Kr⁻¹k∇uikL²(Br/2). (2.35) Forψi, we recall that using theL^p-regularity theory for the Laplacian and a scaling argument, the solution v∈H₀¹(BR)of the equation−∆v=hsatisfies

k∇vkL^∞(BR)≤K(p)R^1−2/pkhkL^p(BR), for allp >2.

Applying this estimate withp= 3to (2.34), we get k∇ψikL^∞(B_r/2)≤Cr^−2/3

k∇uk²L⁶(Br/2)+kf˜kL³(B_r/2)+C3k∇ukL³(B_r/2)

. (2.36)

Also, by the Sobolev embedding theorem, we have k∇ukL⁶(B_r/2)≤K

kD²ukL²(B_r/2)+r^−2/3k∇ukL²(B_r/2)

. (2.37)

Therefore, by putting together (2.23), (2.35), (2.36), (2.37) and the interpolation inequality k∇ukL³(B_r/2)≤ k∇uk^1/3_L²_(Br/2)k∇uk^2/3_L⁴_(Br/2),

we deduce (2.24).

Now we turn back to equation (TWc). By setting Ex,r(u) =

Z

B(x,r)

e(u),

we obtain the following result.

Corollary 2.3. There exist ε0 > 0 and a positive constant K(ε0), such that for any c ≥ 0 and any u∈ E(R²)solution of (TWc)satisfying

Ex,r(u)≤ε0, for somex∈R² andr∈(0,1], we have

B(x,r/2)osc u≤K(ε0)(1 +c)Ex,r(u)^1/2, (2.38) k∇ukL^∞(B(x,r/4))≤K(ε0)(1 +c)Ex,r(u)^1/4r^−2/3. (2.39) In particular, if E(u)≤ε0, then (1.20) and (1.21) hold.

Proof. Estimates (2.38) and (2.39) follow from Lemmas 2.1 and 2.2. Then, taking r= 1, we conclude that (1.21) holds. Now we turn to (1.20). For anyy∈R² we have

2E(u)≥ Z

B(y,1/2)

u²₃≥π 4 min

B(y,1/2)|u3|². (2.40)

On the other hand, by Lemma 2.1,

B(y,1/2)max |u3| ≤ osc

B(y,1/2)u3+ min

B(y,1/2)|u3| ≤K(1 +c)E(u)^1/2+ min

B(y,1/2)|u3|. (2.41) By combining (2.40) and (2.41), we are led to (1.20).

Proof of Proposition 1.4. Since uhas finite energy, for everyε >0, there existsρ >0 such that for all y∈R²

Ey,ρ(u)≤ε. (2.42)

In fact, sincee(u)∈L¹(R²), by Lemma A.3, for every ε >0 we can decomposee(u) =e1,ε(u) +e2,ε(u) such that

ke1,ε(u)kL¹(R²)≤ε/2 and ke2,ε(u)kL^∞(R²)≤Kε, for some constantKε. Then for anyy∈R²,

ke2,ε(u)kL¹(B(y,ρ))≤Kεπρ². Taking

ρ= ε

2Kεπ 1/2

,

we obtain (2.42). Thus, invoking Corollary 2.3, with ε=ε0andr= min{1, ρ}, we conclude that k∇ukL^∞(B(y,r/4))≤K(ε0)(1 +c)E(u)^1/4, for ally∈R².

Thereforeu∈W^1,∞(R²), withk∇ukL^∞(R²)≤K(ε0)(1 +c)E(u)^1/4.Differentiating (TWc), we find that v=∂ju,j= 1,2, satisfies

Lλ(v)≡ −∆v−2(∇u:∇v)u−c(u×∂1v) +λv=|∇u|²v+ 2u3v3u+u²₃v−v3e3+c(v×∂1u) +λv, in R². Since∇u∈L^∞(R²)∩L²(R²), we deduce that the r.h.s. of the formula above belongs toL²(R²).

Therefore takingλ >0 large enough, we can invoke the elliptic regularity theory for linear systems and deduce that v∈W^2,2(R²). Then, by the Sobolev embedding theorem, D²u∈L^p(R²), for all p∈[2,∞) and a bootstrap argument allows us to conclude that ∇u∈W^k,p(R²)for allk∈Nandp∈[2,∞].

