On nonlinear Schrödinger equations in exterior domains

On nonlinear Schrödinger equations in exterior domains Équations de Schrödinger non linéaires dans des domaines

extérieurs

N. Burq, P. Gérard

, N. Tzvetkov

Université Paris Sud, Mathématiques, bât 425, 91405 Orsay cedex, France Received 13 December 2002; received in revised form 4 March 2003; accepted 7 March 2003

Available online 4 October 2003

Abstract

We prove a local smoothing effect and Strichartz type estimates for the Schrödinger equation on the exterior of a non- trapping obstacle. As a consequence we deduce global existence and uniqueness results for the Cauchy problem for nonlinear Schrödinger equations in these particular geometries.

2003 Elsevier SAS. All rights reserved.

Résumé

On démontre un effet de régularisation local et des inégalités de type Strichartz pour l’équation de Schrödinger à l’extérieur d’un obstacle non captant. On en déduit des résultats d’existence globale et d’unicité pour l’équation de Schrödinger non linéaire.

2003 Elsevier SAS. All rights reserved.

MSC: 35Q55; 35BXX; 37K05; 37L50; 81Q20

Keywords: Nonlinear Schrödinger; Smoothing effect; Non-trapping; Dispersive equations

1. Introduction

LetΘ⊂R^d,d 2, be a compact smooth obstacle. Denote byΩ the complementary ofΘ. In this paper we shall suppose that the obstacleΘis non-trapping which means that any light ray reflecting on the boundary ofΘ according to the laws of the geometric optics leaves any compact set in finite time. In other words any generalized bicharacteristic in the boundary cotangent bundle^bT^∗Ω(see Melrose and Sjöstrand [23,24] for a precise definition)

* Corresponding author.

E-mail addresses: [email protected] (N. Burq), [email protected] (P. Gérard), [email protected] (N. Tzvetkov).

URLs: http://www.math.u-psud.fr/ burq, http://www.math.u-psud.fr/ tzvetkov.

0294-1449/$ – see front matter 2003 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2003.03.002

leaves any compact set in finite time. Our goal here is to study the existence of global strong solutions for the nonlinear Schrödinger equation, posed onΩ,

(i∂_t+)u=F (u), inR×Ω (1.1)

with initial data

u(0, x)=u0(x), x∈Ω, (1.2)

subject to Dirichlet boundary conditions

u(t, x)=0, (t, x)∈R×∂Ω. (1.3)

The nonlinear interactionF is supposed to be of the formF =∂V /∂z¯withF (0)=0, where the “potential”V is real valued and satisfiesV (e^iθz)=V (z)for everyz∈C, θ∈R. Moreover we suppose thatV is of classC³and

D_z,^k_z¯V (z)C_k

1+ |z|2+α−k

, k=0,1,2,3.

Some phenomena in Physics turn out to be modeled by exterior problems and moreover one may expect rich dynamics under various boundary conditions. A first step in that direction is to establish well defined dynamics in the natural spaces determined by the conservation laws associated to (1.1). Ifu(t,·)∈H₀¹(Ω)∩H²(Ω)is a solution of (1.1) then (see Cazenave [13, Theorem 4.1.1]) it enjoys the conservation laws

d dt

Ω

u(t, x)²dx=0 (charge conservation), (1.4a)

d dt

Ω

∇u(t, x)²dx+

Ω

V u(t, x)

dx

=0 (energy conservation) (1.4b)

and therefore one can obtain via the Gagliardo–Nirenberg inequalities that for a large class of potentialsV the quantityu(t,·)_H¹

0(Ω)remains finite along the trajectory starting fromu₀∈H₀¹(Ω)∩H²(Ω). This fact makes the study of (1.1) in the spaceH₀¹(Ω)of particular interest and motivates us to callH₀¹(Ω)the energy space for (1.1).

It is clearly also of interest to study of (1.1) inL²(Ω), the space associated to the conservation law (1.4a). The main issue in the analysis is that the regularities ofH¹orL²are a priori too poor to be achieved by the “classical methods” (see, e.g., Segal [26], Lions [22]) for establishing local existence and uniqueness for (1.1)–(1.2)–(1.3).

