HAL Id: hal-03060388
https://hal.archives-ouvertes.fr/hal-03060388v2
Preprint submitted on 16 Jun 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
DISPERSION FOR THE WAVE AND
SCHRÖDINGER EQUATIONS OUTSIDE A BALL
AND COUNTEREXAMPLES
Oana Ivanovici, Gilles Lebeau
To cite this version:
Oana Ivanovici, Gilles Lebeau. DISPERSION FOR THE WAVE AND SCHRÖDINGER
EQUA-TIONS OUTSIDE A BALL AND COUNTEREXAMPLES. 2021. �hal-03060388v2�
DISPERSION FOR THE WAVE AND SCHR ¨ODINGER EQUATIONS OUTSIDE A BALL AND COUNTEREXAMPLES
OANA IVANOVICI1 AND GILLES LEBEAU2
Abstract. We consider the wave and Schr¨odinger equations with Dirichlet boundary conditions in the exterior of a ball in Rd. For d = 3 a sharp global in time parametrix is constructed to derive
sharp dispersive estimates, matching the R3 case, for all frequencies (low and high). For d ≥ 4, we
provide an explicit solution to the wave equation with data a smoothed out δQ0 at large frequency
1/h, where Q0 is a point at large distance s ∼ h−1/3 from the center of the ball: the decay rate
of that solution exhibits a (t/h)d−34 loss with respect to the boundary less case, that occurs for all t ∼ 2s with corresponding observation points around the mirror image of Q0 with respect to the
center of the ball (at the Poisson Arago spot). A similar counterexample is also obtained for the Schr¨odinger flow.
1. Introduction
We consider the linear wave equation on an exterior domain Ω = Rd\ Θ where Θ is a smooth, compact, strictly convex obstacle and ∆D is the Laplacian with constant coefficients and Dirichlet boundary conditions: (1.1) (∂2 t − ∆D)u(x, t) = 0, x ∈ Ω, u(x, 0) = u0(x), ∂tu(x, 0) = u1(x), u(x, t) = 0, x ∈ ∂Ω .
We will also address the Schr¨odinger equation (1.2) (i∂t+ ∆D)v(x, t) = 0, x ∈ Ω, v(x, 0) = v0(x), v(x, t) = 0, x ∈ ∂Ω.
Heuristically, dispersion relates to how waves spread out with time, while retaining their energy: it quantifies decay for waves’ amplitude. In Rd, the half-wave propagator e±it
√
−∆ can be computed explicitly, yielding the dispersion estimate, for χ ∈ C0∞(]0, ∞[) and Dt= −i∂t,
(1.3) kχ(hDt)e±it √ −∆k L1(Rd)→L∞(Rd)≤ C(d)h−dmin{1, (h/|t|) d−1 2 } .
For the Schr¨odinger propagator, dispersion follows at once from its explicit Gaussian kernel:
(1.4) ke±it∆k
L1(Rd)→L∞(Rd)≤ C(d)|t|−d/2. These estimates, together with the propagator being unitary on L2
(Rd), provide all the necessary tools to obtain the whole set of Strichartz estimates (although the endpoints are more delicate, see [13]). Strichartz estimates for Rd and manifolds without boundary have been understood for some time (see [29],[7], [8], [12], [18] for Rd and also [26], [30] for low regularity metrics). Even though the boundary-less case has been well understood, obtaining results for the case of manifolds with boundary has been surprisingly elusive.
Our aim in the present paper is to prove dispersion for wave and Schr¨odinger equations outside a ball in Rd
: for d = 3 we prove that both linear flows satisfy the same dispersion bounds as in R3. In higher dimensions d ≥ 4 we prove that these estimates cannot hold as in Rd as losses do appear at the so called Poisson-Arago spot.
1
Sorbonne Universit´e, CNRS, LJLL, F-75005 Paris, France
2
Universit´e Cˆote d’Azur, CNRS, Laboratoire JAD, France
E-mail addresses: oana.ivanovici@math.cnrs.fr, gilles.lebeau@univ-cotedazur.fr. Oana Ivanovici was supported by ERC grant ANADEL 757 996.
While several positive results on dispersive effects on exterior domains have been established since the mid-90’s, the question about whether or not dispersion did hold remained completely open, even for the exterior of a ball. We recall that (local in time) Strichartz estimates were proved to hold as in Rd for the wave equation in [28] and for the Schr¨odinger equation in [11], with arguments that did not require the dispersion estimate to be known (both estimates are known to extend globally in time using local energy decay or local smoothing estimates). Since there is no obvious concentration of energy, like in the case of a generic non-trapping obstacle, where concave portions of the boundary can act as mirrors and refocus wave packets, one would expect dispersive estimates to hold outside strictly convex obstacles (for spherically symmetric functions this was proved, outside a sphere in [17]). Our main positive result is the following:
Theorem 1.1. Let Θ = B3(0, 1) ⊂ R3
be the unit ball in R3
and set Ω = R3\ Θ. Let ∆D denote the Laplace operator in Ω with Dirichlet boundary condition and let χ ∈ C0∞(0, ∞).
(1) Dispersion holds for the wave flow in Ω like in R3: kχ(hDt)e±it √ −∆Dk L1(Ω)→L∞(Ω). h−3min{1, h |t|}. (2) Dispersion holds for the classical Schr¨odinger flow in Ω:
ke±it∆DkL1
(Ω)→L∞(Ω). |t|−3/2.
In Theorem1.1and in the remaining of the paper, A . B means that there exists a constant C such
that A ≤ CB and this constant may change from line to line but is independent of all parameters. It will be explicit when (very occasionally) needed. Similarly, A ∼ B means both A . B and B . A. Remark 1.2. Having the full dispersion in 3D we can immediately obtain the endpoint Strichartz estimates, following [13]. Theorem1.1also helps in dealing with the non-linear Schr¨odinger equation (see e.g. the recent work [32], which makes use of our dispersion estimate to reprove the main result from [15], on global well-posedness for the defocusing energy critical NLS in 3D).
Remark 1.3. We claim that Theorem 1.1 still holds with Ω replaced by the exterior of an obstacle Θ with smooth, strictly geodesically concave boundary. By the last condition we mean that the second fundamental form on the boundary of Θ is positive definite. The general case of Theorem1.1
requires arguments that may be seen as perturbative of those, mostly explicit, used in the exterior of a ball, and we will deal with it elsewhere. However, we stress out that the proof of the dispersive bounds in the general case, at least when both the source and the observation points are close to the boundary, is essentially the same as outside a ball: hence the proof of Theorem1.1in this particular case is a crucial step toward the general result. Moreover, its explicit parametrix construction provides counterexamples in dimensions d ≥ 4.
A loss in dispersion may be related to a cluster point: such clusters occur when optical rays, sent along different directions, are no longer diverging from each other. If a point source illuminates a disk or a ball, Huygens’s principle says that every point of the obstacle acts as a new point source: diffraction by the obstacle deviates light on the boundary which then arrives at the center of the shadow behind the obstacle in phase and constructively interferes. This results in a bright spot at the shadow’s center. Therefore, our intuition tells us that if there is a location where dispersion could fail, this could happen at the Poisson-Arago spot (which really is part of the line joining the source and the center of the obstacle: in physical experiments, one may place a screen somewhat symmetrically to the source of light, hence the choice of wording): it turns out that, indeed, that region is brighter than the illuminated regime when d ≥ 4.
Theorem 1.4. Let d ≥ 4 and let Θ = Bd(0, 1) be the unit ball in Rd. Set Ωd
= Rd\ Bd(0, 1) and let ∆D denote the Laplace operator in Ωd with Dirichlet boundary condition. Let Q±(s) ∈ Ωd be two points of Ωd, symmetric with respect to the center of the ball Bd(0, 1), at distance s > 1 from 0 ∈ Rd. Let χ ∈ C0∞ be supported in a sufficiently small neighborhood of 1 and equal to 1 near 1 : there exist h0 < 1 and constants 0 < γ1 < γ2 . 1 (independent of h) such that, for all 0 < h ≤ h0 and all arcsin(1/s) = γh1/3 with γ ∈ [γ1, γ2], the following holds at t = 2(arcsin1s+√s2− 1) ∼ 2h−1/3
(χ(hDt)e it√−∆D(δ Q+(s)) (Q−(s)) ∼ h −dh t d−12 h−d−33 .
For the classical Schr¨odinger operator, let 0 < h ≤ h20 and let arcsin(1/s) = γh1/6 for all γ ∈ [γ1, γ2]. Then at t = h1/3, the following holds
(χ(hDt)e it∆D(δ Q+(s)) (Q−(s)) ∼ h −d 6− d−3 6 .
For the semi-classical Schr¨odinger flow associated to the operator ih∂t− h2∆D, replace t = h1/3above by t = h−2/3. For d ≥ 4, these estimates contradict the usual (flat) ones.
Remark 1.5. For d ≥ 4, Theorem 1.4 provides a first example of a domain on which global in time Strichartz estimates do hold like in Rd (see [28], [11]) while dispersion fails.
Remark 1.6. The proof of Theorem1.4is based on the explicit parametrix (for source and observation points at equal distance from the ball) that is obtained in the proof of Theorem1.1and then generalized to higher dimensions.
