Energy decay for a locally undamped wave equation

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ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

MATTHIEULÉAUTAUD ANDNICOLASLERNER

Energy decay for a locally undamped wave equation

Tome XXVI, n^o1 (2017), p. 157-205.

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pp. 157-205

Energy decay for a locally undamped wave equation

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Matthieu Léautaud⁽¹⁾ and Nicolas Lerner⁽²⁾

ABSTRACT. — We study the decay rate for the energy of solutions of a damped wave equation in a situation where theGeometric Control Conditionis violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in transversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semi-classical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

RÉSUMÉ. — Nous étudions le taux de décroissance de l’énergie des solutions de l’équation des ondes amorties dans une situation où la Condi- tion de Contrôle Géométrique n’est pas satisfaite. Nous supposons que l’ensemble des trajectoires non amorties forme un sous-tore plat, et que la métrique est localement plate dans un voisinage. Nous supposons aussi que la fonction d’amortissement est localement homogène dans les directions transverses. Nous démontrons la décroissance à un taux polynomial optimal, qui dépend de l’homogénéité de la fonction d’amortissement.

Notre méthode repose sur une procédure de deuxième microlocalisation, qui consiste à découper l’espace des phases en toutes petites régions res- pectant le principe d’incertitude, mais bien trop petites pour entrer dans

(*)Reçu le 27 novembre 2015, accepté le 24 mars 2016.

Keywords:Damped wave equation, polynomial decay, torus, second microlocalization, geometric control condition, non-selfadjoint operators, resolvent estimates.

(1) Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Diderot (Paris VII) Bâtiment Sophie Germain 75205 Paris Cedex 13 (France) — leautaud@math.univ-paris-diderot.fr

The first author is partially supported by the Agence Nationale de la Recherche under grant GERASIC ANR-13-BS01-0007-01.

(2) Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris Cedex 05 (France) —

nicolas.lerner@imj-prg.fr

Article proposé par Nalini Anantharaman.

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le cadre de l’analyse microlocale semi-classique standard. Une méthode de multiplicateurs nous permet, dans chaque région, d’obtenir des estimées d’énergie, que nous recollons finalement.

1. Introduction and main results

1.1. Introduction

We consider a smooth connected compact Riemannian manifold (M, g) of dimensionn, and denote by ∆gthe associated negative Laplace–Beltrami operator. Givenb∈L^∞(M), we study the decay rates for the damped wave equation onM:

(∂_t²u−∆gu+b(x)∂tu= 0 in R⁺×M,

(u, ∂tu)|t=0= (u0, u1) in M. (1.1) The energy of a solution is defined by

E(u(t)) = 1

2(k∇_gu(t)k²_L2(M)+k∂_tu(t)k²_L2(M)), (1.2) (see for instance Appendix A for a definition of ∆g and∇g) and evolves as

d

dtE(u(t)) =− Z

M

b|∂tu|²dx.

The energy is thus actually damped whenb>0 a.e. onM, what we assume from now on. We define the subset ofM on which the damping is effective as

ωb :=[

{U ⊂M, U open,essinfU(b)>0}. (1.3) Notice that ωb is an open set included in the interior of suppb and thus ωb⊂suppb.⁽¹⁾ In the usual case wherebis continuous, we haveωb={b >0}

andωb= suppb. As soon asωb6=∅one hasE(u(t))→0 ast→+∞(see for instance [28, 29]). Moreover, a criterion for uniform (and hence exponential) decay is due to Rauch–Taylor [37] (see also [3] and Lemma 5.1 below): there existC >0, γ >0 such that for all data,

E(u(t))6Ce^−γtE(u(0)),

if the Geometric Control Condition (GCC) holds: every geodesic starting from S^∗M (see Appendix A for a precise statement) enters the set ωb in finite time. Conversely, if there is a geodesic that never meets supp(b), then uniform decay does not hold (see for instance [36]). In the caseb∈C⁰(M), the situation is simpler since uniform decay is equivalent to the fact that

(1)Remark that the equality may fail, taking for instanceb=1KwhereKis a compact Cantor set with positive measure satisfying ˚K=∅, in which case supp(b) =Kandωb=∅.

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ω_b(={b >0}) satisfies (GCC), as remarked by Burq and Gérard [9]⁽²⁾. As a consequence, when (GCC) is not satisfied, we cannot expect a uniform decay of the energy with respect to all data inH¹(M)×L²(M). However, Lebeau [28, 29] proved that there is always a uniform decay rate of the energy, with respect to smoother data, say inH²(M)×H¹(M). This motivates the following definition.