The Cauchy problem associated to (1.1) with Ω =R^d attracted much attention during last 20 years (see the books by Bourgain [3], Cazenave [13], Sulem and Sulem [27] and the references therein) and the theory of existence of finite energy (or L²) solutions to (1.1) for potentialsV of polynomial growth has been much developed (for a discussion on this issue and open problems we refer to Bourgain [4]). Roughly speaking the argument for establishing finite energy solutions of (1.1) consists of combiningH¹ local well-posedness with conservation laws (1.4a), (1.4b) which eventually provide a control on theH¹norm. The local well-posedness is carried out by the classical Picard iteration scheme and the nonlinearity is controlled in the iteration process due to some smoothing properties of the free evolution. In the caseΩ=R^d the crucial fact on the free evolution is the family of so called Strichartz estimates which can be deduced from an explicit formula for the free solution and the Tomas–Stein restriction argument from harmonic analysis. Unfortunately in the case of exterior problem no suitable explicit representation of the free evolution is available and therefore the problem of establishing Strichartz estimates for the solution of (1.1) withF =0 meets serious difficulties. However as it was shown by our experience with NLS on compact manifolds (see [9]) one may approach the problem of the existence of finite energy solutions for (1.1) even with weaker linear estimates than the whole family of Strichartz inequalities. That is exactly what we are going to do here.

In 2d, local well-posedness inH_D^s(Ω)(see the next section for definition of that space),s >1, for the initial boundary value problem (1.1)–(1.2)–(1.3) can be obtained by “classical methods” and therefore one barely misses the key regularityH¹. Nevertheless it is known that for α2 (see Cazenave [13, Theorem 4.5.1], Brézis and

Gallouet [5], Vladimirov [38], Ogawa and Ozawa [25]) one can obtain the global existence ofH¹solution to (1.1) for a suitable class of potentials V. The work of M. Tsutsumi [32] shows that one could extend the result to α∈ ]2,3]if the data u0∈H₀¹(Ω) is such that such thatu0∈H₀¹(Ω). Here we will be able to extend these results to much more general nonlinearities. Even whenα3 we have a stronger result comparing to the above mentioned works since we obtain that the flow map is Lipschitz continuous on bounded sets ofH₀¹(Ω). On the other hand the considerations in [5,38,25,32] are valid on any domainΩ with smooth boundary without any geometric assumption.

The main difficulty in higher dimensions is that one needs to “gain at least 1/2 derivative” with respect to the classical well-posedness results. We will be able to do this as far asα < _d−²₂ which does not cover all possible nonlinearities forH¹theory in the caseΩ=R^d. Recall that (see, e.g., Kato [18]) whenΩ=R^dthe critical order of the nonlinearity for the well-posedness in the energy spaceH¹ turns out to be α= _d−⁴₂. It seems however that here we obtain the first global existence and uniqueness results in dimensionsd3 for (1.1)–(1.2)–(1.3) with large initial data. It should be mentioned that “small data techniques” can be applied to (1.1)–(1.2)–(1.3) under some geometric assumptions (which imply our non-trapping assumption) onΘ(see Y. Tsutsumi [34], M. Tsutsumi [33]).

That approach yields the global existence of small amplitude solutions to (1.1)–(1.2)–(1.3) in any dimension for nonlinearities of sufficiently high order (and initial data sufficiently smooth).

We now state our result concerning finite energy solutions.

Theorem 1. Suppose thatα <_d−2² ,V (z)−C(1+ |z|)^β,β <2+⁴_d and thatΘis non-trapping. Then

(1) For any u₀∈H₀¹(Ω) the initial boundary value problem (1.1)–(1.2)–(1.3) has a unique global solution u∈C(R;H₀¹(Ω))satisfying the conservation laws (1.4a), (1.4b).

(2) Ifd=2,3,4, for anyT >0 the flow mapu₀→uis Lipschitz continuous from any bounded set ofH₀¹(Ω)to C([−T , T];H₀¹(Ω)).

(3) Whend=3 andα=2 statements (1) and (2) hold providedu_0H¹

0(Ω)be sufficiently small.

Our proof of Theorem 1 strongly relies on a local smoothing effect for the free evolution exp(itD), where _D is the Laplace operator acting on L²(Ω), with domain D=H²(Ω)∩H₀¹(Ω). This phenomenon has been first observed in the case ofR^d in the works of Constantin and Saut [14], Sjölin [28] and Vega [37]. It was later generalized by many authors to different perturbations of the flat Laplacian (see Ben Artzi and Klainerman [1], Constantin and Saut [15], Doi [17] . . . ). It is important to realize that the local smoothing can be reduced to bounds on the cut-off resolvent of the corresponding stationary operator. Since such resolvent estimates are fortunately available for the exterior problem of non-trapping obstacle we will be able in Section 2 below to derive a local smoothing estimate for exp(it_D) and hence to extend the above mentioned results to the case of boundary value problems, a fact which seems to be of independent interest. Following a strategy suggested by Staffilani and Tataru [30], we shall also be able to prove that away from the obstacle the free evolution enjoys the Strichartz estimates exactly as for the flat space. Once we have the linear estimates we perform the usual Picard iteration to getH¹ well-posedness for the nonlinear problem. Let us mention that the assumptionV (z)−C(1+ |z|)^β in Theorem 1 is crucial for the global existence of solutions. For example, ifα=2, d =2 andV (z)= −|z|⁴, regular solutions can develop singularities in finite time (see [11], Remark 1.1). Blow up phenomena for boundary problems with more general nonlinearities are displayed in Kavian [19] by using viriel type identities, however it is not clear to us whether these arguments can be applied to exterior domains. Note that despite of the fact that the functionalF is not Lipschitz continuous on bounded sets ofH₀¹(Ω), due to the “dispersive properties” of the linear part of the equation, the flow map turns out to have that property at least ford4. It is an interesting problem to check that property in dimensions higher than 4.

