Smoothing estimates for the variable coefficients Schrödinger equation with electromagnetic potentials

J. Math. Anal. Appl. 402 (2013) 286–296

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Smoothing estimates for the variable coefficients Schrödinger equation with electromagnetic potentials Federico Cacciafesta SAPIENZA — Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy

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abstract

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Article history: Received 20 August 2012 Available online 30 January 2013 Submitted by Jie Xiao

In this paper we develop the classical multiplier technique to build up a virial identity for the electromagnetic variable coefficients Schrödinger equation. Following the strategy of (Fanelli and Vega, 2009) [13] we shall use such an identity to prove smoothing estimates for the associated flow in a perturbative setting. © 2013 Elsevier Inc. All rights reserved.

Keywords: Smoothing estimates Variable coefficients Schroedinger Perturbative

1. Introduction and main results We consider the electromagnetic Schrödinger equation with variable coefficients



iut (t , x) = Hu(t , x) u(0, x) = f (x),

(1.1)

where u : R1+n → C and the Hamiltonian H is defined as Hu = − ∂jb ajk (x)∂kb u







+ V (x)u.

Here the covariant derivatives ∂ b = (∂1b , . . . , ∂nb ) associated to the (divergence-free) magnetic potential b = (b1 , . . . , bn ) : Rn → Rn are given by

∂kb =

∂ + ibk (x), ∂ xk

while a = a(x) = [ajk (x)]1,n is a symmetric matrix of real valued functions satisfying C −1 Id ≤ a(x) ≤ C Id,

x ∈ Rn

(1.2)

for some C > 0. Using the summation convention over repeated indices, we associate to the matrix a(x) the bilinear form a(v, w) = ajk (x)v j zk and the operator Aψ = ∂j (ajk (x)∂k ψ).

E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2013.01.040

(1.3)

F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

287

A well-known phenomenon for the Schrödinger equation is the gain of regularity of the solutions with respect to the initial data: indeed, the free flow satisfies the following classical estimate sup R >0

1 R

+∞





|∇ eit ∆f |2 ≤ ∥f ∥

|x|≤R

0

(1.4)

1

˙2 H

(see [7,17,18]). In recent years the case of potential perturbations or variable coefficients equations has attracted increasing interest, and several generalizations of estimate (1.4) have been investigated (see [5,6,8,4,10,15,16,9] and references therein). The aim of this paper is to develop the multiplier method in order to prove virial identities for Eq. (1.1), and to use such identities to prove smoothing estimates for the electromagnetic Schrödinger flow in dimension n = 3 with variable coefficients, in analogy with the results in [1,2], at least in a perturbative setting. Although the smoothing estimates for the Schrödinger equation with variable coefficients are not entirely new (see [16]), to the best of our knowledge our virial identities are original and may be interesting in view of the applications to more general dispersive estimates (see Remark 1.6). In order to ensure self-adjointness for the Hamiltonian H, we will make the following abstract assumptions:

   • The principal part H0 u = − ∂jb ajk (x)∂kb u of the Hamiltonian H is essentially self-adjoint on L2 (Rn ) with form domain    2 D(H0 ) = f : f ∈ L , a(∇b f , ∇b f ) < ∞ . • The potential V is a perturbation of H0 in the Kato–Rellich sense, i.e. there exists an ε > 0 such that ∥Vf ∥L2 ≤ (1 − ε)∥Hf ∥L2 + C ∥f ∥L2 , for all f ∈ D(H ). These assumptions, together with (1.2), allow us to define via spectral theorem the linear propagator in a standard way; moreover the perturbed norms s

∥f ∥H˙ s = ∥H 2 f ∥L2

(1.5)

are conserved by the respective flows, and so they yield the conservation laws

∥eitH f ∥H˙ s = ∥f ∥H˙ s ,

s ≥ 0.

The first result we prove is a general virial identity for the variable coefficients Schrödinger equation. In order to state it we introduce the following notations, where we use implicit summation over repeated indices:

 • xˆ = |xx| , ⟨x⟩ = 1 + |x|2 • Vra = a(ˆx, ∇ V ), Baτ = a(ˆx, B), D2a φ = akj alm ∂j ∂m φ . Theorem 1.1. Let φ : Rn → R be a radial, real valued multiplier, and let

ΘS (t ) =

 Rn

φ|u|2 dx.