Definition 1.1. — Givena∈Rand a decreasing functionf: [a,+∞)→ R^∗+ such thatf(t)→0 ast→+∞, we say that the solutions of (1.1)decay at ratef(t)if there existsC >0such that for all(u0, u1)∈H²(M)×H¹(M), for allt>a, we have

E(u(t))¹² 6Cf(t) ku0k_H2(M)+ku1k_H1(M)

.

Note that decay at a ratef(t) depends only on (M, g) and on the damping functionb. Note also thatf(t)² characterizes the decay of the energy and f(t) that of the associated norm. Lebeau [28, 29] proved that decay at rate 1/logtalways holds, independently of (M, g) andb as soon asω_b6=∅.

As noticed for instance in [5], decay at a rate f(t) implies faster decay for “smoother” data: taking for example b ∈ C^∞(M), decay at rate f(t) implies that for all s >0, there exists Cs > 0 such that for all (u0, u1) ∈ H^s+1(M)×H^s(M), we have for larget,

E(u(t))¹² 6Csf(t/s)^s(ku0k_Hs+1(M)+ku1k_Hs(M)).

In view of the Rauch-Taylor theorem mentioned above, it is convenient to introduce the subset of phase-space consisting in points-directions that are never brought into the damping regionωb by the geodesic flow. Namely, theundamped set is defined by

S={ρ∈S^∗M, for allt∈R, φ_t(ρ)∩T^∗ω_b=∅},

whereφ_tis the geodesic flow (see for instance Appendix A). With this definition, (GCC) is equivalent toS=∅. In this article, we are concerned with the damped wave equation in a geometric situation where the undamped setS is the cotangent space to aflat subtorusofM (of dimension 16n⁰⁰6n−1) under two main additional assumptions: the metric is locally flat around this subtorus; the damping functionbonly depends on variables transverse to this torus and enjoys locally a prescribed homogeneity. As a particular case, we can consider situations where the geodesic flow has asingle undamped trajec- toryif the metric is locally flat around this trajectory; the damping function b only depends on variables transverse to the flow and enjoys a prescribed

(2)This is no longer the case in general ifbis not continuous, as proved in [27].

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homogeneity in a neighbourhood of the undamped trajectory. Such situations may for instance occur on the torusM = Tⁿ = (R/2πZ)ⁿ endowed with the flat metric, as in the following examples. One of our motivations is to understand the optimal decay rate in the following model problems.

Example 1.2. — Let M = T² = (R/2πZ)² ≡ [−π, π]², endowed with the flat metric, letγ >0, and let b(x₁, x₂) =x^2γ₁ nearx₁= 0, positive else- where, depending only onx₁. The undamped set consists in two undamped trajectories:

S={0}x₁×T¹x2× {0}ξ₁× {±1}ξ₂ =S^∗({0} ×T¹).

For the case whereb= sin²x1, Wen Deng communicated to us a direct study in [16].

Example 1.3. — LetM =T³= (R/2πZ)³≡[−π, π]³, endowed with the flat metric, letγ >0, and letb(x1, x2, x3) = (x²₁+x²₂)^γ. The undamped set consists in two undamped trajectories:

S ={0,0}_x₁_,x₂×T¹x₃× {0,0}_ξ₁_,ξ₂× {±1}_ξ₃ =S^∗({0,0} ×T¹).

Example 1.4. — LetM =T³≡[−π, π]³, endowed with the flat metric, letγ >0, and let

b(x1, x2, x3) =x^2γ₁ .

The undamped set is a 2-dimensional subtorus given by:

S={0}x₁×T²x2,x3× {0}ξ₁× {(ξ2, ξ3)∈R², ξ₂²+ξ₃²= 1}=S^∗({0} ×T²).

Decay rates for the damped wave equation on a flat metric with a lack of (GCC) have already been studied in [1, 10, 31, 34]. In [1] it is proved that, on M =Tⁿ, decay at a ratet^−1/2always occurs ifωb6=∅. On the other hand, the decay cannot be better thant⁻¹as soon as (GCC) is strongly violated, i.e. as soon as there exists a neighbourhoodN of a geodesic such thatN ∩supp(b) =

∅, see [1]. In this paper, we are studying the opposite situation, i.e. the case of a weak lack of damping on M =Tⁿ: only a positive codimension invariant torus is undamped. In the situation of Examples 1.2, 1.3, and 1.4, for instance, we may expect (and we shall prove) a decay at a stronger polynomial rate thant⁻¹. Functions onTⁿ

00 shall be identified in the whole paper with 2πZⁿ

00-periodic functions onRⁿ

00.