Our second global well-posedness result deals withL²solutions.

Theorem 2. Suppose thatα <2/d and thatΘ is non-trapping. Then for anyu₀∈L²(Ω)the initial boundary value problem (1.1)–(1.2)–(1.3) has a unique global solution in the following classX:

•Ifα <¹_d thenX=C(R;L²(Ω)).

• If α _d¹ thenX is any of the spaces C(R;L²(Ω))∩L^p_loc(R;L^q(Ω)) where (p, q) satisfy ¹_p +^d_q = ^d₂, 2< p <^2(α_αd⁺₋¹⁾₁.

Moreover

(1) The solutionusatisfies the conservation law (1.4a).

(2) For any pair(p, q)satisfying 2< p∞, ¹_p+^d_q=^d₂ one hasu∈L^p_loc(R;L^q(Ω)).

(3) For any T > 0 the flow map u₀ → u is Lipschitz continuous from any bounded set of L²(Ω) to C([−T , T];L²(Ω)).

Remark 1.1. The result of Theorem 2 is in strong contrast with the case of a bounded open setΩ. Indeed, in [12], we proved that, ifΩis a ball, there exists someα0>0 such that, for everyα∈ ]0, α0], the Cauchy problem for

i∂_tu+_Du=

1+ |u|²_α/2 u

is not well-posed onL²(Ω)in the sense of Theorem 2.

Remark 1.2. For the sake of conciseness, we have chosen to restrict the study to the case of Dirichlet boundary con- ditions. However, the case of Neumann conditions could be handled using the same ideas (see Remarks 2.3 and 2.9).

Remark 1.3. The structural assumptions on the nonlinear interactionF are needed to establish the global well- posedness. If one is interested only in local in time results then we can assume only the following growth conditions,

F (z₁)−F (z₂)|z₁−z₂|

1+ |z₁| + |z₂|α

(D_z,z¯F )(z₁)−(D_z,z¯F )(z₂)|z₁−z₂| ,

1+ |z₁| + |z₂|max{α−1,0}.

The rest of the paper is organized as follows. We complete this section by introducing some notation. In Section 2, we first state the Sobolev embeddings we need for the sequel. Then we state some estimates for the cut-off resolvent of_D. Further we prove local smoothing estimates in the form needed for the proof of the crucial nonlinear estimate. We complete Section 2 by proving Strichartz type inequalities for exp(itD). We distinguish the cases when we evaluate the free wave away from the obstacle. Section 3 is devoted to the proof of Theorem 1 while Section 4 deals with the proof of Theorem 2.

Notations. ForT >0,p∈ [1,+∞], ifXis a Banach space, we denote byL^p_TXthe Banach space ofXvalued functions on[0, T]equipped with the following norm

fL^p_TX= T

0

f (t)^p

Xdt

1/p

with the usual modification forp= +∞. For any positiveAandBthe notationAB(respectivelyAB) means that there exists a positive constantcsuch thatAcB(respectivelyAcB).

2. Linear estimates

2.1. Functional spaces and embeddings

LetΩ⊂R^d,d2, be a smooth domain. Fors0,p∈ [1,+∞], we denote byW^s,p(Ω)the Sobolev spaces on Ω. We write L^p(Ω)and H^s(Ω) instead of W^0,p(Ω) andW^s,2(Ω) respectively. For s∈Z+ the norm in

W^s,p(Ω)can be expressed in an explicit way while for non-integer values ofsmore care is needed and one can define in this case the spacesW^s,p(Ω)by suitable interpolation (see [2]). ByW₀^1,q(Ω), we denote the closure of C₀^∞(Ω)inW^1,q(Ω). The spaceW₀^1,2(Ω)is usually denoted byH₀¹(Ω). IfΩ has compact boundary then we have