(1.6)

Then the solution u of (1.1) with initial condition f ∈ L2 , Hf ∈ L2 satisfies the following virial-type identity:

¨ S (t ) = 4 Θ

 Rn

∇b uD2a φ∇b udx −

 +2 Rn

 Rn

|u|2 A2 φ dx − 2

 Rn

φ ′ Vra |u|2 dx + 4I

 Rn

uφ ′ a(∇b u, Baτ )dx

[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx.

(1.7)

Remark 1.1. Notice that the quantities appearing in (1.7) are natural generalizations of the analogous quantities that appear in the flat case, i.e., the standard Jacobian D2 φ , the radial component of the electric potential (which in the unperturbed case is simply Vr = xˆ · ∇ V ) and the tangential vector field Bτ = xˆ B. On the other hand, the last two terms containing derivatives of the coefficients are new and do not appear in the classical computation. We can prove a similar result for the electromagnetic, variable coefficients wave equation, i.e. utt (t , x) = Hu(t , x) u(0, x) = f (x) ut (0, x) = g (x)



with the same notations as before. We can prove the following

(1.8)

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F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

Theorem 1.2. Let φ, ψ : Rn → R be two radial, real valued multipliers, and let

ΘW ( t ) =

   1 2 2 φ|ut | + φ a(∇b u, ∇b u) − (Aφ)|u| dx +



2

Rn

Rn

|u|2 (V φ + ψ)dx.

(1.9)

Then for the solution u of (1.8) with initial data f , g ∈ L2 , Hf , Hg ∈ L2 the following virial-type identity holds:

   1 ∇b uD2a φ∇b udx − |u|2 A2 φ dx + 2 |ut |2 ψ dx − 2 a(∇b u, ∇b u)ψ dx n 2 Rn Rn Rn R   + |u|2 Aψ dx + (2ψ V − φ ′ Vra )|u|2 dx + 2I uφ ′ a(∇b u, Baτ )dx

¨ W (t ) = 2 Θ



Rn

Rn

 +2 Rn

Rn

[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx.

Remark 1.2. We point out that, to the best of our comprehension, the term the analogous computation in [13].



(1.10)

2ψ V |u|2 that appears here was overlooked in

As a standard application of the above virial identities (see [13,3]) we prove smoothing estimates for the Schrödinger equation in dimension 3 in a perturbative setting. Similar estimates can be proved for the wave equation with minor modifications, but we prefer not to pursue this topic here. The assumptions on the potentials will be expressed as usual in terms of the Morrey–Campanato norms ∞

 ∥|f ∥C α =

ρ α sup |f (x)|dρ.

(1.11)

|x|=ρ

0

Theorem 1.3. Let n = 3, assume the matrix [ajk ] has the form ajk = δjk + ε˜ajk , satisfies (1.2) and the following assumption

 sup j,k

sup |∂ γ a˜ jk |

 .

|γ |=p

C

⟨x⟩p+

,

p = 0, . . . , 3, γ ∈ Nn .

(1.12)

Assume moreover that the potentials are such that

∥|Baτ |2 ∥C 3 + ∥(Vra )+ ∥C 2 ≤

1−ϵ 2

,

|V (x)| .

cε

|x|2

(1.13)

for some cε sufficiently small. Then the solution u of (1.1) corresponding to f ∈ L2 , Hf ∈ L2 satisfies the estimate sup

1 − Cε

R >0

R



+∞



0

|∇b u|2 dxdt . ∥f ∥2 |x|≤R

1

(1.14)

H˙ 2

for some constant Cε < 1. Remark 1.3. The second assumption of smallness on the potential in (1.13), not necessary in the unperturbed case, seems to be removable at the cost of some additional technicalities. Remark 1.4. Hypothesis (1.12) is naturally satisfied if for instance a˜ jk = ⟨x⟩−p δjk , with p > 0. Remark 1.5. Following the strategy of [13], it is possible to obtain smoothing estimates for both the Schrödinger and wave equation in higher dimensions n ≥ 4 using a suitable modifications of the multiplier used in the proofs of the previous theorems, under slightly different assumptions on the potential. This problem will be pursued in forthcoming papers. Remark 1.6. The non perturbative case remains an interesting open problem: we do not know whether virial identities (1.7) and (1.10) could be used to prove smoothing estimates in cases when the matrix a is not close to the identity. Even the radial case, in which the choice of the multiplier seems to be forced to the standard one, seems to require some new tools. Furthermore, it would be very interesting to try to adapt the choice of multiplier to the scalar product a, regarding it as a