According to [1, 5, 8, 29], proving a decay rate for solutions of (1.1) reduces to proving a high-energy estimate for the operators

P_λ=−∆g−λ²+iλb, λ∈R^∗, D(P_λ) =H²(M). (1.4) The latter are for instance obtained by performing a Fourier transform in the time variable of the damped wave operator∂_t²−∆g+b(x)∂t,λbeing the frequency variable dual to the timet. More precisely, concerning polynomial

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decay, the optimal result was proved by [8] (see also [4] for generalizations) and can be stated as follows (see [1, Proposition 2.4]).

Proposition 1.5. — Given α >0, the solutions of (1.1)decay at rate t⁻^α¹ if and only if there exist C, λ0 positive, such that for all u∈H²(M), for allλ>λ₀, we have

CkPλuk_L2(M)>λ^1−αkuk_L2(M). (1.5) Recall that uniform decay is equivalent to the estimate (1.5) withα= 0.

1.2. Main results

We first have a negative result.

Theorem 1.6. — Assume that there exists 16n⁰⁰6n−1,ε0>0, and C1>0 such that withn⁰=n−n⁰⁰, we have

•

B_Rn0(0, ε0)×Tⁿ

00,|dx⁰₁|²+· · ·+|dx⁰_n0|²+|dx⁰⁰₁|²+· · ·+|dx⁰⁰_n00|²

⊂(M, g), (1.6)

• ∇x⁰⁰b= 0 inN =B_Rn0(0, ε0)×Tⁿ

00, (1.7)

• 06b(x⁰)6C1|x⁰|^2γ inN. (1.8)

Then, there existC₀>0 and(u_k)_k∈_N∈H²(M)^N withku_kk_L2(M) = 1such that

kP_ku_kk_L2(M)6C₀k^γ+1¹ , fork∈N^∗. As a consequence, the best estimate we could expect is

CkP_λuk_L2(M)>λ^γ+1¹ kuk_L2(M), (1.9) i.e. (1.5) with α = 1− _γ+1¹ . Moreover, (see also [5, Proposition 3]), our Theorem 1.6 prevents decay at a rateo t⁻⁽¹⁺^γ¹⁾

: the best expected decay rate is

t⁻⁽¹⁺¹^γ⁾.

Let us now state our partial converse of Theorem 1.6: under some additional global assumptions on M and b, decay at rate t⁻⁽¹⁺¹^γ⁾ indeed holds. We first provide a simpler result in the case n⁰⁰ = 1 under a global invariance assumption onb. We then give our more general result in Theorem 1.8.

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Theorem 1.7. — Let (M, g) = (M⁰×T¹, g⁰+|dxn|²)where(M⁰, g⁰)is a smooth compact Riemannian manifold of dimension n−1. Assume that there existy⁰∈M⁰,C₁>1and a neighbourhood N⁰ of y⁰ such that

•g⁰=|dx1|²+· · ·+|dxn−1|² is flat inN⁰, (1.10)

•b=b⊗1does not depend on the variablex_n∈T¹,andb∈L^∞(M⁰), (1.11)

•C₁⁻¹|x⁰−y⁰|^2γ6b(x⁰)6C1|x⁰−y⁰|^2γ forx⁰∈ N⁰, (1.12)

•b>C₁⁻¹ a.e. onM⁰\ N⁰. (1.13)

Then, Property (1.5) holds with α = 1− _γ+1¹ , i.e. decay occurs at rate t⁻⁽¹⁺^γ¹⁾.

This theorem tackles in particular the case of Examples 1.2 and 1.3.

Note that simple examples of functions satisfying the assumptions are given by b(x⁰) = Q(x⁰ −y⁰)^γ locally around y⁰, where Q is a positive definite quadratic form. We note that very little regularity is required for the damping coefficient b: its “vanishing rate” is prescribed here (1.12) in a relatively weak sense. One may however discuss its global invariance property in the xn-direction. It can indeed be removed: Theorem 1.7 is a particular case of the following result, where T¹ is replaced by Tⁿ

00 (adding no significant difficulty) andbis not supposed to be globally invariant anymore, but instead satisfies (GCC) outside the undamped trajectory. We presented Theorem 1.7 separately as its proof is simpler and contains nevertheless the key ideas for the next result.