The estimates (1.20) and (1.21) are given by Corollary 2.3.

Proposition 1.4 shows thatu3is uniformly continuous, so that u3(x)→0, as|x| → ∞. In particular ubelongs to the spaceE˜(R²), where

E˜(R^N) ={v∈ E(R^N) :∃R≥0 s.t. kv3kL^∞(B(0,R))^c)<1}.

In the caseN ≥3, we always suppose thatu∈ E(R^N)∩U C(R^N)and then it is immediate thatu∈E˜(R^N).

Now we recall a well-known result (see e.g. [34, Proposition 2.5]) that provides the existence of the lifting for any function inE˜(R^N).

Lemma 2.4. Let N ≥2 andv∈E˜(R^N). Then there existsR≥0such that v admits the lifting ˇ

v(x) =̺(x)e^iθ(x), onB(0, R)^c, (2.43) where ̺ = p

1−v₃² and θ is a real-valued function. Moreover, ̺, θ ∈ H_loc¹ (B(0, R)^c) and ∇̺,∇θ ∈ L²(B(0, R)^c)).

Corollary 2.5. Let c ≥0 and u∈ E(R²) be a solution of (TWc). Then there is R ≥0 such that the lifting u(x) =ˇ ̺(x)e^iθ(x) holds on B(0, R)^c and satisfies ∇̺,∇θ ∈ W^k,p(B(0, R)^c) for any k ≥ 2 and p∈ [2,∞]. Moreover, there exists a constant ε(c)>0, depending only on c, such that if E(u) ≤ε(c), then we can take R= 0.

Proof. By Proposition 1.4, u∈ E˜(R²) and then Lemma 2.4 gives us the existence of the lifting, whose properties follow from Proposition 1.4 and the identity

|∇uˇ|²=̺²|∇θ|²+|∇̺|², onB(0, R)^c, (2.44) noticing that 1− kv3k²L^∞(B(0,R)^c) = inf{̺(x)² : x∈B(0, R)^c} >0. The last assertion is an immediate consequence of (1.20).

In the caseN ≥3, some regularity for the solutions of the equation (2.1) can be obtained considering that uis a stationary solution in the sense introduced by R. Moser in [37].

Definition 2.6. LetΩ⊂R^N be a smooth bounded domain andg∈L^p(Ω;R³). A solutionu∈H¹(Ω;S²) of (2.1)is calledstationary if

div(|∇u|²ej−2∇u.∂ju) = 2∂ju·g, in Ω, (2.45) for all j∈ {1, . . . , N}in the distributional sense.

If we suppose thatuis a smooth solution of (2.1), then

div(|∇u|²ej−2∇u.∂ju) =−2∆u·∂ju= 2g·∂ju,

so it is a stationary solution. However not every solution u∈ H¹(Ω;S²) of (2.1) satisfies (2.45). The advantage of stationary solutions is that they satisfy a monotonicity formula that allows to generalize some standard results for harmonic maps. However, when g belongs only to L²(Ω), the regularity estimates hold only forN ≤4.

Lemma 2.7 ([37]). Let N ≤4 and y ∈ R^N . Assume that u ∈ H¹(B(y,1);S²)∩W^1,4(B(y,1)) is a stationary solution of (2.1), with Ω =B(y,1) andg∈L²(B(y,1)). Then there existK >0 andε0 >0, depending only on N,such that if

k∇ukL²(B(y,1))+kgkL²(B(y,1))=ε≤ε0, we have

k∇ukL⁴(B(y,1/4))≤Kε²¹.

Applying this result to equation (TWc), we are led to the following estimate.

Lemma 2.8. LetN≤4. There existK >0andε0>0, depending only onN, such that for any solution u∈ E(R^N)∩C^∞(R^N)of (TWc), withc∈[0,1], satisfyingE(u)≤ε0,we have

k∇ukL⁴(B(x,1))≤KE(u)^1/2.

Now we are in position to complete the regularity result in higher dimensions stated in the introduc- tion.