Remark 1.7. Notice that a ”twisted” sphere in higher dimensions is still a good model to provide counterexamples. However, in physical experiments the Arago spot is very sensitive to small-scale deviations from the ideal circular cross-section: if there does not exist a ”source” point whose apparent contour (defined as the boundary of the set of points that can be viewed from the source) may produce constructive interferences, then the odds of obtaining a bright spot in the shadow region drastically decrease. If the cross-section of the obstacle deviates slightly from a circle, then the shape of the point-source Poisson-Arago spot changes and becomes a caustic. In particular, if the object has an ellipsoidal cross-section, the Poisson-Arago spot has the shape of an evolute (i.e. the locus of the center of curvature while moving along the ellipsoid).
Remark 1.8. The discussion above illustrates that a loss in dispersion occurs for obstacles which ”look very much” like even a tiny portion of the sphere Sd−2(at least viewed from specific locations). One may conjecture that there are many generic convex obstacles in Rd, d ≥ 4 for which dispersion does hold. However, in higher dimensions d ≥ 5, we may expect obstacles for which losses in dispersion may occur with different powers (less than h−(d−3)/3 and related to lower dimensional spheres within the cross-section). These interesting issues will be addressed elsewhere.
In optics, the Poisson-Arago spot (also known as the Fresnel bright spot) is a bright point that appears when a circular object is illuminated by a source of light, at the center of a screen located in a circular object’s shadow, and due to Fresnel diffraction; it played an important role in the discovery of the wave nature of light (now a common way to demonstrate that light behaves as a wave in undergraduate physics). In his report from 1819 to the Academy of Sciences, Arago mentions: ”L’un de vos commissaires, M.Poisson, avait d´eduit des int´egrales rapport´ees par l’auteur, le r´esultat singulier que le centre de l’ombre d’un ´ecran circulaire opaque devait, lorsque les rayons y p´en´etraient sous des incidences peu obliques, ˆetre aussi ´eclair´e que si l’´ecran n’existait pas. Cette cons´equence a ´et´e soumise `
a l’´epreuve d’une exp´erience directe, et l’observation a parfaitement confirm´e le calcul”. Arago later noted that the phenomenon had already been observed by Delisle [6] and [20], a century earlier. In his work [6] from 1715, Delisle mentions that when a small ball was illuminated by sunlight, the ball’s shadow contained alternating bright and dark rings concentric with the center of the ball’s shadow.
The mathematical methods used to investigate scattering problems depend heavily on the frequency of the wave motion. In particular, if the wavelength is very small, the scattering obstacle produces a shadow with an apparently sharp edge. Closer examination reveals that the edge of a shadow is not sharply defined but breaks into fringes; this phenomenon is know as diffraction. At the other end of the scale, obstacles that are very small compared to the wavelength disrupt the incident wave without producing an identifiable shadow. In ”Diverses exp´eriences d’optiques”, Maraldi states that ”la lumiere plus grande au milieu des boules plus petites, fait voir qu’elle circule en plus grande abondance et plus facilement autour des petites boules qu’autour des grandes” and [20, Fig.8 following page 142] shows light at the center of a ball’s shadow.
Investigating scattering of waves by obstacles has a long tradition in physics and applied mathemat-ics, stretching back to the work of Fresnel on diffraction (based on Huygens-Fresnel principle), which was the turning point for the wave theory of light. Since then, many researchers contributed to both qualitative and quantitative understanding of the subject, among which we should mention Kirchhoff, Watson, Fock, Frieldlander, Keller, Buslaev, Nussenzweig, Ludwig, Babich, etc. H¨ormander [10] made
geometrical optics a branch of mathematics, providing powerful tools and clarifying relevant concepts: ”a wealth of others ideas” from geometrical optics (referred to in the introduction to [10]) was later on exploited by Melrose, Taylor, Andersson, Eskin, Sj¨ostrand and Ivrii in dealing with propagation of singularities for mixed problems. Of particular importance in recent works is the Melrose and Taylor parametrix for the diffractive Dirichlet problem which gives the form of the solution to (1.1) near diffractive points (see also Zworski [33]).
The main difficulty in the exterior of convex obstacles is represented by the behavior of the diffracted wave, both near the shadow boundary and near the boundary of the obstacle; to quote (the opening line of) Ludwig [19], ”the asymptotic behavior of the field scattered by a convex object at high frequencies is extremely complicated”... The study of this particular regime (a neighbourhood of a diffractive - or grazing - ray, which hits the boundary without being deviated) started with the work of Keller [14] on the geometrical theory of diffraction - which can be seen as an extension of geometrical optics which accounts for diffraction. When the obstacle is regular, this theory shows that the shadow region is reached by rays that are ”creeping” on the boundary surface. These rays follow Fermat’s principle and therefore creep on the surface following the shortest possible path, in other words following boundary geodesics, before joining an observation point near the shadow region. After Keller’s work [14], it had been conjectured that the decreasing rate of the intensity of light in the shadow region had to be exp(−Cτ1/3), where τ is the frequency and where the constant C depends on the geometry of the geodesic flow on the boundary of the obstacle. Assuming an analytic boundary, this was proved by Lebeau [16] (and used as a crucial tool in the work [2]). The C∞ boundary case was proved a decade later by Harg´e and Lebeau [9] with a different argument.
The result of [9] was the starting point of this work, providing all the necessary tools to tackle the diffractive regime. Indeed, to deal with the diffractive part we will combine several key elements: the parametrix obtained by Melrose and Taylor near the diffractive regime, Keller’s asymptotic on the energy decreasing rate in the shadow region and Melrose’s theorem [21] regarding the decay for outgoing solutions to the wave equation in non-trapping domains.
As we recalled earlier, even though the dispersive estimates remained unknown for a long while, global in time Strichartz estimates are known to hold like in the flat case (in every dimension). Indeed, on a manifold for which the boundary is everywhere strictly geodesically concave (no multiply-reflected rays, no gliding rays - such as the complement in Rd of a strictly convex obstacle), the Melrose and Taylor parametrix was used in [28] to prove that Strichartz estimates for the wave equation do hold as in Rd (except for the endpoints). Later on, the first author showed in [11] the same result for the classical Schr¨odinger flow with Dirichlet boundary condition, for which an additional difficulty, related to the infinite speed of propagation of the flow, had to be overcome. The first step in [11] consisted in proving sharp, scale-invariant Strichartz for the semi-classical Schr¨odinger equation (e.g. on a time interval of size the wavelength for the classical equation) on compact manifolds with strictly concave boundaries of dimension d ≥ 2 (an example of which is provided by the so-called Sina¨ı billiard, e.g. a punctured torus) that we combined with the approach of [3]. What is worth mentioning here is that in order to obtain Strichartz in semi-classical time in [11] we bypassed dispersion: in fact, we side-stepped this issue by taking advantage, as in [28], of the L2 continuity of certain operators to reduce consideration to operators like those on a manifold without boundary: we want to stress out that this approach is very unlikely to work when one is interested in obtaining dispersion (for either wave or Schr¨odinger equations).
The paper is organized as follows: in the second section we recall some well known results that will be useful in the proofs of Theorems1.1 and 1.4. The third section, split into three main parts, is devoted to the proof of Theorem 1.1 for the wave equation in the high frequency regime: near transverse or elliptic points this follows from classical results; near the diffractive regime and when the distance between the source point and the boundary is not too small, a parametrix is obtained using the Melrose and Taylor parametrix (directly, if the observation point is sufficiently close to the boundary, otherwise in combination with the Kirchhoff’s integral representation formula). When both source and observation points are close to the boundary, a parametrix is obtained using spherical harmonics. In the fourth section we explain how one can derive sharp dispersion bounds for the Schr¨odinger flow using the Kana¨ı transform and taking advantage of the previous results obtained in Section 3 for
waves. In the fifth section we deal with the low frequency regime, using classical results on the exterior Dirichlet problem for the Helmholtz equation. In the last section we prove Theorem1.4, first in the case of the wave equation, by explicit computations at the Poisson Arago spot, and then we deduce similar results for the Schr¨odinger flow using again the Kana¨ı transform. The Appendix contains some useful properties of the Airy functions together with a general description of the Melrose and Taylor parametrix (modulo smoothing operators) near diffractive points.
2. General setting
To investigate exterior problems we need to take into account the magnitude of τ |Θ|, where |Θ| denotes the size of the obstacle. More specifically, the set of τ such that τ |Θ| . 1 will be the low frequency regime (which, in scattering problems divides into the Rayleigh region τ |Θ| 1 and the resonance region τ |Θ| ∼ 1) and the set of values τ such that τ |Θ| 1 is called the high frequency regime. Mathematical methods used to study scattering phenomena in the Rayleigh or resonance region differ sharply from those used in the high frequency regime.
We now recall some known results that will be useful in the proofs of Theorems 1.1 and 1.4 in the high-frequency regime τ 1, which is by far the most intricate. The low frequency regime will be dealt with in Section5.1, using classical results for the exterior Dirichlet problem for Helmholtz equation.
2.1. The fundamental solution in Rd
. Let d ≥ 2, let ∆ denote the Laplacian in Rd and consider (2.1)
(∂2
t − ∆)Uf ree= F, x ∈ Rd, Uf ree|t=0= u0, ∂tUf ree|t=0= u1. Proposition 2.1. The solution to (2.1) reads, for all t 6= 0, as
Uf ree(t) = ∂tR(t) ∗ u0+ R(t) ∗ u1+ Z t
0
R(t − s) ∗ F (s)ds,
where, for every d ≥ 2, the Fourier transform in space of R is given by \R(t, ξ) = sin(t|ξ|)|ξ| .