W^1,q(Ω)∩H₀¹(Ω)⊂W₀^1,q(Ω), q2. (2.1)

In order to obtain (2.1), we can use thatW₀^1,q(Ω)can be identified with the kernel of the natural trace map from W^1,q(Ω)toL^q(∂Ω)and then use thatL^q(∂Ω)⊂L²(∂Ω),q 2, which follows from the compactness of the boundary. By_Dwe denote the Dirichlet Laplacian onΩ. The domain of_DisH₀¹(Ω)∩H²(Ω). Fors0, we can define(−D+1)^s/2via the functional calculus of self-adjoint operators. We denote byH_D^s(Ω)the domain of (−_D+1)^s/2. It is known that (see, e.g., [31])

H_D¹(Ω)=H₀¹(Ω) (2.2)

and we will make often use of (2.2) without explicit mention. We next defineH_D⁻¹(Ω)(this space is often denoted in the literature simply byH⁻¹(Ω)) as the dual ofH_D¹(Ω). Then we defineH_D^s(Ω)fors∈ [−1,0]via interpolation and due to Corollary 4.5.2 in [2], we have the duality betweenH_D^s(Ω)andH_D^−s(Ω)fors∈ [0,1]. We now state the Sobolev embeddings that will be used in that paper.

Proposition 2.1. LetΩ⊂R^d,d2 be smooth domain. Then the following continuous embeddings hold H₀¹(Ω)⊂L^p(Ω), 2p 2d

d−2 (p <+∞ifd=2), (2.3)

H_D^s(Ω)⊂L^p(Ω), 1 2− 1

p= s

d, s∈ [0,1[, (2.4)

H_D^s+1(Ω)⊂W^1,p(Ω), 1 2− 1

p =s

d, s∈ [0,1[, (2.5)

W^1,p(Ω)⊂L^q(Ω), 1 p −1

q =1

d,1p < q <+∞, (2.6)

W^s,p(Ω)⊂L^∞(Ω), s >d

p, p1, (2.7)

H^s+

1 p

D (Ω)⊂W^s,q(Ω), 1 p+d

q =d

2, p2, s∈ [0,1]. (2.8)

The proof of Proposition 2.1 follows from the standard Sobolev embeddings and the use of extension operators.

2.2. Resolvent estimates

Since−_Dis a positive self-adjoint operator the resolvent(−_D−λ)⁻¹is analytic inC\R⁺. In this section we collect several bounds for(−_D−λ)⁻¹whenλapproachesR⁺. We first state the high frequencies bound.

Proposition 2.2. For everyχ∈C₀^∞(R^d),d2, there exists a positive constantCsuch that for every|λ|1 and 0< ε1 one has

χ

−D−(λ±iε)²₋1

χ

L²(Ω)→L²(Ω)C|λ|⁻¹.

The result of Proposition 2.2, for which the non-trapping assumption plays a crucial role, is proven for|λ| 1 in greater generality by Lax and Phillips [21], Melrose and Sjöstrand [23,24], Vainberg [35], Vasy and Zworski [36].

We also refer to [8] for a self contained proof which, joined with the results in [6], would relax the smoothness

assumption. The boundedness of the cut-off resolvent onL²(Ω)for finite|λ| =0 results from Rellich uniqueness theorem (see [21] or [7, Annexe B.1]). Proposition 2.2 can be also stated as a weightedL²estimate for the operator (−_D−(λ±iε)²)⁻¹.

Remark 2.3. Proposition 2.2 is also true for the resolvent associated to Neumann boundary conditions. The proof in this case is the same, using propagation of singularities arguments.

Next we state the small frequencies bound.

Proposition 2.4. Assume thatΘ= ∅. Then for everyχ∈C₀^∞(R^d),d2, the cut-off resolventχ (−_D−(λ± iε)²)⁻¹χ,|λ|1, 0< ε1 is a bounded operator onL²(Ω)with an operator norm independent ofλandε.

For the proof of Proposition 2.4, we refer to [7, Annexe B.2]. Remark that this latter proof breaks down if Θ= ∅since the Poincaré inequality is used to control the localL²-norm of a function by the local L²-norm of its gradient (that is is whyΘ= ∅is required). Propositions 2.2, 2.4 can be used to prove the boundedness of the cut-off resolvent between Sobolev spaces as shows the next proposition.