F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

289

new metric (see for instance [14] in the case of the hyperbolic space). We recall also that very few results exist when the coefficients depend also on time (and then the estimates obtained are only local). It is possible to adapt the above techniques to study this case, at least in special situations, including some equations with nonlocal nonlinearities (see e.g. [11,12]). These topics will be the object of future work. The paper is organized as follows: in Sections 2 and 3 we prove respectively Theorems 1.1–1.3, while the final section is devoted to the proof of a quite general weighted magnetic Hardy’s inequality that is needed in several steps of the paper. 2. Proofs of virial identities 2.1. Schrödinger equation: proof of Theorem 1.1 3

Following the standard well-known strategy, we can easily compute for a solution u ∈ H 2 of (1.1)

˙ S (t ) = −i⟨u, [H , φ]u⟩ Θ

(2.1)

¨ S (t ) = −⟨u, [H , [H , φ]]u⟩ Θ

(2.2) 2

where [·, ·] is the standard commutator and ⟨·, ·⟩ is the Hermitian product in L . Denoting with T = −[H , φ] it is trivial to compute explicitly, by the Leibnitz formula, T = Aφ + 2a(∇φ, ∇b ).

(2.3)

Hence we can rewrite (2.2) as

¨ S (t ) = ⟨u, [H , T ]u⟩ Θ

(2.4)

with T given by (2.3). We thus need to compute the commutator [H , T ]; we divide it into three parts as follows

[H , T ]u = [H0 , 2a(∇φ, ∇b )]u + [H0 , A2 ψ]u + [V , T ]u = −I − II + III .

(2.5)

The term III is trivial: indeed we immediately have

with

[V , T ]u = 2[V , a(∇φ, ∇b )]u = −2a(∇φ, ∇ V )u = −2φ ′ Vra u

(2.6)

= a(ˆx, ∇ V ), so that the corresponding term in (2.2), obtained integrating (2.8) multiplied by u gives  −2 φ ′ Vra |u|2 dx.

(2.7)

Vra

Rn

We now turn to II. Writing the components in details we have

  [H0 , A2 φ]u = −∂jb ajk ∂kb (∂l (alm ∂m φ)u) + ∂l (alm ∂m φ)∂jb (ajk ∂kb u) = −II1 + II2 .

(2.8)

For the term II1 we have II1 = ∂jb ajk ∂k Aφ u + ajk Aφ∂kb u





= A2 φ · u + 2ajk ∂k (Aφ)∂jb u + Aφ(∂j ajk )∂kb u + (Aφ)ajk ∂jb ∂kb u.

(2.9)

Term II2 reads instead as II2 = Aφ(∂j ajk )∂kb u + (Aφ)ajk ∂jb ∂kb u.

(2.10)

Hence plugging (2.9) and (2.10) into (2.8) yields

[H0 , A2 φ]u = −A2 φ · u − 2ajk ∂k (Aφ)∂jb u.

(2.11)

Finally we handle the term I. We have

  [H0 , 2a(∇b , ∇φ)]u = −2∂jb ajk ∂kb (alm ∂l φ∂mb u) + 2alm ∂l φ∂mb [∂jb (ajk ∂kb u)] = −2(I1 − I2 ).

(2.12)

We treat separately the two terms. Starting with I1 we have b b b I1 = ∂jb · {ajk (∂k alm )∂l φ∂m u + ajk alm ∂k ∂l φ∂m u + ajk alm ∂l φ∂kb ∂m u}.

We now turn to the last term I2 , that is the most tricky to handle. We need the following remark

∂mb ∂jb = ∂jb ∂mb + i(∂m bj − ∂j bm )

(2.13)

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F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

to write b I2 = alm ∂l φ∂jb [∂m (ajk ∂kb u)] + ialm ∂l φ(∂m bj − ∂j bm )ajk ∂kb u.