2.2. Free solution in Rd. Let Q0∈ Rd, d ≥ 2, and let uf ree(Q, Q0, t) denote the solution to the free wave equation (2.1) with F = 0, u0= δQ0 and u1= 0, where δQ0 is the Dirac distribution at Q0: (2.2) uf ree(Q, Q0, t) :=
1 (2π)d
Z
ei(Q−Q0)ξcos(t|ξ|)dξ.
Let win(Q, Q0, τ ) :=1t>0\uf ree(Q, Q0, τ ) denote the Fourier transform in time of uf ree(Q, Q0, t)|t>0, (2.3) win(Q, Q0, τ ) = τd−12 e
−iτ |Q−Q0|
|Q − Q0|d−12
Σd(τ |Q − Q0|),
where, if d = 3, Σ3=4πi , while if d ≥ 4 and |Q − Q0| τ−1, there exists constants Σj,dwith Σ0,d6= 0 such that
(2.4) Σd(τ |Q − Q0|) =X
j≥0
Σj,d(|Q − Q0|τ )−j.
2.3. The Neumann operator. Let Θ ⊂ Rd, d ≥ 2, be a bounded obstacle with smooth boundary ∂Θ and set Ω = Rd\ Θ. We assume that ∂Θ has positive curvature. We define K : D0
(R+× ∂Ω) → D0
(R+× Ω) as follows: for a distribution f ∈ D0(R × ∂Ω) with support in {t ≥ 0}, K(f ) is defined as the solution to the linear wave equation with data on the boundary equal to f , that is
(2.5) K(f ) = g, where
(∂2
t − ∆)g = 0, x ∈ Ω, supp(g) ⊂ {t ≥ 0}, g|∂Ω= f. We then define the Neumann operator N : D0(R+× ∂Ω) → D0(R+× ∂Ω) as
N (f ) = ∂νK(f )|∂Ω, where ~ν is the outward unit normal to ∂Ω pointing towards Ω.
2.4. The wave equation outside a strictly convex obstacle. Recall Θ ⊂ Rd, d ≥ 2, is a strictly convex, bounded, obstacle with smooth boundary and Ω = Rd\ Θ. We consider the wave equation (1.1) with initial data (δQ0, 0), where Q0 is an arbitrary point of Ω:
(2.6) (∂2 t − ∆D)U = 0 in Ω × R, U |t=0= δQ0, ∂tU |t=0= 0, U |∂Ω= 0. Let U (Q, Q0, t) = cos(t √
−∆D)(δQ0)(Q) denote the solution to (2.6). Let uf ree be the solution to the
free wave equation (2.1) with F = 0, u0= δQ0 and u1 = 0 given in (2.2). The Fourier transform in time of Uf ree+ = 1t>0uf ree equals win defined in (2.3).
As Q06∈ ∂Ω, for any sufficiently small time |t| d(Q0, ∂Ω), the solution to (2.6) in Ω is just uf ree. From (2.5), U+ := 1t>0U may be written as a sum of the incoming wave, Uf ree+ , and the reflected wave, K(−Uf ree+ |∂Ω),
(2.7) U+= Uf ree+ − K(U+
f ree|∂Ω). Moreover, ∂νU+|∂Ω= ∂νUf ree+ |∂Ω− N (Uf ree+ |∂Ω). We also introduce
(2.8) U (Q, Q0, t) :=
U (Q, Q0, t), if Q ∈ Ω, 0, if Q ∈ Θ.
Then, using Duhamel formula, U reads as follows (2.9) U |t>0= Uf ree+ − −1+
(∂νU+)|∂Ω,
where we set, for F such that supp(F ) ⊂ {t0≥ 0} and with R defined in Proposition2.1,
(2.10) −1+ F (t) =
Z t
−∞
R(t − t0) ∗ F (t0)dt0. Remark 2.2. For d = 3, using the explicit form of R yields
(2.11) −1+ (∂νU+)|∂Ω(Q, Q0, t) = 1 4π Z ∂Ω ∂νU+(P, Q0, t − |Q − P |) |Q − P | dσ(P ).
Definition 2.3. Consider normal coordinates (x, Y ) ∈ R+ × ∂Ω such that x → (x, Y ) is the ray orthogonal to ∂Ω at Y . Any point Q ∈ Ω writes Q = Y + x~νY, where Y ∈ ∂Ω is the orthogonal projection of Q on ∂Ω and ~νY is the outward unit normal to ∂Ω pointing towards Ω. For (0, Y ) =: P ∈ ∂Ω, let Σ+P be the half space of Rd defined by Σ+P := {Q ∈ Ω|, ~νP(Q − P ) ≥ 0}. For Q ∈ Ω, let ∂Ω+Q be the part of ∂Ω that can be viewed from Q, ∂Ω+Q := {P ∈ ∂Ω|Q ∈ Σ+P}, which we call the illuminated region from Q. We define the apparent contour CQ as the boundary of the set ∂Ω+Q. 2.5. Finite speed of propagation.
Definition 2.4. A domain Ω is said to be non-trapping if for some R > 0 such that |P | < R for every P ∈ ∂Ω there exists TR such that no generalized geodesic of length TRlies completely within the ball BR= {Q ∈ Ω||Q| ≤ R}. TRis called escape time.
The next result, due to Melrose [21], holds in odd dimensions.
Theorem 2.5. ([21, Theorem 1.6 ]) Let Ω be a non-trapping domain in Rd with odd d ≥ 3, let ∆ D denote the Dirichlet Laplacian on Ω and let R, TR as in Definition2.4. Then there exists a sequence λj ∈ C, Im (λj) < 0, (Im (λj))j→∞ & −∞ and associated generalized eigenspaces Vj ⊂ C∞(Ω) with dimensions mj< ∞ such that
(1) v ∈ Vj ⇒ v|∂Ω= 0;
(2) (∆D+ λ2j)Vj⊂ Vj, (∆D+ λ2j)mj−1Vj= {0};
(3) if (U0, U1) are supported in {|Q| ≤ R}, there exists vj,k∈ Vj, such that for all > 0, N ∈ N and multi-index α and for some constant C = C(R, N, , α), the solution U to (1.1) with Dirichlet condition on ∂Ω and initial data (U0, U1) satisfies for t > TR:
sup {|Q|≤R} D α (t,Q) h U (Q, t) − N −1 X j=1 e−iλjt mj−1 X k=0 tkvj,k(Q)i ≤ Ce −(t−TR)(|Im λN|−).
We now turn to Ω = R3\Θ with Θ a smooth, bounded, strictly convex obstacle. Let U be the solution to (2.6) with data (U0, U1) = (δQ0, 0) for some Q0 ∈ Ω. Then, according to (2.9), the extension U
from (2.8) writes U |t>0= Uf ree+ |t>0− U
#(Q, Q0, t), where U#(Q, Q0
, t) := −1+
(∂νU+)∂Ω(Q, Q0, t) is defined in (2.11) (with d = 3). The integral in (2.11) is supported in t ≥ dist(Q, ∂Ω) + dist(Q0, ∂Ω) and by strong Huygens principle, U#(Q, Q0, t)|∂Ω = 0 for t > dist(Q0, ∂Ω) + diam(Θ). We define T0= dist(Q0, ∂Ω) + diam(Θ) + 1, and we got
(2.12) support(U#(·, Q0, T0)) ∪ support(∂tU#(·, Q0, T0)) ⊂ {Q, dist(Q, ∂Ω) ≤ diam(Θ) + 1}. Let R > 0 be chosen such that {Q, dist(Q, ∂Ω) ≤ diam(Θ) + 1} ⊂ B3(0, R), where B3(0, R) denotes the ball of center 0 and radius R in R3 (in particular, all boundary points P ∈ ∂Ω satisfy |P | < R). For t ≥ T0, one has ∂νU+(., t)|∂Ω = ∂νU#(., t)|∂Ω and U#(Q, Q0, t)|∂Ω = 0. We apply Theorem2.5 to U#(Q, Q0, t) whose initial data at time T0 satisfies (2.12) : as such, for every multi-index α, there exists Cα such that ∀t > TR+ T0, supP ∈∂Ω|∂(t,Q)α U
+(Q, Q
0, t)||Q=P . e−Cα(t−(TR+T0)) In particular, ∂νU+(P, Q0, t) is smooth for t > TR + T0; moreover, there exists C > 0 such that ∀t > TR+ T0 supP ∈∂Ω|∂νU+(P, Q0, t)| . e−C(t−(TR+T0)). Hence, Theorem 1.1 for the wave equation will follow if
we prove fixed time bounds for t such that t−|Q−P | ≤ T0+TR= dist(Q0, ∂Ω)+diam(Θ)+TR+1. Here, for some fixed R such that {Q, dist(Q, ∂Ω) ≤ diam(Θ) + 1} ⊂ B3(0, R), we have TR≤ 2R + diam(Θ). Moreover, using finite speed of propagation, we must have t − |Q − P | ≥ dist(Q0, ∂Ω), as otherwise the contribution for P ∈ ∂Ω of ∂νU+(P, Q0, t − |Q − P |) is trivial. We have obtained the following lemma: Lemma 2.6. There exists C0= C(Θ) > 0 independent of Q, Q0, such that, to prove dispersion bounds for the solution to (2.6) (with d = 3), it is enough to consider only t such that
d(Q0, ∂Ω) + d(Q, ∂Ω) ≤ t ≤ d(Q0, ∂Ω) + d(Q, ∂Ω) + C0.