Proposition 2.5. Assume thatΘ= ∅. Then for everyχ∈C₀^∞(R^d), d2, χ0, everys−1 there exists a positive constantC such that for everyλ∈Rand 0< ε1 one has

χ

−D−(λ±iε)²₋1

χ

H_D^s(Ω)→H_D^s+1(Ω)C. (2.9)

Remark 2.6. For Rez <0, an integration by parts gives χ (−_D−z)⁻¹χ

H_D^s(Ω)→H_D^s+1(Ω) C

(1+ |Rez|)^1/2 (2.10)

which, in the region−ε²<Rez <0, implies the same estimate as in (2.9) (one can get even better).

Proof of Proposition 2.5. Setµ=λ±iεand letuandf be such that

(D+µ²)u=χf. (2.11)

We multiply (2.11) byχu¯and after integration onΩ, we get

−

χ|∇u|²+µ²

χ|u|²−

(∇u,∇χ )χ₁u¯=

χ²fu,¯

whereχ₁∈C₀^∞(R^d),χ₁0, is equal to one on the support ofχand(·,·)denotes the scalar product inC^d. Since χχ₁²and using that|µ||λ| +1 we obtain that for everyδ >0,

χ|∇u|²

|λ| +12

χ₁²|u|²+δ

|∇χ|²|∇u|²+(4δ)⁻¹

|χ1u|²+

χ²fu¯ . Since|∇χ|²χ and by choosingδsmall enough, we get

χ|∇u|²

|λ| +1₂

χ₁u²_L2(Ω)+ χf²_L2(Ω). Using Propositions 2.2, 2.4, we deduce that

|λ| +1

χ1uL²(Ω)

|λ| +1χ₁(D+µ²)⁻¹(χ1χf )

L²(Ω)χfL²(Ω)

and therefore

χ|∇u|²χf²_L2(Ω).

Using again Propositions 2.4 and 2.2, we get χ u_H¹

D(Ω)χf_L²_(Ω).

This completes the proof of Proposition 2.5 fors=0, i.e., χ

−_D−(λ±iε)²₋1

χ

L²(Ω)→H_D¹(Ω)C. (2.12)

Dualizing (2.12), we obtain, χ

−_D−(λ±iε)²₋1

χ

H_D⁻¹(Ω)→L²(Ω)C (2.13)

which yields Proposition 2.5 withs= −1.

We next prove it fors=1. Let againuandf be such that (2.11) holds andχ1∈C₀^∞(R^d),χ10, be equal to one on the support ofχ. Write

χ u_H²

D(Ω)≈ χ u_H¹

D(Ω)+D(χ u)

L²(Ω). Sinceχ u_H¹

D(Ω)can be estimated by means of (2.12), we only need to bound_D(χ u)L²(Ω). Further we write _D(χ u)=χ _Du+ [_D, χ]χ₁u

and using that the commutator[D, χ]is bounded fromH_D¹(Ω)toL²(Ω), we get [D, χ]χ₁u

L²(Ω)χ₁u_H¹

D(Ω). Using (2.12), we obtain

χ1u_H¹

D(Ω)=χ1(D+µ²)⁻¹(χ1χf )

L²(Ω)χfL²(Ω).

It remains to boundχ DuL²(Ω). SinceDu=χf−µ²u, we deduce thatDu∈H_D¹(Ω). SinceDusolves the equation

(D+µ²)(Du)=D(χf ), a use of (2.13) yields

χ _Du_L²_(Ω)χ_1D(χf )_H−1

D (Ω)χf_H¹

D(Ω),

where we used that χ1D is bounded fromH_D¹(Ω) to H_D⁻¹(Ω). This proves the result for s=1. Since we obtained (2.9) fors= −1 ands=1 we can use an interpolation argument to get it fors∈ [−1,1]. Applying the operatorDto the equation and an induction argument give the result for anys∈N. Finally we use interpolation to get it for anys1. 2

2.3. Local smoothing

Now we are going to use the resolvent bounds of the previous section to deduce several estimates for the linear Schrödinger equation posed onΩ with Dirichlet boundary conditions. This procedure is known in the literature at least for the homogeneous estimates (see, for example, [1]). The proof presented here is based on the observation that it is sufficient to establish the non-homogeneous bound and then all other estimates follow by the so called T Targument together with a simple symmetry consideration. Finally the nonhomogeneous estimate is proven by

performing Fourier transform in time and applying Proposition 2.5. From now on we shall work on positive time intervals only. Of course similar considerations apply to negative time intervals.

Proposition 2.7. Assume thatΘ= ∅. Then for everyT >0, for everyχ∈C₀^∞(R^d),d2, χ u_L²

TH_D^s+1(Ω)Cχf_L²

TH_D^s(Ω), (2.14)

wheres∈ [−1,1]andu(t)=_t

0e^{i(t−τ )D}χf (τ )dτ, χ v_L2

TH_D^s+1/2(Ω)Cv₀H_D^s(Ω), (2.15)

wheres∈ [0,1]andv(t)=e^{it D}v₀.