We expand the first term beyond again to have b alm ∂l φ∂jb [∂m (ajk ∂kb u)] = ∂jb · alm ∂l φ(∂m ajk )∂kb u + alm ∂l φ ajk ∂mb ∂kb u − ∂j (alm ∂l φ)∂mb (ajk ∂kb u)





  = ∂jb · alm ∂l φ(∂m ajk )∂kb u + alm ∂l φ ajk ∂mb ∂kb u − ∂mb · {(∂j alm )∂l φ ajk ∂kb u + alm ∂j ∂l φ ajk ∂kb u} + ∂j (∂m (alm ∂l φ))ajk ∂kb u. Now we integrate I + II over Rn and use the integration by parts to rearrange the terms in a suitable way, as follows. First of all using again (2.13) we have







2

b b ajk alm ∂l φ∂m ∂k u − ajk alm ∂l φ∂kb ∂mb u · ∂jb u − ialm ∂l φ(∂m bj − ∂j bm )ajk ∂kb u· udx

Rn





= 4I Rn

 = 4I Rn



ajk alm ∂l φ(∂k bm − ∂m bk )∂jb u · udx

uφ ′ a(∇b u, Baτ )dx

(2.14)

where we have defined Baτ = a(ˆx, B). Two other terms give

 Rn

    ∂jb · ajk alm ∂k ∂l φ∂mb u + ∂mb · ajk alm ∂j ∂l φ∂kb u u dx = 4



∇b uD2a φ∇b u dx

(2.15)

 ∂jb · {ajk (∂k alm )∂l φ∂mb u − alm (∂m ajk )∂l φ∂kb u} + ∂mb · {ajk (∂j alm )∂l φ∂kb u} u dx.

(2.16)



2

Rn

where we have defined the distorted Jacobian D2a φ = ajk alm ∂k ∂l φ . The remaining terms are given by





2 Rn

Denoting with ∇ a(w, x) =

 =2 Rn



l

∂l ajk (wj zk ) we can rewrite (2.16) as

[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx.

(2.17)

Thus putting all together, (2.7) and (2.14)–(2.16) and the integration of (2.11) gives (1.7). 2.2. Wave equation: proof of Theorem 1.2 The proof of Theorem 1.2 is analogous to the previous one. We consider the following quantity

ΘW ( t ) =

 Rn

   1 2 2 φ|ut | + φ a(∇b u, ∇b u) − (Aφ)|u| dx + 2

Rn

φ V |u|2 dx +

 Rn

|u|2 ψ dx.

Differentiating (2.18) in time, integrating by parts and using Eq. (1.8) yield, term by term, d



dt  d

φ|ut |2 dx = −2R⟨ut , φ H0 u⟩ − 2R⟨ut , φ Vu⟩;

dt Rn  1 d 2 dt  d

Rn

dt Rn  d dt

φ a(∇b u, ∇b u)dx = +2R⟨ut , φ H0 u⟩ − 2R⟨ut , a(∇φ, ∇b u)⟩;

Rn

(Aφ)|u|2 dx = −R⟨ut , (Aφ)u⟩;

φ V |u|2 dx = 2R⟨ut , φ Vu⟩; |u|2 ψ dx = 2R⟨ut , ψ u⟩.

Recalling the definition of T = −[H , φ] = Aφ + 2a(∇φ, ∇b )

(2.18)

F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

291

we thus immediately have

˙ W (t ) = −R⟨ut , Tu⟩ + 2R⟨ut , ψ u⟩. Θ

(2.19)

Differentiating the first term on the RHS of (2.18) and using Eq. (1.8) yields d

−

dt

⟨ut , Tu⟩ = ⟨u, HTu⟩ − ⟨ut , Tut ⟩.

(2.20)

Notice that the antisymmetry of T yields

R⟨ut , Tut ⟩ = 0;

(2.21)

moreover we have

⟨u, HTu⟩ = ⟨u, tHu⟩ + ⟨u, [H , T ]u⟩ = −⟨HTu, u⟩ + ⟨u, [H , T ]u⟩ and hence

R⟨u, HTu⟩ =

1 2

⟨u, [H , T ]u⟩.