Remark 2.7. Due to Lemma2.6, if |t| is sufficiently large, Q0 and the observation point Q cannot be simultaneously very close to ∂Ω. If Q0is very close to ∂Ω, dist(Q0, ∂Ω) 1, we use the symmetry of the Green function with respect to Q0 and Q to replace Q0 by Q : we are therefore left to consider only data Q0 such that dist(Q0, ∂Ω) > c > 0 for some constant c, in which case the Melrose and Taylor parametrix can be used. For bounded |t|, we will consider separately dist(Q0, ∂Ω) 1 and dist(Q, ∂Ω) 1, where the Melrose and Taylor construction does not hold.
3. Proof of Theorem 1.1 in the high-frequency regime : the wave equation We start with the general form of a parametrix for the wave flow inside Ωd= Rd\ Bd(0, 1) for any d ≥ 3 : this construction will be particularly useful in Section3.2, in order to prove Theorem1.1(for d = 3) when both the source and the observation points stay away from a fixed, small neighborhood of ∂Ω3, as well as in Section6, in order to construct a counterexample when d ≥ 4.
3.1. General form of the wave flow in Rd\ Bd(0, 1). Let d ≥ 3, let Bd(0, 1) denote the unit ball in Rd
with centre O = {0} ∈ Rd and let Ωd
= Rd\ Bd(0, 1). Then ∂Ωd
= Sd−1is the unit sphere in Rd. Let N and S denote the North and the South pole, respectively.
Consider the following change of coordinates :
(3.1)
xd= r cos ϕ, xd−1= r sin ϕ cos ω1
xk = r sin ϕ × Πd−k−1j=1 sin ωjcos ωd−k, 2 ≤ k ≤ d − 2, x1= r sin ϕ × Πd−kj=1sin ωj,
then Ωd= Rd\ Bd(0, 1) = {r ≥ 1, (ϕ, ω) ∈ [0, π]d−2× [0, 2π]}.
In these spherical coordinates the Laplace operator in Rd takes the form
(3.2) ∆ = 1 rd−1 ∂ ∂r rd−1 ∂ ∂r + 1 r2∆Sd−1,
where ∆Snis the Laplace operator on Sn and ∆ Sd−1 = 1 (sin ϕ)d−2 ∂ ∂ϕ (sin ϕ)d−2 ∂ ∂ϕ +(sin ϕ)1 2∆Sd−2. Let r = 1 + x, y = π2 − ϕ, then Ωd= {x ≥ 0, y ∈ [−π/2, π/2], ωk ∈ [0, π], ωd−2∈ [0, 2π]} and (3.3) ∆ = ∂ 2 ∂x2 + (d − 1) (1 + x) ∂ ∂x+ 1 (1 + x)2 ∂2 ∂y2 + (d − 2) tan y ∂ ∂y + 1 (cos y)2∆Sd−2 .
Let Q0∈ Ωd be the source point : we can assume without loss of generality that Q0∈ ON . Denote by s the distance between Q0 and the centre of the ball O and write Q0= QN(s); the distance between Q0 and any given point P ∈ Sd−1 with coordinates P = (0, y, ω), depends only on s and on y. In particular, the free wave flow (2.2) starting at Q0enjoys rotational symmetry with respect to the d − 2 tangential variables ω describing Sd−2.
Recall (2.2), u+f ree(Q, Q0, t) := 1t>0uf ree(Q, Q0, t) and win(Q, Q0, τ ) = [u+f ree(Q, Q0, τ ) ((2.3)). If u solves (2.6) in Ωd, let u+ := 1t>0u and let u be the extension of u to Rd. Using (2.9), for Q ∈ Ωd, u(Q, Q0, t)|t>0reads u(Q, Q0, t)|t>0= u+f ree(Q, Q0, t)−−1+ ∂xu+|
Sd−1
(Q, Q0, t), where −1+ has been defined in (2.10) and where, for P ∈ Sd−1, we have
(3.4) ∂xu+(P, Q0, t) = (∂xu+f ree|Sd−1− N (u+f ree|Sd−1))(P, Q0, t).
Let h ∈ (0, 1) and let χ ∈ C0∞([12, 2]) be a smooth cutoff equal to 1 on [34,32] and such that 0 ≤ χ ≤ 1. As we are interested in evaluating χ(hDt)u(Q, Q0, t), we set
(3.5) u#:= −1+ ∂xu+|Sd−1
, u+f ree,h:= χ(hDt)u+f ree, u#h := χ(hDt)u#(Q, Q0, t).
As the (frequency localized) free wave flow uf ree,h does satisfy (1.3) everywhere, in order to prove Theorems1.1or1.4for the wave equation we are reduced to evaluating χ(hDt)u#(Q, Q0, t).
Proposition 3.1. If Q ∈ Ωd is such that τ dist(Q, Sd−1) 1, we have (3.6) u#h(Q, Q0, t) = χ(hDt)u#(Q, Q0, t) = Z eitτχ(hτ ) Z P ∈Sd−1 F (∂xu+| Sd−1)(P, Q0, τ ) × τ d−3 2 |Q − P |d−12
Σd(τ |Q − P |)e−iτ |Q−P |+ O((τ |Q − P |)−∞)dσ(P )dτ, where Σ3 := 4πi and for d ≥ 4, Σd is given in (2.4) and where F (∂xu+|
Sd−1)(P, Q0, τ ) denotes the Fourier transform in time of ∂xu+|Sd−1(P, Q0, t) given in (3.4) for P ∈ Sd−1.
Proof. We compute u# using the fundamental solution R from Proposition2.1: u#(Q, Q0, t) = Z t −∞ R(t − t0) ∗ ∂xu+|Sd−1(., t0)dt0 = Z P ∈Sd−1 h I(0,∞)R(., Q − P ) ∗ ∂xu+(P, Q0, .) i (t)dσ(P ),
where in the first line the convolution is taken in space while in the last line it is taken in time. Here I(0,∞) is the characteristic function of the positive half line. Taking the Fourier transform in time of u#, that we denote F (u#), yields, with F (u#
h) = χ(hτ )F (u #), F (u#h)(Q, Q0, τ ) = χ(hτ ) Z P ∈Sd−1 FI(0,∞)R(., Q − P )(τ )F (∂xu+)(P, Q0, τ )dσ(P ). We first compute, for τ > 1 large on the support of χ(hτ ) and for τ |Q − P | 1
FI(0,∞)R(., Q − P )(τ ) = Z ∞ 0 e−it0τR(t0, Q − P )dt0= Z ∞ 0 e−it0τ Z Rd sin(t0|ξ|) |ξ| e i(Q−P )ξdξdt0 =X ± ±1 2i Z ∞ 0 e−it0τ Z Rd 1 |ξ|e i(Q−P )ξ±it0|ξ|dξdt0 = τ d−3 2 |Q − P |d−12 e−i|Q−P |τΣd(τ |Q − P |) + O((τ |Q − P |)−∞), (3.7)
where Σ3 = 4π1 and for d ≥ 4, Σd is the asymptotic expansion defined in (2.4) (as (2.3) is obtained in this way from R). Setting ρ = |ξ|, θ = ξ/|ξ| in the third line of (3.7), the phase function becomes −t0τ − |z|ρ cos θ
1± t0ρ, where z = Q − P 6= 0 and where we consider an orthogonal basis of Rdsuch that z = |z|(0, ..., 0, z
|z|). Changing variables ρ = ˜ρ and t0= |z|t00, the phase is now τ |z|(−t00− ˜ρ cos θ1± t00ρ):˜ applying stationary phase with respect to ˜ρ, t00(with Hessian equal to 1) yields 1 = ± ˜ρ and cos θ1= ±t00. Using τ > 1, τ |z| 1, the contribution from − sign in the third line of (3.7) is O((τ |z|)−∞). To obtain the last line in (3.7) we use τ |z| 1 to apply stationary phase in θ1. In order to obtain dispersive bounds (for all Q ∈ Ω3 when d = 3 or for d ≥ 4 at Q = QS(s) for s > 1 sufficiently large) for the left hand side term in (3.6) we compute F (∂xu+|Sd−1): it follows from
classical results for P outside a small neighborhood of the apparent contour CQ0 but constructing F (∂xu+)(P, Q0, τ ) is an intricate process for P near CQ
0. Let first describe CQ0 explicitely (see
Defi-nition2.3): for Q ∈ Ωd= Rd\ Bd(0, 1), write Q = (x, Y ), where we identify Y = (y, ω) with a point on the boundary Sd−1. Let Q0 = QN(s) with coordinates (x(QN(s)), y(QN(s))) = (s − 1,π
2) and let φ(x, y, Q0) := |Q − Q0|, then
(3.8) φ(x, y, Q0) := (s2− 2s(1 + x) sin y + (1 + x)2)1/2.