Remark 2.8. Remark that in the estimate abovev₀is not assumed to have compact support. Remark also that the proof will show that the constantsC do not depend onT, i.e., the estimates are global in time.

Proof of Proposition 2.7. We first prove (2.14). Extendf (τ,·)by zero forτ /∈ [0, T]. According to the support properties off andutheir Fourier transforms (in time) are holomorphic in the domain{Imz <0}and satisfy the equation

(−z+_D)u(z,ˆ ·)=χf (z,ˆ ·).

Takingz=λ−iε,λ∈R,ε >0, lettingεtend to zero, using Proposition 2.5 and Remark 2.6, we get χuˆ_L2(R;H_D^s+1(Ω))χfˆ_L²_(R;H^s

D(Ω)), s∈ [−1,1].

The proof of (2.14) is completed by observing that the Fourier transform of any function fromRto a Hilbert space Hdefines an isometry onL²(R;H ).

Now we turn to the proof of (2.15). We first prove it fors=0, i.e. if we denote byAthe operator which to givenu0∈L²(Ω)associatesχe^{it D}u0, we need to prove thatAis bounded fromL²(Ω)toL²_TH_D^1/2(Ω). But the continuity ofAfromL²(Ω)toL²_TH_D^1/2(Ω)is equivalent to the continuity of its adjoint

(Af )(t)= T

0

e^{−iτ D}χf (τ )dτ

from L²_TH_D^−1/2(Ω) to L²(Ω), which in turn is equivalent to the continuity of AA from L²_TH_D^−1/2(Ω) to L²_TH_D^1/2(Ω). Write

(AAf )(t)= T

0

χe^{i(t−τ )D}χf (τ )dτ

= t

0

χe^{i(t−τ )D}χf (τ )dτ+ T

t

χe^{i(t−τ )D}χf (τ )dτ

and it suffices to apply (2.14) withs= −¹₂(together with time inversion for the second term) in order to conclude thatAAis bounded fromL²_TH_D^−1/2(Ω)toL²_TH_D^1/2(Ω). This completes the proof of (2.15) fors=0.

We now prove (2.15) for s=1. Observe that the boundedness of χe^{it D} from H_D¹(Ω) to L²_TH_D^3/2(Ω) is equivalent to the continuity of(−_D+1)χe^{it D}fromH_D¹(Ω)toL²_TH_D^−1/2(Ω). Write

(−D+1)χe^{it D}=χ (−D+1)e^{it D}− [D, χ]e^{it D}.

Letχ˜∈C₀^∞(R^d)be such thatχ˜=1 on the support ofχ. Then [_D, χ]e^{it D}u₀

L²_TH_D^−1/2(Ω)[_D, χ] ˜χe^{it D}u₀

L²_TH_D^−1/2(Ω)

χe˜ ^{it D}u₀

L²_TH_D^1/2(Ω)

u0L²(Ω),

where in the last line we used that (2.15) fors=0 is already established. Therefore[_D, χ]e^{it D} is bounded fromL²(Ω)toL²_TH_D^−1/2(Ω)and in particular fromH_D¹(Ω)toL²_TH_D^−1/2(Ω). Hence it remains to prove that the operator

B:=χ (−_D+1)e^{it D}

is bounded fromH_D¹(Ω)toL²_TH_D^−1/2(Ω)or equivalently thatB(−D+1)⁻¹Bis bounded fromL²_TH_D^1/2(Ω)to L²_TH_D^−1/2(Ω). An easy computation yields

B(−_D+1)⁻¹Bf (t)=

T

0

χ (−_D+1)e^{i(t−τ )D}χf (τ )dτ

=(−_D+1)χ T

0

e^{i(t−τ )D}χf (τ )dτ+ [χ , _D] T

0

e^{i(t−τ )D}χf (τ )dτ.

Observe that (−_D+1)χ

T

0

e^{i(t−τ )D}χf (τ )dτ=(−_D+1)(AAf )(t)

and therefore using (2.14) withs= ¹₂ together with a splitting of the integration on[0, T] as shown above, we readily get that(−D+1)AAis bounded fromL²_TH_D^1/2(Ω)toL²_TH_D^−1/2(Ω). Next we write

[χ , _D] T

0

e^{i(t−τ )D}χf (τ )dτ = [χ , _D](AAf )(t)

and again due to (2.14) withs=¹₂we obtain the boundedness of[χ , D]AAfromL²_TH_D^1/2(Ω)toL²_TH_D^−1/2(Ω).