Theorem 1.8. — Take 1 6 n⁰⁰ 6 n−1 and assume that (M, g) = (M⁰×Tⁿ

00, g⁰+|dx⁰⁰₁|²+· · ·+|dx⁰⁰_n00|²)where (M⁰, g⁰)is a smooth compact Riemannian manifold of dimension n⁰ = n−n⁰⁰ and (x⁰⁰₁, . . . , x⁰⁰_n00) denote variables inTⁿ

00. Assume that there exist y⁰∈M⁰,C₁>1 and a neighbour- hoodN⁰ of y⁰ such that

• g⁰=|dx⁰₁|²+· · ·+|dx⁰_n0|² is flat inN⁰, (1.14)

• b∈L^∞(M),∇_x⁰⁰b∈L^∞(M),and∇_x⁰⁰b= 0 inN⁰×Tⁿ

00, (1.15)

• C₁⁻¹|x⁰−y⁰|^2γ 6b(x⁰)6C₁|x⁰−y⁰|^2γ forx⁰∈ N⁰, (1.16)

• any geodesic starting from S^∗M\S^∗({y⁰} ×Tⁿ

00)intersects ω_b in

finite time. (1.17)

Then, Property (1.5)holds with α= 1−_γ+1¹ , i.e. decay at ratet⁻⁽¹⁺^γ¹⁾. Remark 1.9. — The proof of this theorem (as well as those of the previous ones) also holds without significant modification if the torus Tⁿ

00 =

(R/2πZ)ⁿ⁰⁰is replaced by any compact connected Riemannian manifoldM⁰⁰.

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In this situation, Fourier series onTⁿ

00 have to be replaced by the spectral decomposition for the Laplace–Beltrami operator ∆_M⁰⁰ onM⁰⁰, and Fourier multipliers at the end of Section 5.2 have to be replaced by functional calculus for ∆_M⁰⁰ (for this argument, more smoothness than (1.15) forbin the x⁰⁰-direction is required however). The results also remain essentially un- changed ifbvanishes nearfinitely manypointsy⁰₁, y⁰₂,· · · (instead of a single oney⁰) assuming Assumptions (1.14)-(1.17) around each point (with possi- bly different vanishing ratesγ1, γ2,· · ·, in which case the decay rate is given byt⁻⁽¹⁺^max¹^γi⁾).

This result applies for instance on the torus: assumeM =Tⁿ and that there is a single undamped trajectory Γ. Assume that there exists a neigh- bourhoodN of this trajectory such thatb is invariant inN in the direction of Γ, and that it is positive homogeneous of degree 2γ in N in variables orthogonal to Γ. Then, we have the property (1.5) with α= 1− _γ+1¹ , i.e.

decay at ratet⁻⁽¹⁺¹^γ⁾.

Since the work of Lebeau [29] (see also the introduction of [1] and the references therein), it is quite well established that the main parameters gov- erning the decay rates when (GCC) fails are the global and local dynamics of the geodesic flow. Our results confirm the idea, raised in [10, 1], that once the geometry (and hence the dynamics) is fixed, the next relevant feature when regarding the best decay rate is the rate at which the damping coefficientb vanishes.

Observe that the biggerγ, the worse is Estimate (1.9). This is consistent with the fact that for largeγ, the functionbis very flat on{y⁰} ×Tⁿ

00so that much energy may keep concentrated on the set whereb is small. Note that formally, when taking γ → 0⁺ in Estimate (1.9) (and forgetting that the constantC we obtain depends onγ), we recover the uniform decay estimate (i.e. (1.5) withα= 0), equivalent to (GCC). Indeed, if b is positive homogeneous of degree zero, it does not vanish aty⁰ so that (GCC) is satisfied.

It would certainly be interesting to prove Estimate (1.9) with a constantC uniform with respect toγ to make this remark rigorous.

The plan of the article is as follows. Taking advantage of the homogeneity of b, the sought estimate near the undamped set may be reduced to an estimate onRⁿ

0 for a non-selfadjoint Schrödinger operator (with purely imaginary potential) of the form

−∆ +iW(x), W(x)∼ |x|^2γ.