(2.22)

Plugging (2.21) and (2.22) into (2.20) yields d dt

R⟨ut , Tu⟩ =

1 2

⟨u, [H , T ]u⟩.

(2.23)

We now turn to the derivative of the second term on the RHS of (2.19) to write 2

d dt

R⟨ut , ψ u⟩ = 2⟨ut , ψ ut ⟩ + 2R⟨Hu, ψ u⟩.

(2.24)

Integrating by parts immediately yields

R⟨Hu, ψ u⟩ = −



a(∇b u, ∇b u)ψ dx − R



a(∇b u, ∇ψ)u dx +



ψ V |u|2 dx.

(2.25)

Note now that



a(∇b u, ∇ψ)u dx =



|u|2 Aψ dx − R



a(∇ψ, ∇b u)udx

so that

 R

a(∇b u, ∇ψ)u dx = −

1



2

|u|2 Aψ dx.

(2.26)

Hence by (2.24)–(2.26) we obtain 2

d dt

R⟨ut , ψ u⟩ = 2



|ut |2 ψ dx − 2



a(∇b u, ∇b u)ψ dx +



|u|2 Aψ dx + 2



ψ V |u|2 dx.

(2.27)

In conclusion, by (2.19), (2.23) and (2.27) we have obtained the following identity

¨ W (t ) = Θ

1 2

⟨u, [H , T ]u⟩ + 2



|ut |2 ψ dx − 2



a(∇b u, ∇b u)ψ dx +



|u|2 Aψ dx + 2



ψ V |u|2 dx

(2.28)

which, together with the explicit calculation of the term ⟨u, [H , T ]u⟩ already discussed in the proof of Theorem 1.1, yields (1.10). 3. Proofs of the smoothing estimates The proof of Theorem 1.3 is based on the standard multiplier method, and in particular it follows closely the strategy of [13]; the main tool consists in the choice of a suitable radial function φ to be put in virial identity (1.7) and the explicit evaluation of the integrals that come into play. The handling of the generalized bilaplacian term will be simplified by the introduction of a new easy identity involving a further small multiplier ϕ . First of all, using integration by parts and relation (2.1) we notice that

˙ S (t ) = 2I Θ

 Rn

a(∇φ, ∇b u)u dx,

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F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

so that we can rewrite virial identity (1.7) as

 ∇

2 Rn

2 b uDa

 + Rn

φ∇b u dx −

1



2

|u| A φ dx − 2 2

Rn

 Rn

φ

′

Vra



2

|u| dx + 2I Rn

uφ ′ a(Baτ , ∇b u)dx

[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx = K (t )

(3.1)

for all t > 0, where we have defined the quantity K (t ) =

d



 

a(∇b u, ∇φ)u dx .

I

dt

Rn

We thus need the following interpolation lemma, valid for every dimension n ≥ 3, to control this term. Lemma 3.1. Let n ≥ 3, [ajk (x)] be a symmetric real matrix of smooth functions satisfying (1.2), and φ : Rn → R be a radial function such that φ ′ is bounded and Aφ = ∂j (ajk ∂k φ) .

1

(3.2)

| x|

Then the following estimate holds:

   

 

Rn

a(∇b u, ∇φ)udx ≤ ∥f ∥2

1

H˙ 2

.

(3.3)

Remark 3.1. Condition (3.2) is ensured if for instance φ ′ (r ), r φ ′′ (r ) are bounded and hypothesis (1.12) holds, or as well if the matrix [ajk (x)] is of the form h(|x|)Id and the functions h(|x|) and |x|h′ (|x|) are bounded. Proof. Consider the quadratic form T (f , g ) =

 Rn

f ajk ∂jb g ∂k φ dx =



1

Rn

1

f ajk2 ∂jb g · ajk2 ∂k φ dx

which is well defined since the matrix [ajk (x)] is symmetric and positive definite. From the boundedness of φ ′ and a we immediately have the first estimate, by Hölder’s inequality, 1

|T (f , g )| . ∥f ∥L2 ∥ajk2 ∂jb g ∥L2 = ∥f ∥L2 ∥g ∥H˙ 1 .