This completes the proof of (2.15) fors=1. We finally obtain (2.15) fors∈ [0,1]via an interpolation argument which ends the proof of Proposition 2.7. 2

Remark 2.9. If one considers the Neumann Laplacian_N, we can obtain a similar result as in Proposition 2.7, with constants depending on the time interval. Indeed takeΨ ∈C₀^∞(R)equal to 1 close to 0 and decompose

u=Ψ (−N)u+(1−Ψ )(−N)u,

f=Ψ (−_N)f+(1−Ψ )(−_N)f, (2.16)

v₀=Ψ (−_N)v₀+(1−Ψ )(−_N)v₀.

Taking into account Remark 2.3, we can apply the strategy of the proof of Proposition 2.7 to(1−Ψ )(−_N)u, (1−Ψ )(−_N)f and(1−Ψ )(−_N)v₀to obtain estimates similar as (2.14), (2.15) for the contributions of these terms. To deal with the contributions of the other terms, we simply use the conservation of theL²norms and the fact that for these parts, theL²andH_N^k norms are equivalent (due to the spectral cut-off). This argument gives an L^∞in time estimate for these terms which can be converted (using Hölder inequality) into anL²in time estimate.

2.4. Strichartz type estimates

In the next proposition we show that away from the obstacle the free evolution satisfies the usual Strichartz bounds. We will use a strategy of [30] where similar considerations are performed in the context ofC²short range perturbation of the free Laplacian onR^d.

Proposition 2.10. For everyT >0, for everyχ∈C^∞₀ (R^d),χ=1 close toΘthere existsC >0 such that (1−χ )u

L^p_TW^s,q(Ω)Cu₀H_D^s(Ω), (2.17)

wheres∈ [0,1],u(t)=e^{it D}u₀and(p, q),p >2, is any Strichartz admissible pair, i.e.

2 p +d

q =d

2. (2.18)

Proof. Setv(t)=(1−χ )e^{it D}u₀. Thenvsatisfies the equation (i∂t+)v= [D,−χ]u,

v(0)=(1−χ )u0. (2.19)

Sinceχ=1 close toΘ, Eq. (2.19) can be regarded in the whole spaceR^d. Hence v(t)=e^{it 0}(1−χ )u₀+

t

0

e^{i(t−τ )0}[_D,−χ]u(τ )dτ,

where₀ is the free Laplacian onR^d and therefore the contribution of(1−χ )u₀ satisfies the usual Strichartz estimate and we have reduced the problem to the study of

w(t):=

t

0

e^{i(t−τ )0}[_D,−χ]u(τ )dτ. (2.20)

Using Proposition 2.7, we get [D,−χ]u

L²_TH^−1/2(R^d)u0L²(Ω).

Let Λ₀ϕ(t, x):=e^{it 0}ϕ(x). We proceed by using the smoothing effect for Λ₀. Applying inequality (1.10) in Corollary 2 and inequality (3.4) in Proposition 2 from [1], we have, for every cutoff functionχ₀inR^d,

(1−0)^1/4(χ0Λ0ϕ)

L²([0,T]×R^d)ϕL²(R^d). The dual inequality reads

Λ^∗₀

χ0(1−0)^1/4ψ

L²(R^d)ψL²([0,T]×R^d). Combining with Strichartz estimates onR^dforΛ0, this yields

Λ₀Λ^∗₀

χ₀(1−₀)^1/4ψ

L^p_TL^q(R^d)u₀L²(Ω) (2.21)

Notice that

Λ0Λ^∗₀(f )(t)= T

0

e^{i(t−τ )0}f (τ )dτ.

However we are interested in estimatingw(t)defined by (2.20) rather than Λ₀Λ^∗₀

[_D,−χ]u .

For this it suffices to use the following result due to Christ and Kiselev [16].

Theorem (M. Christ and A. Kiselev). Consider a bounded operator T:L^p(R;B₁)→L^q(R;B₂)

given by a locally integrable kernelK(t, s)with values in bounded operators fromB₁toB₂whereB₁andB₂are Banach spaces. Suppose thatp < q. Then the operator

T ψ(t)=

s<t

K(t, s)ψ(s)ds

is bounded fromL^p(R;B₁)toL^q(R;B₂)and TL^p(R;B₁)→L^q(R;B₂)

1−2^{−(p−1−q−1)}₋1

TL^p(R;B₁)→L^q(R;B₂).