This key and optimal estimate, which is of independent interest, is proved in Section 2. In turn, it provides a bound on the size of the pseudospectrum

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for this operator, generalizing results of E. B. Davies [14] and K. Pravda- Starov [35] in the case of the 1D complex harmonic oscillator,−_dx^d²2+e^iθx². Section 3 is devoted to the proof of two simple technical lemmata, one of them being the scaling argument. The proof of Theorem 1.7 is given in Section 4. The proof of the main result, namely Theorem 1.8, is completed in Section 5 in two steps: first, we prove a geometric control lemma in Sec- tion 5.1. Then, in Section 5.2, we patch together the estimates obtained in the different microlocal regions. Section 6 provides a proof of the lower bound of Theorem 1.6. In Section 7, we discuss the spirit of the proof, which relies on some kind of second microlocalization. In particular, our proof could not work with a standard semi-classical localization procedure: we are left with a region in the phase space, near the undamped set S, where further cutting of the phase space is necessary, with a stopping procedure linked to the Heisenberg Uncertainty Principle. To patch together the estimates, we use implicitly a metric which should satisfy some admissibility properties.

Although we have avoided in the main part of the text to resort to very general tools of pseudodifferential calculus, we hope that Section 7 could bring a more conceptual vision of the technicalities included in the previous sections.

The paper ends with three appendices recalling some facts of geometry and pseudodifferential calculus.

Note. — This article was written in 2014 [24] and a short version presented in [25]. Some time after this article was submitted, N. Burq and C. Zuily [11] and W. Deng [17] managed to weaken some of the assumptions of our Theorem 1.8.

2. A sharp estimate for a non-selfadjoint operator onR^d

2.1. Statements

After a Fourier transformation in the periodic direction and a scaling argument (see the following sections), our main result is reduced to the following theorem. We define onL²(R^d) (below, we shall take d=n⁰) the unbounded operator

Q^λ₀ =−∆ +iW_λ(x), λ >0, (2.1) whereWλ is a family of real-valued measurable functions and

D(Q^λ₀) ={u∈H²(R^d), Wλu∈L²(R^d)}.

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Theorem 2.1. — Suppose thatW_λ is a family of real-valued measurable functions on R^d and that there exist C₁ > 1 and γ > 0 such that for all λ >0, we have

C₁⁻¹|x|^2γ 6Wλ(x)6C1hxi^2γ =C1(1 +|x|²)^γ. (2.2) Then, there existsC₀>0such that for allµ∈R, allλ >0andu∈Cc²(R^d), we have

C₀k(Q^λ₀ −µ)uk_L2(R^d)>

µ^2γ+1^γ 1(µ>1) +|µ|1(µ6−1) + 1

kuk_L2(R^d). (2.3) We stress the fact that the sole uniform Assumption (2.2) yields the uniform estimate (2.3). The power_2γ+1^γ is optimal in this estimate. Although not needed for the application to the damped wave equation, we provide for completeness a direct proof of this fact in Lemma C.1. The papers by E.B. Davies [14] and K. Pravda-Starov [35] gave a version of the above estimates in the case of the 1D complex harmonic oscillator,−_dx^d²2 +e^iθx².

To prove Theorem 2.1, we need the following lemma.

Lemma 2.2. — Suppose that W_λ satisfy the uniform Assumption (2.2) and letabe a smooth function onR^2d, bounded as well as all its derivatives.

Then, there exists C > 0 such that for all λ > 0 and all u ∈ Cc⁰(R^d), we have

kVλa^wuk_L2(R^d)6C kuk_L2(R^d)+kVλuk_L2(R^d)

,

whereV_λ=W_λ^1/2 anda^w stands for the Weyl quantization of the symbola.

Proof of Lemma 2.2. — Using the upper bound in Assumption (2.2) yields

kV_λ(x)a^wuk²_L2(R^d)= Z

R^d

V_λ²(x)|a^wu|²dx 6C1

Z

R^d

hxi^2γ|a^wu|²dx=C1khxi^γa^wuk²_L2(R^d). Then, we notice thathxi^γ andhxi^−γ are admissible weight functions for the metric|dx|²+|dξ|² in the sense of [30, Definition 2.2.15]. As a consequence of symbolic calculus, we have

hxi^γ]a]hxi^−γ ∈S(1,|dx|²+|dξ|²),

whereS(1,|dx|²+|dξ|²) is the space of smooth functions onR^2d which are bounded as well as all their derivatives (see Section B.1 in the Appendix for more on this topic). Calderón–Vaillancourt Theorem (see e.g. [30, Theo- rem 1.1.4]) yields

hxi^γa(x, ξ)^whxi^−γ∈ L(L²(R^d)),

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which implieskhxi^γa^wuk_L2(R^d).khxi^γuk_L2(R^d).This finally gives kVλ(x)a(x, ξ)^wuk²_L2(R^d).khxi^γuk²_L2(R^d)

.kuk²_L2(R^d)+k|x|^γuk²_L2(R^d)

.kuk²_L2(R^d)+kVλuk²_L2(R^d),

according to the uniform lower bound in Assumption (2.2). This concludes

the proof of the lemma.