(3.4)

Now we integrate by parts to have T (f , g ) = −



1

1

g ajk2 ∂jb f · ajk2 ∂k φ dx −



gf Aφ dx.

Using hypothesis (3.2), Hölder and Magnetic Hardy’s inequality (cfr. Proposition 4.1) we have

    1/2 1/2  |T (f , g )| ≤ sup(ajk )g  · ∥ajk ∂jb f ∥L2  j ,k L2

. ∥g ∥L2 ∥f ∥H˙ 1 .

(3.5)

Interpolation between (3.4) and (3.5) yields

|T (f , g )| . ∥f ∥

1

H˙ 2

∥g ∥

1

H˙ 2

and thus the conservation of the H˙ s norm concludes the proof.



Now we turn to the RHS of (1.7). Before fixing the multipliers, we state an easy lemma that will be very helpful to us. Lemma 3.2. For every u solution of (1.1) and ϕ : Rn → R the following equality holds



Aϕ|u| − 2



ϕ a(∇b u, ∇b u) −



ϕ V |u|2 = 0.

Proof. Multiply Eq. (1.1) by φ u, integrate by parts and take the real part.

(3.6)



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Putting together (1.7) and (3.6) we thus obtain a new virial identity involving both φ and ϕ

   ∇b uD2a φ∇b udx − |u|2 A(Aφ + ϕ)dx − 2 φ ′ Vra |u|2 dx + ϕ a(∇b u, ∇b u) n Rn Rn R  + ϕ V |u|2 + 4I uφ ′ a(∇b u, Baτ )dx

¨ S (t ) = 4 Θ



Rn



[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx.

+2 Rn

(3.7)

Now we choose an explicit radial multiplier φ , first. Let us consider, for some M > 0 to be fixed later, the function r



φ0 (r ) =

φ ′ (s)ds

(3.8)

0

where

 1  M + r ,

φ0′ (r ) =

 M +

3 1 2

−

r ≤1 1 6r 2

,

r > 1.

We then define the scaled function r  φ(r ) = φR (r ) = Rφ0 R for which we can easily compute

φ ′ (r ) =

 r  M + ,  M +

φ ′′ (r ) =

R

2

6r

−

 1   , R

3r

R r   1 + 2M r

,

, 2

r > R,

(3.9)

(3.10)

, 3

 1 2M   + ,

2

r ≤R

3R 3 1 R

 ·

∆φ =

r ≤R

3R 1

r > R, r ≤R (3.11) r > R.

∆ φ(r ) = −4π M δx=0 − 2

1 R2

δ|x|=R

(3.12)

(notice that this function φ satisfies the hypothesis of Lemma 3.1). We pick instead the second multiplier to be ϕ = −ε A˜ φ . It is easy to verify that this function is well defined, continuous and such that

|ϕ(x)| ≤

Cε

⟨x⟩1+

.

(3.13)

The introduction of this new multiplier takes the advantage of making the term Aφ + ϕ radial, and this will facilitate some steps of our proof. We begin with the following important relation, valid in any dimension.

 2 φ ′  a,τ 2 ∇ u ∇b uD2a φ∇b u = φ ′′ ∇ba,r u + b r

a ,r

(3.14)

a,τ

where ∇b and ∇b denote respectively the radial and tangential component of the covariant gradient with respect to the Hermitian product a(·, ·), so that

 2 |∇ba,r u|2 = a xˆ , ∇b u

(3.15)

 a,τ 2     ∇ u = ajk · ∂ b u2 − a xˆ , ∇b u 2 . j b

(3.16)

and

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F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

Notice moreover that our choice of ϕ yields A(Aφ + ϕ) = A(∆φ). Thus plugging (3.12) and (3.14) into (3.7) and neglecting the negative part Vra− of the electric potential yield



2

3R |x|≤R

  1 |∇ba,τ u|2 2 dx − |u| A(∆φ)dx − φ ′ Vra+ |u|2 dx j |x| 2 Rn Rn Rn   uφ ′ a(∇b u, Baτ )dx + ϕ a(∇b u, ∇b u)dx + ϕ V |u|2 dx

  ajk · ∂ b u2 dx + 2M 

+ 2I Rn

 + Rn



Rn

Rn

[2a (∇b u, ∇ a(∇φ, ∇b u)) a (∇φ, ∇ a(∇b u, ∇b u))] dx ≤

 

d

I

dt

Rn

a(∇b u, ∇φ, )u dx

 (3.17)

for all R > 0. Now we estimate the LHS of (3.17) term by term. First of all notice that

  ϕ a(∇b u, ∇b u)dx ≥ −Cε  n



R

a(∇b u, ∇b u)

⟨x⟩1+

Rn

≥ −Cε sup R >0

1



R |x|≤R

  dx

|∇b u|2 dx.