The apparent contour CQ0 is the set of points P ∈ Sd−1 such that the ray Q0P is tangent to Sd−1: in other words CQ0 = {P ∈ S
d−1with coordinates (0, y, ω) such that ∂
xφ(0, y, Q0) = 0}. As ∂xφ(x, y, Q0) = (1 + x − s sin y)/φ vanishes at x = 0 when sin y = 1s,
CQN(s)= {P ∈ S
d−1with coordinates (0, y
0, ω), where y0 satisfies sin(y0) = 1 s}.
Let P ∈ Sd−1 be an arbitrary point on the boundary with coordinates (0, y, ω), then |P − QN(s)| = ψ(y, s) (and also |P − QS(s)| = ψ(−y, s)) where we have set
(3.9) ψ(y, s) := φ(0, y, Q0) = (s2− 2s sin y + 1)1/2.
We split the integral over Sd−1of (3.6) as a sum of two integrals, according to whether P is sufficiently close to CQ0 or not. Fix ε ∈ (0, 1) small enough and let χ0∈ C∞
0 ((−2, 2)) be equal to 1 on [−1, 1] and such that 0 ≤ χ0 ≤ 1. Write u#h = u#h,χ
0 + u
#
h,1−χ0 where u
#
h,χ is like (3.6) with an additional factor χ(dist(P, CQ0)/ε) into the symbol.
Proposition 3.2. If s > 1 + ε0 for some fixed ε0> 0, then |u#h,1−χ
0(Q, Q0, t)| . 1 hd( h t) d−1 2 .
Remark 3.3. We assume s > 1 + ε0 for small ε0 > 0 for (3.6) to hold true. In Section 3.2 we will deal only with the case s > 1 + ε0 and dist(Q, S2) > 1 + ε0: Proposition 3.2 will reduce matters to obtaining bounds for |u#h,χ
0(Q, Q0, t)| only. In Section6 we will take Q0 = QN(s), with large s > 1:
Proposition3.2will apply as well, and we will be left with the diffractive part of u#h.
Proof. Let χ+(l) := (1 − χ0(l))|l>0 and χ−(l) := (1 − χ0(l))|l<0, then 1 − χ0 = χ+ + χ−. Write u#h,1−χ 0 = u # h,χ++ u # h,χ−, where u # h,χ+, u #
h,χ− correspond to the illuminated and shadow parts of u
# h. For y on the support of χ+(y−yε0), i.e. such that y − y0 ≥ ε, a point P = (0, y, ω) belongs to the illuminated region ∂Ω+d,Q
0 (recall Definition2.3). From classical results, for such P , the phase function
of F (∂xu+)(P, Q0, τ ) is −iτ |P −Q0| (and the symbol is τ τd−12
|P −Q0| d−1
2
a(τ, P, Q0) with a a classical symbol of order 0 with respect to τ ). Therefore, the phase of u#h,χ
+(Q, Q0, t) is τ (t − |Q − P | − |P − Q0|). Let
Ω+d,Q
0⊂ Ω be the open set of Q ∈ Ωdsuch that the segment [Q0, Q] is contained in Ωd. Critical points
of P → |P − Q| + |P − Q0| are such that, for some λ ∈ R \ {0}, P − Q |P − Q|+ P − Q0 |P − Q0| = λ~νP with ~νP the unit normal to ∂Ωdpointing towards Ωd. If Q ∈ Ω+d,Q
0, by assumption of strict convexity
of Θ = Bd(0, 1), the restriction of this phase to ∂Ω+d,Q
0 has a unique critical point at P = P−(Q, Q0),
and this critical point is non degenerate.More precisely, for Q ∈ Ω+d,Q
0, there exists exactly two optical
broken trajectory [Q0, P ] ∪ [P, Q] where P = P−(Q, Q0) ∈ ∂Ω+d,Q
0 is the critical point of the phase of
u#h,χ
+(Q, Q0, t). If Q /∈ Ω
+
d,Q0, by strict convexity of Θ = Bd(0, 1) there is a unique non-degenerate
critical point at P+(Q, Q0) ∈ ∂Ω+d,Q
0 which is the intersection of the segment [Q0, Q] with ∂Ω
+ d,Q0. As
P±(Q, Q0) stay away from a small neighborhood of CQ0 (due to the support property of χ+), one has
a lower bound on ∇2
P(|Q − P | + |P − Q0|) which is moreover uniform with respect to Q, Q0. Stationary phase yields usual dispersion bounds for u#h,χ
+(Q, Q0, t).
For y on the support of χ−, y − y0 < −ε so P = (0, y, ω) belongs to the shadow region, and the contribution of u#h,χ−(Q, Q0, t) is O(h∞) simply because the symbol a(τ, P, Q0) of F (∂xu+)(P, Q0, τ ) is such that a(τ, P, Q0) ∈ O(τ−∞) uniformly with respect to Q0and for |P − Q0| ∼ t. This eventually yields |F (u#h,χ
−)(Q, Q0, τ )| ≤ CNτ
−N/t(d−1)/2with CN independent of Q, Q0and for τ > 1 large. In the remaining of this section we only deal with u#h,χ
0, for which we have
F (u#h,χ 0)(Q, Q0, τ ) =χ(hτ ) Z χ0(dist(P, CQ0)/ε)F (∂xu +)(P, Q 0, τ ) × τ d−3 2 |Q − P |d−12
Σd(τ |Q − P |)e−iτ |Q−P |+ O((τ |Q − P |)−∞)
dσ(P ). (3.10)
We will provide an explicit representation of χ0(dist(P, CQ0)/ε)F (∂xu
+)(P, Q
0, τ ) that enables proving both Theorem1.1(for d = 3 and dist(Q, ∂Ω3) ≥ 1+ε0) and Theorem1.4(for d ≥ 4 and dist(Q, Sd−1) ∼ h−1/3) for the wave equation. At this point, we advise the reader to review Section7.2 (which gives an overview of the known results and the parametrix construction for the wave flow near diffractive points), as we will often refer to it. The next result (stated in a more general setting in Proposition
7.5) will be particularly useful in order to compute F (∂xu+)(Q, Q0, τ ) for Q near CQ0.
Proposition 3.4. Microlocally near a glancing point (0, y0, ω), ω ∈ [0, π]d−3× [0, 2π], there exist smooth functions θ(x, y, α) and ζ(x, y, α) such that the phase functions θ ±23(−ζ)3/2 satisfy the eikonal equation associated to (3.3); moreover, there exist symbols a(x, y, α), b(x, y, α) satisfying transport equations such that, for any α ∈ R near 1,
Gτ(x, y, α) := eiτ θ(x,y,α) aA+(τ2/3ζ) + bτ−1/3A0+(τ 2/3ζ)A−1 + (τ 2/3ζ 0)
is such that, for τ > 1, (τ2+ ∆)Gτ ∈ OC∞(τ−∞). Moreover: the system of eikonal equations reads
(3.11) ( (∂xθ)2+ 1 (1+x)2(∂yθ)2− ζ (∂xζ)2+ 1 (1+x)2(∂yζ)2 = 1, ∂xθ∂xζ +(1+x)1 2∂yθ∂yζ = 0.
The system (3.11) admits a pair of solutions θ(y, α), ζ(x, α) of the form (3.12)
θ(y, α) = yα, ζ(x, α) = α2/3ζ(˜ 1+x
α ),
where ˜ζ is the solution to (− ˜ζ)(∂ρζ)˜2+ρ12 = 1, with ˜ζ(1) = 0, ∂ρζ(1) < 0. The glancing set is defined˜ by ζ|x=0 = 0 : this corresponds to α = 1. The symbols a, b ∈ S0
(1,0) are asymptotic expansions of the form a 'P
j≥0aj(iτ )−j, b 'Pj≥0bj(iτ )−j. Moreover, a|∂Ωis elliptic near (0, y0, 1) and has essential support included in a small neighborhood of (0, y0, 1) while b|∂Ω= 0.
Remark 3.5. Proposition3.4is Proposition7.5 for the particular case of the exterior of a ball in Rd. In the general setting, existence of solutions θ(x, Y, Υ) and ζ(x, Y, Υ) to the eikonal equation in a neighbourhood of a glancing point was proved by Melrose and Taylor, Eskin, Zworski etc; ζ can be chosen such that its trace on the boundary is independent of the tangential variable, i.e. such that ζ|x=0= ζ0(Υ), for deep geometrical reasons; while for the exterior of a ball, it may be checked easily.
Lemma 3.6. The equation (− ˜ζ)(∂ρζ)˜2+ρ12 = 1, with ˜ζ(1) = 0 has an unique solution for ρ near 1.
Moreover ˜ζ has an explicit representation for ρ > 1: (3.13) 2 3(− ˜ζ) 3/2(ρ) = Z ρ 1 √ w2− 1 w dw. = p ρ2− 1 − arccos(1 ρ) = 2 3 √ 2(ρ − 1) 3/2 ρ1/2 1 + ρ − 1 8 + O((ρ − 1) 2). For ρ < 1, we also have
(3.14) 2 3 ˜ ζ3/2(ρ) = Z 1 ρ √ 1 − w2 w dw = ln[(1 + p 1 − ρ2)/ρ] −p 1 − ρ2.
We note that at ρ = 1, we have ˜ζ = 0 and limρ→1(ρ−1)(− ˜ζ) = 21/3. Lemma 3.7. Recall a is elliptic, of the form a ' P
j≥0aj(iτ )
−j; also, b ' P
j≥0bj(iτ )
−j where bj|x=0= 0 for all j ≥ 0. Moreover, a0, b0 can be taken such that
(3.15) a0= (cos y) d−2 2 γ 1 + x α , b0= 0 everywhere , where γ satisfies 2∂ρζ∂ργ + γ˜ ∂2 ρζ +˜ (d−1) ρ ∂ρζ˜ = 0, γ(1) = 1.