In view of (2.21), we apply Christ–Kiselev’s theorem to K(t, s)=1_[0,T](t)1_[0,T](s)e^i(t−s)0χ₀(1−₀)^1/4

and we setψ=(1−₀)^−1/4[_D,−χ]uwithχ₀=1 near the support ofχ. This yields, forp >2, wL^p_TL^q(R^d)[_D,−χ]u

L²_TH^−1/2(R^d)u₀L²(Ω). This completes the proof fors=0.

The cases=1 can be treated similarly simply by differentiating the first equation of (2.19), considered as equation on the whole spaceR^d. Since we established (2.17) fors=0 and s=1 an interpolation argument completes the proof of Proposition 2.10. 2

Now we state a Strichartz estimate (with loss of derivative) for e^{it D}. Proposition 2.11. For everyT >0 there existsC >0 such that

u_L^p

TW^s,q(Ω)Cu_0Hs+1/p

D (Ω), (2.22)

wheres∈ [0,1],u(t)=e^{it D}u₀and(p, q),p >2, satisfies (2.18).

Remark 2.12. In [9], Strichartz inequalities as (2.22) are proven for the free Schrödinger equation posed on a compact Riemannian manifold (without boundary). Although the estimates are the same, the ideas behind are very different. In [9], the loss of derivatives (optimal for the endpoint cases on the sphere) came from the fact that we were able to prove the usual estimates (without loss) only for small time intervals (depending on the frequency).

Here the loss (certainly not optimal . . . ) comes from the fact that close to the boundary, we perform simply Sobolev embeddings together with the local smoothing. The gain arising from the smoothing effect tells us that the wave spends few time close to the obstacle.

Remark 2.13. In [29] Smith and Sogge prove that the wave equation posed on the exterior of strictly convex obstacle satisfies the same Strichartz estimates as the solution of the wave equation posed onR^d. It is natural to expect that the techniques of [29] combined with the semi-classical approach of [9] can provide the full set of Strichartz inequalities, at least locally in time, for the Schrödinger equation posed on the exterior of strictly convex

obstacle. Such a result would extend the well-posedness theory of the flat space to the case of the exterior of a strictly convex obstacle.

Proof of Proposition 2.11. Considerχ∈C₀^∞(R^d)equal to 1 close toΘand decompose u(t)=χe^{it D}u₀+(1−χ )e^{it D}u₀:=v(t)+w(t).

Due to Proposition 2.10, we obtain thatw(t)satisfies the usual Strichartz estimates (without losses) and therefore we only need to evaluatev(t). Using Proposition 2.7, we get

v_L²

TH_D¹(Ω)u_0H^1/2

D (Ω). (2.23)

Next we use an energy argument to deduce, v_L∞

TL²(Ω)u_0L²_(Ω). (2.24)

Interpolating between (2.23) and (2.24) with weights_p² and 1−_p² respectively gives v_L^p

TH_D^2/p(Ω)u_0H^1/p

D (Ω).

Since _p² + ^d_q = ^d₂, using Proposition 2.1 we have that H_D^2/p(Ω)⊂L^q(Ω) and the embedding is continuous, therefore

v_L^p

TL^q(Ω)u0_H^1/p

D (Ω)

which completes the proof of Proposition 2.11 whens=0.

Next we consider the cases=1. Applying an energy argument, we get v_L∞

TH_D^p/(p−2)(Ω)u0_H^p/(p−2)

D (Ω). (2.25)

Interpolation between (2.23) and (2.25) with weights_p² and 1−_p² respectively gives v_L^p

TH_D^1+2/p(Ω)u_0H1+1/p

D (Ω). (2.26)

Due to Proposition 2.1 the continuous embeddingH_D^1+2/p(Ω)⊂W^1,q(Ω)holds which together with (2.26) ends the proof fors=1. The cases∈ [0,1]can now be treated by interpolation. 2

Now we state the main result of this section.

Proposition 2.14. For everyT ∈ ]0,1]there existsC >0 such that u_L^p

TW^s,q(Ω)Cu₀H_D^s(Ω), (2.27)

wheres∈ [0,1],u(t)=e^{it D}u0and(p, q),p2 satisfies 1

p +d q =d

2. (2.28)

Moreover u_L^p

TW^s,q(Ω)Cf_L¹

TH_D^s(Ω), (2.29)

wheres∈ [0,1],u(t)=_t

0e^{i(t−τ )D}f (τ )dτ and(p, q),p >2 satisfies (2.28).

Proof. Letχ∈C₀^∞(R^d)such thatχ=1 close toΘ. The triangle inequality yields, u_L^p

TW^s,q(Ω)χ u_L^p

TW^s,q(Ω)+(1−χ )u

L^p_TW^s,q(Ω)