Now, the proof of Theorem 2.1 follows from the next two lemmata.

Lemma 2.3. — There existsC >0 andµ0>0 such that for allµ>µ0, allλ >0 andu∈Cc²(R^d), we have

Ck(Q^λ₀−µ)uk_L2(R^d)>µ^2γ+1^γ kuk_L2(R^d). (2.4) Lemma 2.4. — For any µ₀ > 0, there exists C > 0 such that for all µ6µ₀, all λ >0andu∈Cc²(R^d), we have

Ck(Q^λ₀−µ)uk_L2(R^d)> |µ|1(µ6−1) + 1

kuk_L2(R^d). (2.5) Let us first prove the simpler Lemma 2.4, the proof of the more involved Lemma 2.3 being postponed to the end of the section.

Proof of Lemma 2.4. — We start with the caseµ6−1. We have then k(Q^λ₀−µ)uk_L2(R^d)kuk_L2(R^d)>Reh(Q^λ₀−µ)u, ui_L2(R^d)

>−µhu, ui_L2(R^d)=|µ|kuk²_L2(R^d), so that Estimate (2.5) holds forµ6−1.

Next, let us prove that there existsC >0 such that for allµ∈[−1, µ0], allλ >0 andu∈Cc²(R^d), we have

Ck(Q^λ₀−µ)uk_L2(R^d)>kuk_L2(R^d). (2.6) If not, there exist sequences (µk)k∈N ∈ [−1, µ0]^N, (λk)k∈N ∈ (R^∗+)^N, and (uk)_k∈N∈Cc²(R^d)^Nsuch that

k(Q^λ₀^k−µk)ukk_L2(R^d)< 1

k+ 1, kukk_L2(R^d)= 1. (2.7)

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This implies

0← k(Q^λ₀^k−µk)ukk_L2(R^d)kukk_L2(R^d)>Reh(Q^λ₀^k−µk)uk, uki_L2(R^d)

=k∇u_kk²_L2(R^d)−µ_kku_kk²_L2(R^d)

>k∇ukk²_L2(R^d)−µ0kukk²_L2(R^d)

0← k(Q^λ₀^k−µk)ukk_L2(R^d)kukk_L2(R^d)>ImhQ0u, ui_L2(R^d)=kVλukk²_L2(R^d)

>C₁⁻¹k|x|^γu_kk²_L2(R^d). (2.8) SinceH_γ¹(R^d) :={u∈H¹(R^d),|x|^γu∈L²(R^d)}injects compactly inL²(R^d) and (uk)k∈N is bounded in H_γ¹(R^d), we may extract a subsequence such that (u_k)k∈N converges strongly in L²(R^d), u_k →u_∞ ∈L²(R^d). According to (2.8), we also obtain u_k *0 weakly in L²(R^d): in fact we have for φ∈ Cc⁰(R^d\{0}),

Z

u∞(x)φ(x)dx

← Z

uk(x)φ(x)dx

6k|x|^γukk_L2(R^d)k|x|^−γφk_L2(R^d)→0, proving that suppu∞ ⊂ {0} and thus the L² functionu∞ = 0. This con- tradicts (2.7) which impliesku∞k_L2(R^d)= 1. This proves (2.6). As a consequence, Estimate (2.3) is now proven to hold for allµ∈(−∞, µ0].

We are now left to proving Lemma 2.3, i.e. to study the most substantial case whereµ > µ₀, but we may keep in mind that we can freely choose the large fixed constantµ₀. We set

ν =µ^1/2, Q^λ_ν =Q^λ₀−ν², (2.9) and study the asymptotics when ν → +∞. From the above remarks, we have only to prove the estimate (2.3) for ν > ν₀, where ν₀ can be chosen arbitrarily large. First of all, we note that

kQ^λ_νuk_L2(R^d)kuk_L2(R^d)>ImhQ^λ_νu, ui_L2(R^d)=hW_λu, ui_L2(R^d)

>C₁⁻¹h|x|^2γu, ui_L2(R^d), (2.10) which will be used several times during the proof. In particular, this estimate provides the right scaling in the region|x|>ν^1/(2γ+1), according to the lower bound in Assumption (2.2). Next, we split the phase space in two different regions.