Proof of Proposition 1.5. Recalling again a classical results for elliptic systems with quadratic growth (see [5, 31, 28]),u∈U C(R^N)yields thatu∈C^∞(R^N). This is due to the fact that now we are assuming that uis uniformly continuous and then we can choose r >0 small such that the oscillation ofuon the ballB(y, r)is small, uniformly iny. Then we can make the quantityA(u, r)defined in (2.22) as small as needed and repeat the first part of the proof of Lemma 2.2 to conclude that for ally∈R^N

kD²ukL²(B(y,r/2))+k∇uk²L⁴(B(y,r/2))≤K(N)r⁻¹ k∇ukL²(B(y,r))+ku3kL²(B(y,r))

, (2.46)

for some constantK(N)andr >0small enough, independent ofy. At this stage we note that we cannot follow the rest of the argument of Lemma 2.2, since it relies on the two-dimensional Sobolev embeddings.

However, it is well-known that using (2.46) it is possible to deduce that ∇u ∈ L^p(R^N), for all p≥ 2.

More precisely, as discussed before, there exists r∈(0,1]such that

B(y,2osc^Nr)u≤ 1

8(1 +c)(2N−1), for ally∈R^N. (2.47)

Then, by iterating Lemma A.5, we have Z

B(y,r)|∇u|^2N+2≤K(N)(1 +c)^2NE(u)

r^2N , for ally∈R^N. (2.48) By proceeding as in the proof of Lemma 2.2, we decomposeui asui=φi+ψi, where

(−∆φi= 0, inB(y, r), φi=ui, on∂B(y, r),

(−∆ψi=|∇u|²ui+u²₃ui−δi,3u3+cu×∂1u, in B(y, r),

ψi= 0, on∂B(y, r).

In view of (2.48), elliptic regularity estimates imply thatψi∈W^2,N+1(B(y, r))and then by the Sobolev embedding theorem we can establish an upper bound for k∇ψikL^∞(B(y,r)) in terms of powers of E(u).

Sinceφiis a harmonic function, we obtain a similar estimate forφi as in the proof of Lemma 2.24. Then we conclude that∇u∈L^∞(R^N), so that, by interpolation, ∇u∈L^p(R^N), for allp∈[2,∞]. Proceeding as in the proof of Proposition 1.4, we conclude that∇u∈W^k,p(R^N), for allk∈Nandp∈[2,∞].

Now we turn to (1.22) and (1.23). Let us first takeN = 3and ε0 given by Lemma 2.8, such that E(u)≤ε0. Then, by the Morrey inequality,

B(y,1/2)osc u≤Kk∇ukL⁴(B(y,1))≤KE(u)^1/2, (2.49) for all y∈R³ and for all c∈[0,1]. Takingε0 smaller if necessary, (2.47) holds withr= 1/16and then so it does (2.48) (with r = 1/16). Hence the previous computations give a bound for ∇u ∈ L^∞(R³) depending only on E(u), which yields (1.23).

In order to prove (1.22), we estimate the minimum of|u3|onB(y,1/2)as in (2.40), and using (2.49) we conclude that

B(y,1/2)max |u3| ≤KE(u)^1/2, which implies (1.22).

It only remains to consider the case N = 4. Note that the r.h.s. of (2.47) is less than or equal to 1/112, forc∈[0,1]. Letr_∗>0be the maximal radius given by the uniform continuity ofufor this value, i.e.

r_∗= sup

ρ >0 :∀x, z∈R⁴,|x−z| ≤ρ⇒ |u(x)−u(z)| ≤1/112 . (2.50) We claim that r_∗ ≥1/2 for ε0 small. Arguing by contradiction, we suppose that 0 < r_∗ <1/2. Since (A.2) is satisfied for anyy∈R⁴, withr=r_∗ ands= 2, Lemma A.5 implies that

k∇uk⁶L⁶(B(y,r∗/2))≤8

1 +16 r²_∗

k∇uk⁴L⁴(B(y,r∗)).

Since0< r_∗<1/2, the Morrey inequality implies that

B(y,rosc∗/4)u≤K1r_∗¹³k∇ukL⁶(B(y,r∗/2)) ≤K2r_∗¹³

1 +16 r²_∗

¹₆

k∇ukL²³⁴(B(y,r∗))≤K3E(u)¹³, (2.51) where we have used Lemma 2.8 for the last inequality and K3 >0 is a universal constant. Finally we notice that there exists a universal constantℓ∈Nsuch that for anyx∈R⁴, there is a collection of points y1, y2, . . . , yℓ∈R⁴ such that

B(x,2rosc∗)u≤ Xℓ k=1

B(yosck,r∗/4)u.