(3.18)

The magnetic potential term then gives

     ′ a  2I uφ a(∇b u, Bτ )dx ≥ −2 I uφ a(∇b u, Bτ )dx Rn Rn   1 ≥ −2 M + |u| · |Baτ | · |∇ba,τ u|dx 

′

a

2

 ≥ −2 M +  ≥ −2 M +

Rn

   a,τ 2 ∇ u 1 b

|x|

2 1

∞

dx

dρ

 |x|=ρ

0

   a,τ 2 ∇ u b

|x|

2

 21 

 21  dx

sup R>0

1



R2

|x| · |u|2 · |Baτ |2 dσ

|u|2 dσ

 12

|x|=R

 12

1

∥(Baτ )2 ∥C2 3 .

(3.19)

For the electric potential terms we have, in a similar way,

 − Rn

φ( ′

    ′ a 2  ) |u| dx ≥ −  φ (Vr )+ |u| dx n R   +∞   1 dρ ≥− M+

Vra +

2

2

 ≥− M+  ≥− M+

1 2 1

|x|=ρ

0

+∞



dρ



 sup R >0

sup (|(Vra )+ | · |x|2 )

|x|=ρ

0

2

|(Vra )+ | · |u|2 dσ

1 R2



1



ρ2

|u|2 dσ



|x|=ρ



|u|2 dσ ∥(Vra )+ ∥C 2

(3.20)

|x|=R

and, using Proposition 4.1



   |u|2  ϕ V |u|2 dx ≥ −Cε  V dx ⟨x⟩1+  Rn Rn    |u|2  ≥ −Cε  V dx ⟨x⟩1+  Rn     |u|2  dx ≥ −Cε (∥V |x|2 ∥L∞ )   2 1 + Rn |x| ⟨x⟩  2 |∇b u| ≥ −Cε (∥V |x|2 ∥L∞ ) dx 1+ Rn ⟨x⟩  1 ≥ −Cε (∥V |x|2 ∥L∞ ) sup |∇b u|2 dx. R>0

R |x|≤R

(3.21)

F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

295

Now we focus on the term



[2a (∇b u, ∇ a(∇φ, ∇b u)) − a (∇φ, ∇ a(∇b u, ∇b u))] dx.

2 Rn

Since the matrix [ajk (x)] is a perturbation of the identity we can rewrite as



2ε

2a ∇b u, ∇ a˜ (∇φ, ∇b u) − a ∇φ, ∇ a˜ (∇b u, ∇b u)

 Rn









dx

and estimate it, using (1.2) and (1.12) and Proposition 4.1, with

   (∇φ · ∇ u)(ˆx · ∇ u)   |∇ u|2 (∇φ · xˆ )  b b   b   dx −2ε 2 +  1+   n ⟨ x ⟩ ⟨x⟩1+ R   φ′  2|∇br u|2 + |∇b u|2 dx −2ε 1 + Rn ⟨x⟩   1 3|∇b u|2 −2ε M + dx 1+ 2 Rn ⟨x⟩  1 |∇b u|2 dx. −Cε sup 

≥ ≥ ≥ ≥

(3.22) R |x|≤R Finally we turn to the generalized bilaplacian term. Due to (1.12) and (3.12) and the choice of ϕ it is easily seen that R >0



1

−

2

Rn

|u|2 A(∆φ)dx = (2π M − ε)|u(0)|2 +

1−ε



2R2

|u|2 dσ

(3.23)

|x|=R

for some small constant cε . Before going on we introduce the following compact notation, for the seek of simplicity:

 21 |∇ba,τ u|2 ; dx |x|

 C1 :=

 C2 :=

sup R>0

1



R2

|u|2 dσ

 21

.

|x|=R

Plugging now (3.19), (3.20), (3.22) and (3.23) into (3.17), neglecting some positive terms and taking the supremum over R > 0 yields 2 − Cε

sup



3R

R >0

|∇b u|2 dx + 2MC12 +

1−ε 2

|x|≤R

C22 ≤ K (t ) +

 M+

1 2

   1 · 2C1 C2 ∥(Baτ )2 ∥C2 3 + C22 ∥Vra+ ∥C 2

that is, equivalently, sup

2 − Cε



3R

R >0

|∇b u|2 dx + C (C1 , C2 , M , B, V , a) ≤ K (t )

(3.24)

|x|≤R

where the constant C is given by C (C1 , C2 , M , B, V , a) = 2MC12 +



1−ε 2

     1 1 1 − M+ ∥Vra+ ∥C 2 C22 − 2 M + ∥(Baτ )2 ∥C2 3 C1 C2 . 2

2

1 2

To conclude the proof we need to optimize the smallness condition on ∥(Baτ )2 ∥C 3 and ∥Vra+ ∥C 2 under which we can ensure that C (C1 , C2 , M , B, V , a) ≥ 0. We can fix C1 = 1 since C is homogeneous, and impose



1−ε 2

 − M+

1 2

 ∥

Vra+ C 2



∥

C22

 −2 M +

1 2



1

∥(Baτ )2 ∥C2 3 C2 + 2M > 0

for all C2 > 0. This gives the condition



M+ M

1 2 2



  1 ∥(Baτ )2 ∥C 3 + 2 M + ∥Vra+ ∥C 2 ≤ 1 − ε. 2

In view of minimizing the size of B we choose M =

∥(Baτ )2 ∥C 3 + ∥Vra+ ∥C 2 ≤

1−ε

1 ; 2

(3.25)

hence we obtain

⇒ C (C1 , C2 , M , B, V , a) ≥ 0. 2 Thus if (1.13) is satisfied from (3.24) and (3.26) we have  1 sup |∇b u|2 dx ≤ CK (t ) R>0 3R |x|≤R

(3.26)

for some positive constant C . The thesis comes from the integration in time of the last inequality, the application of Lemma 3.1 and the conservation 1

of the H˙ 2 -norm at the right hand side.

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F. Cacciafesta / J. Math. Anal. Appl. 402 (2013) 286–296

4. Hardy’s inequality We here prove a quite general weighted magnetic Hardy’s inequality that is needed in some parts of the paper. Proposition 4.1 (Weighted Magnetic Hardy’s Inequality). Let n ≥ 3, A : Rn → Rn and w = w(|x|) a radial positive function such that

|w(|x|)| ≤ c1 ,

|w ′ (|x|)| ≤

c2

(4.1)

| x|

for some constants c1 , c2 . Then the following inequality holds for any f ∈ D(H )

 Rn

|f |2 w(|x|)dx ≤ C (n, c1 , c2 ) |x|2

 Rn

|∇b f |2 w(|x|)dx.

(4.2)

Proof. We write, for α ∈ R,

 2   (∇b f )w(|x|) 21 + α f x w(|x|) 12  dx   2 |x| Rn    xw(|x|) |f |2 w(| x |) dx + 2 α R f · ∇b fdx. = |∇b f |2 w(|x|)dx + α 2 2 n n n |x|2 R | x| R R 

0 ≤

(4.3)

Integrating by parts the last term beyond becomes 2α R

 f Rn

xw(|x|)

|x|2



2

· ∇b fdx = −α

|f | div Rn





x

|x|2

w(|x|) dx.

From (4.1) we have

     ′     x  = −  (n − 2)w(|x|) + w (|x|)  − div w(| x |)   2 2 |x| |x| |x|  ≥−

(n − 2)c1 + c2 , |x|2

(4.4)

that plugged into (4.3) yields

{−α 2 − (c2 + (n − 2)c1 )α} that gives (4.2).

 Rn

|f |2 w(|x|)dx ≤ |x|2

 Rn

|∇b f |2 w(|x|)dx

(4.5)



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