2.2. The propagative region

Letχ ∈Cc^∞(R⁺; [0,1]), suchχ = 1 on [1/2,3/2] and χ= 0 on [0,1/4].

Letϕ∈Cc^∞(R^d; [0,1]) such thatϕ(x) = ¹₂ if|x|6¹2 andϕ(x) = 0 if|x|>1.

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We define

ψ(x, ξ) = Z 0

−∞

ϕ(x+τ ξ)dτ ∈C^∞ R^d×(R^d\ {0}) , which is bounded onR^d× R^d\B(0,¹₄)

since ϕ is compactly supported.

We set

mν(x, ξ) =χ |ξ|² ν²

ψ( x ν^2γ+1¹

,ξ

ν)∈S(1, |dx|² ν^2γ+1²

+|dξ|² ν² ),

where each seminorm of the symbolmν is bounded above independently of ν > 1; in particular, we get that m^w_ν is bounded on L²(R^d) with sup_ν_>₁km^w_νk_L(L2)<+∞. Next, we have with{a, b}standing for the Poisson bracket of the functionsa, b,

2 RehQ^λ_νu, im^w_νui_L2(R^d)

= i

(|ξ|²−ν²)^w, m^w_ν u, u

L²(R^d)+ 2 RehV_λ²u, m^w_νui_L2(R^d)

=h

|ξ|²−ν², m_ν ^wu, ui_L2(R^d)+ 2 RehV_λ²u, m^w_νui_L2(R^d), (2.11) since the symbol|ξ|²−ν² is quadratic. Moreover, we can compute

|ξ|²−ν², mν = 2ξ·∂xmν= 2χ(|ξ|² ν² )ξ· ∂

∂x

ψ( x ν^2γ+1¹

,ξ ν)

with χ(|ξ|²

ν² )ξ· ∂

∂x

ψ(xν⁻^2γ+1¹ , ξν⁻¹)

= Z 0

−∞

ν⁻^2γ+1¹ χ(|ξ|²

ν² )(ξ·dϕ)(xν⁻^2γ+1¹ +τ ξν⁻¹)dτ

= Z 0

−∞

νν⁻^2γ+1¹ χ(|ξ|² ν² )d

dτ

ϕ(xν⁻^2γ+1¹ +τ ξν⁻¹) dτ

=ν^2γ+1^2γ χ(|ξ|²

ν² )ϕ(xν⁻^2γ+1¹ ), since|ξν⁻¹|²> ¹4 on suppχ(^|ξ|_ν2²). Hence, we obtain

|ξ|²−ν², m_ν = 2ν^2γ+1^2γ χ(|ξ|²

ν² )ϕ(xν⁻^2γ+1¹ )

>

(ν^2γ+1^2γ if||ξ|²ν⁻²−1|61/2 and|x|ν⁻^2γ+1¹ 61/2, 0 onT^∗R^d.

Moreover we have, 2ν^2γ+1^2γ χ(|ξ|²

ν² )ϕ(xν⁻^2γ+1¹ )∈S(ν^2γ+1^2γ , |dx|² ν^2γ+1²

+|dξ|² ν² ).

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As a consequence, using the sharp Gårding inequality in (2.11) yields CkQ^λ_νuk_L2(R^d)kuk_L2(R^d)

>ν^2γ+1^2γ D

χ0(|ξ|²ν⁻²−1)χ0(|x|²ν⁻^2γ+1² )^w u, uE

L²(R^d)

− |2 RehV_λ²u, m^w_νui_L2(R^d)| −Cν⁻^2γ+1² kuk²_L2(R^d), (2.12) where, for some₀∈(0,1/8),

(χ0∈Cc^∞(R; [0,1]) is such that{χ0= 1}= [−0, 0], and

{χ0= 0}= [−20,20]^c, {0< χ0(t)<1}={0<|t|<20}. (2.13) Next, we check the term 2 RehV_λ²u, m^w_νui_L2(R^d). We have