Proof. Explicit computations using the explicit representations of θ, ζ and ˜ζ. As mentioned above, we aim at computing F (∂xu+)(P, Q0, τ ) for P near CQ
0, where u
+= u+ f ree− K(u+f ree|Sd−1) with K defined in (2.5). Let’s consider first the free wave u+f ree, whose Fourier transform
in time win(Q, Q0, τ ) = ˆu+f ree(Q, Q0, τ ) has been defined in (2.3): win(Q, Q0, τ ) depends only on |Q − Q0| and using (3.8), for Q = (x, y, ω), |Q − Q0| = φ(x, y, Q0) depends only on x, y, s. Hence we may write win(x, y, Q0, τ ) instead of win(Q, Q0, τ ). As (τ2+ ∆)win(·, τ ) ∈ OC∞(τ−∞) with ∆ given by (3.3), Lemma 7.8 from Section 7.2 (see also [27, Lemma A.2]) applies and allows to represent it using the phase functions and symbols of Proposition3.4as follows:
Lemma 3.8. There exists a unique Fτ ∈ E0(R), essentially supported near y0, such that, for all Q = (x, y, ω) near CQ0, win(Q, Q0, τ ) may be written, modulo O(τ−∞) terms
(3.16) win(Q, Q0, τ ) = τ 2π
Z
eiτ (θ(y,α)−zα)aA(τ2/3ζ(x, α)) + bτ−1/3A0(τ2/3ζ(x, α))Fτ(z)dzdα, where the phase functions θ, ζ are given in (3.12), and the symbols a, b are as in Proposition3.4with a elliptic with main contribution a0 given in (3.15) and with b|x=0= 0.
Remark 3.9. Due to rotational symmetry, integration is over α ∈ R instead of Υ ∈ Rd−1as in Lemma
7.8and (7.6); for this reason the power of τ
2π in (3.16) is 1 instead of d − 1 as in Section7.2 (in the general case a point (Y, Υ) belongs to T∗Rd−1so the integral in the definition (7.6) of the operator Tτ is taken on Rd−1; here y ∈ (−π2,π2) and α ∈ R).
From (3.16) and (3.4) with Q0= QN(s), we obtain F (∂xu+)(P, Q0, τ ) near the glancing regime: Lemma 3.10. For P ∈ ∂Ωd= Sd−1, dist(P, CQ0) ≤ 2ε, with ε > 0 small enough, we have
(3.17) F (∂xu+)(P, Q0, τ ) = i 2πe −iπ/3 τ 2π Z
eiτ (y−z)ατ2/3b∂(y, α, τ )
Fτ(z)
A+(τ2/3ζ0(α))dzdα,
where Fτ(z) is the unique function satisfying (3.16), ζ0(α) = ζ(0, α) with ζ defined in (3.12) and where b∂ is an elliptic symbol of order 0 that reads as an asymptotic expansion with small parameter τ−1 and main contribution a0(y, α)∂xζ(0, α).
Remark 3.11. Notice that the integral in (3.17) does not depend on the rotational variable ω. This oc-curs because winenjoys rotational symmetry, and because we consider the restriction on the boundary of ∂xu+. In the following we write F (∂xu+)(y, Q0, τ ) instead of F (∂xu+)(P, Q0, τ ) for P = (0, y, ω). Moreover, dist(P, CQ0) = |y − y0|, ∀P ∈ Sd−1.
Proof. The (Fourier transform in time of the) wave operator with Dirichlet conditions is obtained by re-placing aA(τ2/3ζ)+bτ−1/3A0(τ2/3ζ) in (3.16) by aA(τ2/3ζ)−A
+(τ2/3ζ) A(τ2/3ζ 0) A+(τ2/3ζ0) +bτ−1/3A0(τ2/3ζ)− A0+(τ2/3ζ) A(τ2/3ζ0) A+(τ2/3ζ0)
. Taking the normal derivative, followed by the restriction to the boundary yields
F (∂xu+)(y, Q0, τ ) = τ 2π
Z
eiτ θ(y,α)a∂A(τ2/3ζ0(α)) + b∂τ2/3A0(τ2/3ζ0(α)) ˆFτ(τ α)dα − τ
2π Z
eiτ θ(y,α)a∂A+(τ2/3ζ0(α)) + b∂τ2/3A0+(τ2/3ζ0(α))
A(τ2/3ζ0(α))
A+(τ2/3ζ0(α))Fτ(τ α)dα.ˆ Symbols are obtained from the derivatives of ζ, a, b: a∂ = ∂xa(0, y, α), b∂ = a(0, y, α)(∂xζ)(0, α) + τ−1∂xb(0, y, α). Notice that b∂ is elliptic as a0 is elliptic and (∂xζ)(0, α) = −21/3/α1/3(1 + O(1 − α)) near the glancing regime where α − 1 stays small. Using (7.2) achieves the proof.
The next result, proved in section 3.1.1below, provides an explicit representation for Fτ(z). Lemma 3.12. Recall ψ from (3.9). The unique Fτ such that (3.16) holds may be expressed as (3.18) Fτ(z) = τ2/3 τ
ψ(y0, s) d−12 Z
eiτ ((z−y0)α−ψ(y0,s))χ1(α)f (α, τ )dα + O(τ−∞),
where f is a symbol of order 0, elliptic in a neighborhood of α = 1; χ1 is a smooth cut-off supported near 1 and equal to 1 in a small neighborhood of 1, y0= arcsin(1s) ∈ (0,π2) and ψ(y0, s) =
√ s2− 1. Corollary 3.13. Let |y − y0| ≤ 2ε with ε > 0 sufficiently small, then, modulo O(τ−∞),
(3.19)
F (∂xu+)(y, QN(s), τ ) = τ4/3 τ ψ(y0, s)
d−12 Z
eiτ ((y−y0)α−ψ(y0,s)) b∂(y, α, τ )
A+(τ2/3ζ0(α)) ˜
χ1(α) ˜f (α, τ )dα,
where ˜f is a symbol of order 0 that reads as an asymptotic expansion with main contribution −2π1 e−iπ/3f (α, τ ) and small parameter τ−1, and where ˜χ1 is an enlargement of χ1 (identically 1 on the support of χ1). Proof. Replacing (3.18) into (3.17) yields
F (∂xu+)(y, QN(s), τ ) =
ie−iπ/3 4π2 τ
5/3Z Z eiτ (y−z)α b∂(y, α, τ )
A+(τ2/3ζ0(α))Fτ(z)dzdα =ie −iπ/3 4π2 τ 7/3 τ ψ(y0, s) d−12 Z Z
eiτ (y−z)α b∂(y, α, τ ) A+(τ2/3ζ0(α))
Z
eiτ ((z−y0) ˜α−ψ(y0,s))χ1( ˜α)f ( ˜α, τ )d ˜αdzdα.