Thus, using (2.51),oscB(x,2r∗)u≤ℓK3E(u)¹³.Takingε0≤1/(112ℓK3)³, we get thatoscB(x,2r∗)≤1/112, which contradicts the definition of r_∗. Therefore, oscB(x,1/2)u ≤1/112, for all x∈ R⁴. Moreover, the same argument shows that

B(y,1/8)osc u≤KE(u)^1/3, for ally∈R⁴, and then (1.22) and (1.23) follow as before.

3 Properties related to the kernels and the convolution equations

Through this section, we fix u∈E˜(R^N)∩U C(R^N)a solution of (TWc) for a speed c ∈[0,1]. We also use the notation introduced in Subsection 1.3.

We start recalling the following result forLc. Lemma 3.1. [9, 7]. For any c∈(0,1], we have

kLcfbkL^4/3(R²)≤11kfkL¹(R²), (3.1) and

kLcfbkL²(R^N)≤K(N)kfk

L

2(2N−1)

2N+3 (R^N), if N ≥3. (3.2)

Proof. By the Plancherel identity, the estimate (3.2) is exactly [9, Lemma 4.3]. To prove (3.1), we note that

kLcfbkL^4/3(R²)≤ kLckL^4/3(R²)kfbkL^∞(R²)≤ kLckL^4/3(R²)kfkL¹(R²). Then it only remains to compute kLckL^4/3(R²). Using polar coordinates, we have

kLck^4/3_L^4/3_(R²₎= 4 Z _∞

0

Z π/2 0

r dθ dr

(r²+ 1−c²cos²(θ))⁴/3 = 6 Z π/2

0

dθ

(1−c²cos²(θ))^1/3

≤6 Z π/2

0

dθ

(1−cos²(θ))^1/3 = 6 Z π/2

0

dθ

sin^2/3(θ) = 3B 1

6,1 2

,

(3.3)

where B denotes the Beta function. Using thatB(x, y) = Γ(x)Γ(y)/Γ(x+y),we conclude that kLckL^4/3(R²)≤

3Γ(1/6)Γ(1/2) Γ(2/3)

3/4

≤11. (3.4)

From (3.3) and (3.4), (3.1) follows.

Now we are able to prove the exact form of estimate (E1) stated in the introduction and also further integrability foru3.

Proposition 3.2. LetN ≥2andc∈(0,1). Thenu3∈L^p(R^N), for all p∈(1,2). Moreover, ifc∈(0,1]

andku3kL^∞(R^N)≤1/2, we have

ku3kL⁴(R²)≤54ku3kL^∞(R^N)E(u), (3.5) and

ku3kL²(R^N)≤K(N)ku3kL^∞(R^N)

1 +k∇uk

2N−5 2(2N−1)

L^∞(R^N)

E(u)^{2(2N−1)2N+3} , ifN≥3. (3.6) Proof. Let us recall that by Propositions 1.4 and 1.5, and noticing that

G=−u²₃∇θ, onB(0,3R)^c, (3.7)

we infer thatF, G1, G2∈L^p(R^N), for allp∈[1,∞]. On the other hand, from the Riesz-operator theory, the functions ξ7→ξiξj/|ξ|² areL^q-multipliers for anyq∈(1,∞) and1 ≤i, j≤N. Since Lc is also an L^q-multiplier for anyq∈(1,∞)(see [14]), from (1.14) we conclude thatu3∈L^q(R^N), for allq∈(1,∞).

We turn now to the proof (3.5). Using (1.14) and the Hausdorff–Young inequality kηkL^q(R^N)≤p^1/2pq^−1/2qkbηkL^p(R^N), p∈[1,2], q=p/(p−1), withp= 4/3, and (3.1) we obtain

ku3kL⁴(R²)≤ kbu3kL^4/3(R²)≤11

kFkL¹(R²)+kG1kL¹(R²)+1

2kG2kL¹(R²)

, (3.8)

where we have used thatξ₁²/|ξ|²≤1andξ1ξ2/|ξ|²≤(ξ₁²+ξ₂²)/(2|ξ|²)≤1/2 for the last inequality.