2 RehV_λ²u, m^w_νui_L2(R^d)

=

2 RehVλu, Vλm^w_νui_L2(R^d)

62kVλuk_L2(R^d)kVλm^w_νuk_L2(R^d). Recalling that mν ∈S(1, ^|dx|²

ν^2γ+1²

+^|dξ|_ν2²)⊂ S(1,|dx|²+|dξ|²) as ν >1, we may apply Lemma 2.2 to obtain

RehV_λ²u, m^w_νui_L2(R^d)

.kVλuk_L2(R^d) kuk_L2(R^d)+kVλuk_L2(R^d)

, where the constant involved is uniform w.r.t.ν and λ. Next, using (2.10), we obtain

2 RehV_λ²u, m^w_νui_L2(R^d)

.kuk_L³²2(R^d)kQ^λ_νuk_L¹²2(R^d)+kQ^λ_νuk_L2(R^d)kuk_L2(R^d)

.kuk²_L2(R^d)+kQ^λ_νuk_L2(R^d)kuk_L2(R^d). (2.14) Combining this estimate with (2.12), we have, forν>ν0andν0large enough,

Ckuk²_L2(R^d)+CkQ^λ_νuk_L2(R^d)kuk_L2(R^d)

>ν^2γ+1^2γ D

χ0(|ξ|²ν⁻²−1)χ0(|x|²ν⁻^2γ+1² )^w u, uE

L²(R^d). (2.15) 2.3. The elliptic region.

We now check the regions where|ξ|²ν²or|ξ|²ν². Let₀∈(0,1/2);

we consider a functionθ∈C^∞(R; [−1,1]) such that

θ(σ) =











1 forσ>1 + 2₀,

{0< θ <1} forσ∈(1 +0,1 + 20), 0 for 1−₀6σ61 +₀, {−1< θ <0} forσ∈(1−20,1−0),

−1 forσ61−2₀.

(2.16)

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We claim that

∀σ∈R, (σ−1)θ(σ)>|θ(σ)|(σ+ 1) 0

2 +₀. (2.17) In fact, (2.17) is obvious wheneverθ(σ) = 0 and ifθ(σ)>0, i.e. ifσ >1 +₀, since it amounts to verifying

σ(1− 0

2 +0

)>1 + 0

2 +0

i.e. σ>1 +0, which holds true.

Ifθ(σ)<0, i.e. ifσ <1−0, it amounts to verify σ(1 + ₀

2 +0

)61− ₀ 2 +0

i.e. σ6 1 1 +0

,

which holds true since 1−06 1 1 +0

. A consequence of (2.17) is that, withc₀=₂₊⁰

0, we have

∀ξ∈R^d,∀ν >1, (|ξ|²−ν²)θ(|ξ|²ν⁻²)>c0|θ(|ξ|²ν⁻²)| |ξ|²+ν²

. (2.18) We compute

RehQ^λ_νu, θ(|ξ|²ν⁻²)^wui_L2(R^d)

=

(|ξ|²−ν²)^wu, θ(|ξ|²ν⁻²)^wu

L²(R^d)+ Re

iV_λ²u, θ(|ξ|²ν⁻²)^wu

L²(R^d)

>c0

(|ξ|²+ν²)|θ(|ξ|²ν⁻²)|w

u, u

L²(R^d)

− Re

iV_λ²u, θ(|ξ|²ν⁻²)^wu

L²(R^d)

. Following (2.14), we have

RehiV_λ²u, θ(|ξ|²ν⁻²)^wui_L2(R^d)

.kuk²_L2(R^d)+kQ^λ_νuk_L2(R^d)kuk_L2(R^d), so that we finally obtain, asθ(|ξ|²ν⁻²)^w is bounded onL²(R^d),

Ckuk²_L2(R^d)+CkQ^λ_νuk_L2(R^d)kuk_L2(R^d)

>

(|ξ|²+ν²)|θ(|ξ|²ν⁻²)|^w u, u

L²(R^d). (2.19) 2.4. Patching the estimates together.

Combining (2.10), (2.15) and (2.19), we obtain the following estimate Ckuk²_L2(R^d)+CkQ^λ_νuk_L2(R^d)kuk_L2(R^d)

>kV_λuk²_L2(R^d)+ν²

|θ(|ξ|²ν⁻²)|^w u, u

L²(R^d)

+ν^2γ+1^2γ D

χ0(|ξ|²ν⁻²−1)χ0(|x|²ν⁻^2γ+1² )^w u, uE

L²(R^d). (2.20)