We conclude by stationary phase with respect to both z and ˜α. 3.1.1. Proof of Lemma3.12. The proof of Lemma3.12is based on section7.2. With Ωd= Rd\Bd(0, 1), we denote Y = (y, ω) a point on the boundary Sd−1and Q = (x, Y ) ∈ Ωd. We recall that Q0= QN(s) and that win(Q, Q0, τ ) (from (2.3)) is independent of ω and has phase function φ(x, y, Q0) defined in (3.8). For F ∈ E (R) and for (x, y) near (0, y0), define an operator Tτ : E (R) → D0(R2) as follows
Tτ(F )(x, y) = τ1/3 τ 2π
Z
eiτ (θ(x,y,α)−zα+σ3/3+σζ(x,α))(a + bσ/i)F (z)dσdzdα,
where the phase functions θ and ζ are defined in (3.12), θ(x, y, α) = yα and ζ(x, α) = α2/3ζ(˜ 1+xα ). According to Proposition3.4, near the glancing point ((0, y0), (0, 1)) ∈ T∗S1the trace on the boundary of phase function ζ(x, α) cancels only at a glancing direction and the symbols a, b are supported in a small neighborhood of a glancing point; as ζ0(α) = 0 only at α = 1, we can introduce in the integral defining Tτ a smooth cut-off χ1∈ C0∞supported for α near 1 and such that χ1(α) = 1 on the support of a,b, without changing its contribution. Denote B((0, y0), ε) ⊂ R2 the ball of centre (0, y
0) in R2 and radius ε > 0 and let let χ ∈ C0∞(R2) be a smooth cutoff function supported for (x, y) near (0, y0), equal to 1 on B((0, y0), 2ε) and equal to 0 outside B((0, y0), 3ε), with ε > 0 small as before. We let
˜
Tτ(F )(x, y) := τ1/3 τ 2π
Z
For F supported outside a small neighborhood of y0 we have ˜Tτ(F )(x, y) = O(τ−∞): indeed, critical points satisfy σ2 = −ζ(x, α) and y = z − σ3/3 − σζ(x, α); as ζ(x, α) is close to 0 for x and α on the support of χ(x, y)χ1(α), z must stay close to y and hence close to y0, otherwise non-stationary phase applies. The operator ˜Tτ : D0(Rz) → D0(B((0, y0), 3ε)) is well defined microlocally near (y0, 1) and (τ2+ ∆) ˜T
τ(F ) ∈ OC∞(τ−∞) in B((0, y0), 2ε). If ˜Tτ∗: D0(B((0, y0), 3ε)) → D0(Rz) denotes its adjoint,
˜
Tτ∗(w)(z) := τ 1/3 τ
2π Z
eiτ (zα−yα−σ3/3−σζ(x,α))(a − bσ/i)χ(x, y)χ1(α)w(x, y)dxdydσdα . For (x, y, α) on the support of the symbol, the phase may be stationary only for σ close to 0. We therefore introduce a smooth cutoff χ0(σ) supported near 0 without changing the contribution of ˜Tτ∗(w) modulo O(τ−∞). We now define Eτ := (τ1/3T˜τ∗) ◦ ˜Tτ : D(Rz) → D(Rz) and prove that, microlocally near (y0, 1), Eτ is an elliptic pseudo-differential operator of degree 0, hence invertible. Moreover, setting Fτ = E−1τ ◦ (τ1/3T˜τ∗)(win(., τ )) yields ˜Tτ(Fτ) = win(., τ ) microlocally near (y0, 1). Using the definition Eτ := τ1/3T˜τ∗◦ ˜Tτ, we explicitly compute
Eτ(F )(z) = τ 3 4π2 Z eiτ (z−y)α−σ3/3−σζ(x,α)+(y−˜z) ˜α+˜σ3/3+˜σζ(x, ˜α) χ0(σ)χ1(α)χ1( ˜α)
× (χ(x, y))2(a − bσ/i)(x, y, α)(a + b˜σ/i)(x, y, ˜α)F (˜z)d˜zd˜σd ˜αdxdydσdα . Stationary phase applies in y, ˜α, with critical points yc(˜z, ˜σ, α) := ˜z − ˜σ∂αζ(x, α) and ˜αc = α; at the critical value, the phase is (z − ˜z)α − σ3/3 − σζ(x, α) + ˜σ3/3 + ˜σζ(x, α) and the symbol is τ2χ˜
1(α) ˜χ(x, yc)χ0(σ)(a − bσ/i)(x, yc, α)(˜a + ˜b˜σ/i)(x, yc, α), where ˜a, ˜b are asymptotic expansions w.r.t. τ−1 and with main contribution a(x, yc, α), and b(x, yc, α) which depend upon ˜σ through yc= yc(˜z, ˜σ, α); ˜χ1 and ˜χ are smooth cutoffs supported for α near 1 and for (x, yc) near (0, y0).
We further apply stationary phase with respect to x and ˜σ with critical points ˜σ2+ ζ(x, α) = 0 and ∂xζ(x, α)(˜σ − σ) = 0. As ∂xζ(0, 1) = ∂xζ(1) = −2˜ 1/3, ∂xζ(x, α) does not vanish near x = 0, α = 1 (this is the diffractive condition in Proposition3.4), and ˜σc = σ, ζ(xc, α) = −σ2. This yields xc as a smooth function of σ2 and α for σ close to 0 and for α close to 1. The phase function has second derivative ∂2
xxζ(xc, α)(˜σ − σ) and vanishes at ˜σc = σ; hence, at the critical points, the absolute value of the determinant of the Hessian matrix equals |∂xζ(xc, α)|, which is close to 21/3 near x = 0, α = 1. The critical value of the phase is (z − ˜z)α, and the symbol has main contribution a2+ b2σ2 with σ close to 0 on the support of χ0 and with a elliptic near (0, y0, 1). Integrating with respect to σ yields
Eτ(F )(z) = τ 2π
Z
eiτ (z−˜z)αχ˜1(α) ˜χ(˜z)ΣE(˜z, α, τ )F (˜z)d˜zdα ,
where ΣE(˜z, α, τ ) is a symbol of order 0, elliptic at (y0, 1) that reads as an asymptotic expansion with main contribution a2
0(x, ˜z − σc∂αζ(xc, α), α) with a0 defined in (3.15) and with small parameter τ−1; ˜
χ is a smooth cut-off supported for ˜z near y0 and equal to 1 in a small neighborhood of y0 and ˜χ1 is supported near 1. Microlocally near (y0, 1), the operator Eτ has an inverse Eτ−1 of the form
Eτ−1(f )(z) = τ 2π
Z
eiτ (z−˜z)αΣE−1(˜z, α, τ )f (˜z)dαd˜z,
where ΣE−1 is elliptic of order 0 supported for (˜z, α) near (y0, 1). Let Fτ := Eτ−1(τ1/3T˜τ∗)(win), then,
according to Proposition (7.10), Fτ satisfies (3.16) microlocally near CQ0. We obtain Fτ(z) =τ
2π Z
eiτ (z−˜z)αΣE−1(˜z, α, τ )(τ1/3T˜τ∗)(win)(˜z)d˜zdα
=τ 2π Z eiτ (z−˜z)αΣE−1(˜z, α, τ )τ1/3 τ 2π Z eiτ (˜z−˜y) ˜αaA(τ2/3ζ(˜x, ˜α)) − bτ−1/3A0(τ2/3ζ(˜x, ˜α)) χ1(α)χ(˜x, ˜y)win(˜x, ˜y, Q0, τ )d˜xd˜yd ˜αd˜zdα , where one factor τ1/3 was used integrating over σ in ˜T∗
τ, to obtain a linear combination of Airy functions. Applying stationary phase with respect to ˜z and α in the expression of Fτ yields α = ˜α,
˜
z = z, and provides a factor τ−1. We then have (3.20) Fτ(z) = τ4/3
Z Z
eiτ ((z−˜y)α−φ(˜x,˜y,Q0))χ1(α)χ(˜x, ˜y)
× τ φ(˜x, ˜y, Q0) (d−1)/2 ΣF(˜x, ˜y, α, τ ) aA(τ2/3ζ(˜x, α)) − bτ−1/3A0(τ2/3ζ(˜x, α))d˜xd˜ydα, where ΣF is the product between ΣE−1(˜x, ˜y, α, τ ) and Σd(τ φ(˜x, ˜y, Q0), τ ) defined in (2.4) and φ is
given in (3.8), φ(˜x, ˜y, Q0) = (s2− 2s(1 + ˜x) sin ˜y + (1 + ˜x)2)1/2 as Q0 = QN(s). Next, we prove that the phase function of Fτ(z) has degenerate critical points of order two:
∂y˜φ(˜x, ˜y, QN(s)) = − s(1 + ˜x) cos ˜y φ(˜x, ˜y, QN(s)) , ∂y2˜φ(˜x, ˜y, QN(s)) = s(1 + ˜x) sin ˜y − (∂˜yφ)2 φ(˜x, ˜y, QN(s)) , and ∂y2˜φ vanishes when 1+˜sx ∈ { 1
sin ˜y, sin ˜y}. For (˜x, ˜y) near (0, y0) and s > 1 large, the unique solution is ˜y(˜x) := arcsin(1+˜sx), with ˜y(0) ∈ CQN(s). We have ∂x˜φ|˜y(˜x) =
1+˜x−s sin ˜y
φ(˜x,˜y,QN(s))|y(˜˜x) = 0 and
∂˜yφ|˜y(˜x)= −(1 + ˜x) = −∂y3˜φ|˜y(˜x). Write
(3.21) φ(˜x, ˜y, QN(s)) = φ(˜x, ˜y(˜x), QN(s)) − (1 + ˜x)(˜y − ˜y(˜x)) + (1 + ˜x)(˜y − ˜y(˜x)) 3 3! h 1 + (˜y − ˜y(˜x)) Z 1 0 (1 − o)3 1 (1 + ˜x)∂ 4 ˜
yφ(˜x, ˜y(˜x) + o(˜y − ˜y(˜x)), QN(s))do i
. Let ˜ρ = 1+˜sx. We prove that all ∂y˜jφ/(1 + ˜x), 1 ≤ j ≤ 4, are functions of ˜y and ˜ρ: we have ∂˜yφ
(1+˜x) = −√ cos ˜y 1−2 ˜ρ sin ˜y+ ˜ρ2, ∂2y˜φ (1+˜x) = sin ˜y √ 1−2 ˜ρ sin ˜y+ ˜ρ2 − ˜ ρ cos2˜y √
1−2 ˜ρ sin ˜y+ ˜ρ23 and also
∂3y˜φ (1+˜x) = − ∂y˜φ (1+˜x)(1 + 3 ∂y2˜φ φ2 ), with ∂2 ˜ yφ φ2 = ∂2 ˜ yφ (1+˜x) 1 1−2 ˜ρ sin ˜y+ ˜ρ2, hence ∂3 ˜ yφ
(1+˜x) together with all higher order derivatives are functions of ˜y and ˜
ρ only. As ˜y(˜x) = arcsin ˜ρ, then 1 (1+˜x)∂
4 ˜
yφ(˜x, ˜y(˜x) + o˜z, QN(s)) is a smooth function depending only on o˜z and ˜ρ that we denote E (o˜z, ˜ρ). Let
(3.22) ϕ(˜x, ˜y, α) = −˜yα − φ(˜x, ˜y, QN(s)).
The saddle points of ϕ with respect to ˜y satisfy α = −∂˜yφ(˜x, ˜y, QN(s)); using the expansion of φ from (3.21) we obtain ϕ(˜x, ˜y(˜x) + ˜z, α), for ˜z = ˜y − ˜y(˜x), as
−˜y(˜x)α − φ(˜x, ˜y(˜x), QN(s)) + ˜z(1 + ˜x − α) − (1 + ˜x)z˜ 3 6 1 + ˜z Z 1 0 (1 − o)3E(o˜z, (1 + ˜x)/s)do. The saddle points of ϕ, denoted ˜y(˜x) + ˜z±, are such that (for ˜z ∈ {˜z±})
(3.23) z˜ 2 2 1 +z˜ 3 Z 1 0
(1 − o)3(4E + o˜z∂˜zE)(o˜z, ˜ρ)do= 1 − 1 ρ,
where E is a smooth function of ˜z, ˜ρ and ρ, where we have set ρ = (1 + ˜x)/α = ˜ρ × α/s. Therefore, the critical points ˜z±are functions of ρ and α/s. For ρ near 1, the phase ϕ has a degenerate critical points of order 2 at ˜z = 0. As a consequence, the integral with respect to ˜y in (3.20) may be represented as a linear combination of an Airy function and its derivative: we now prove that this Airy function is exactly A(τ2/3ζ(˜x, α)), with ζ as in (3.12). Using [4], there exists an unique change of variables ˜z → σ, smooth and satisfying dσd˜z ∈ {0, ∞}, ˜/ z(σ) = σk(σ, 1/s, α) with k(σ, 1/s, α) ∼σ 1 +P
j≥1kj(1/s, α)σ j, and there exist smooth functions ζ#x, α, s˜ and Γ(˜x, α, s) such that
(3.24) ϕ(˜x, ˜y(˜x) + ˜z, α) = −σ 3 3 − σζ # ˜ x, α, s+ Γ(˜x, α, s),
and ∂˜zϕ(˜x, ˜y(˜x) + ˜z, α) = 0 if and only of σ2= −ζ#(˜x, α, s). Moreover, ζ#= 0 if and only if ρ = 1. Let ζ#(˜x, α, s) = α2/3ζ˜#(1+˜x
α , α, s). We prove that ˜ζ
# depends only on (1 + ˜x)/α and that the equation for ˜ζ holds for ˜ζ#as well.
Lemma 3.14. The function ˜ζ# is such that (− ˜ζ#)(∂ρζ˜#)2+ 1
Proof. We first compute how ζ# and ϕ relate: at the saddle points ˜z± (which occur when ˜σ± = ±p−ζ#), the critical values ϕ(˜x, ˜y(˜x) + ˜z
±, α) in (3.24) are −σ±3/3 − ˜σ±ζ#+ Γ = − ± (−ζ#)3/2/3 ∓ (−ζ#)3/2+ Γ = ±2 3(−ζ #)3/2+ Γ.
Taking difference and sum between the two critical values and setting ˜y± = ˜y(˜x) + ˜z± yields the following formulas for ζ#(˜x, α, s) = α2/3ζ˜#(1+˜x
α , α, s), Γ: (3.25) 4 3α(− ˜ζ #)3 2 1 + ˜x α , α, s = ϕ(˜x, ˜y+, α)−ϕ(˜x, ˜y−, α) , Γ(˜x, α, s) = ϕ(˜x, ˜y+, α) + ϕ(˜x, ˜y−, α) 2 .
Taking the derivative with respect to ˜x of the first equation in (3.25) and using (3.22) gives 2
q
−˜ζ#∂ρ(− ˜ζ#
) = −∂˜xφ(˜x, ˜y+, QN(s)) + ∂˜xφ(˜x, ˜y−, QN(s)) + ∂˜x˜y+∂˜yϕ(˜x, ˜y+, α) − ∂˜xy−∂˜˜ yϕ(˜x, ˜y−, α). As ˜y±are critical points of ϕ, the last two terms in the right hand side of the last equation cancel. As ∂˜yφ(˜x, ˜y±, QN(s)) = −α, using the eikonal equation for φ yields
(∂˜xφ(˜x, ˜y±, QN(s)))2= 1 −(∂y˜φ(˜x, ˜y±, QN(s))) 2
(1 + ˜x)2 = 1 − α2 (1 + ˜x)2, and the sign of ∂˜xφ(˜x, ˜y±, QN(s)) =
1+˜x−s sin ˜y± φ(˜x,˜y±,QN(s)) = s φ(˜x,˜y±,QN(s))( ˜ρ − sin( ˜ρ + ˜z±)) is ∓ depending on ±˜z±= ±(˜y±− ˜y(˜x)) ≥ 0. Introducing ∂˜xφ(˜x, ˜y±, QN(s)) = ∓(1 − α 2
(1+˜x)2)1/2 in the equation for ˜ζ#,
2 q −˜ζ#∂ ρ(− ˜ζ#) = −∂x˜φ(˜x, ˜y+, QN(s)) + ∂x˜φ(˜x, ˜y−, QN(s)) = 2 1 − α 2 (1 + ˜x)2 1/2 , hence (− ˜ζ#)(∂ρζ˜#)2= 1 −ρ12 for ρ = 1+˜x α . As ˜ζ #(1, ·) = 0, Lemma3.6yileds ˜ζ = ˜ζ#. Remark 3.15. The function Γ is independent of ˜x: taking the derivative with respect to ˜x in the second equation in (3.25) and using the previous computations yields
∂˜xΓ(˜x, α, s) = 1 2
∂˜xφ(˜x, ˜y+, QN(s)) + ∂˜xφ(˜x, ˜y−, QN(s))= 0.
It follows that Γ = Γ(α, s). As ϕ(0, ˜y±(0), α) = −˜y(0)α − φ(0, ˜y(0), QN(s)), using again (3.25) gives Γ(α, s) = −y0α − ψ(y0, s), where y0= arcsin(1s) and ψ(y0, s) = φ(0, ˜y(0), QN(s)).
The integration of ˜Fτ(z) with respect to ˜y in (3.20) yields Airy type factors of the form A(τ2/3ζ(˜x, α)) and A0(τ2/3ζ(˜x, α)). Indeed, using that ˜ζ#= ˜ζ depends only on 1+˜αx, ϕ from (3.24) becomes
ϕ(˜x, ˜y, α) = −σ 3
3 − σα
2/3ζ(˜x, α) + Γ(α, s).
The symbol of (3.20) becomes χ1(α) ˜χ(˜x, ˜y(˜x) + ˜z(σ)), with ˜χ supported near (0, y0). Set again aA(τ2/3ζ(˜x, α)) − bτ−1/3A0(τ2/3ζ(˜x, α)) = τ1/3
Z
eiτ (σ3˜3 +˜σζ(˜x,α))(a − b˜σ/i)d˜σ,
then, using Remark3.15, the phase of Fτ(z) becomes (z −y0)α−ψ(y0, s)−σ
3
3 −σζ(˜x, α)+ ˜ σ3
3 + ˜σζ(˜x, α). We further apply stationary phase with respect to ˜σ and ˜x (as we did when we obtained Eτ), with critical points ˜σ = σ and ˜σ2 = −ζ(˜x, α); the last equation can be solved for ˜x near 0 as, for α close to 1, ∂˜xζ(˜x, α) stays close to −21/3. At critical points, the hessian matrix has determinant |∂˜xζ(˜x, α)|. Stationary phase yields a factor τ−1 and the critical phase is (z − y0)α − ψ(y0, s). Hence,
Fτ(z) = τ4/3+1/3−1 τ ψ(y0, s)
(d−1)/2Z Z
eiτ ((z−y0)α−ψ(y0,s))χ0(σ)χ1(α) ˜˜ ΣF(˜xc(σ2, α), α, τ )dσdα,
where ˜ΣF, obtained from φ(0,y0,QN(s))
φ(˜x,˜y(˜x)+˜z(σ),QN(s))
(d−1)/2
(ΣF(a + bσ/i))(˜x, ˜y(˜x) + ˜z(σ), α, τ ), is elliptic near α = 1 and of order 0: indeed, for s > 1 + ε0 for some small but fixed ε0 > 0, the factor φ(0, y0, QN(s))/φ(˜x, ˜y, QN(s)) reads as an asymptotic expansion with parameter 1/s < 1 and main contribution 1 for (˜x, ˜y) near (0, y0). Integration in σ achieves the proof of Lemma3.12.
In the next three sections we prove Theorem1.1for the wave equation with frequencies comparable to 1/h: let d = 3 and set Ω = Ω3 = R3\ B3(0, 1). We first deal with the situation dist(Q0, ∂Ω) ≥ ε0 and dist(Q, ∂Ω) ≥ ε0, where ε0> 0 is a small, fixed constant, where all the arguments of Section3.1 apply. When dist(Q0, ∂Ω) ≥ ε0 and dist(Q, ∂Ω) ≤ ε0, Theorem 1.1 follows using the parametrix as given by Proposition3.4near the glancing regime, and this is dealt with in Section 3.3. Finally, for dist(Q0, ∂Ω) ≤ ε0 and dist(Q, ∂Ω) ≤ ε0, the parametrix from Proposition 3.4does not hold anymore (as the data is localized at a very small distance to ∂Ω). In Section3.4we represent the solution to the wave equation in terms of spherical harmonics to obtain sharp dispersion bounds.
3.2. The case dist(Q0, ∂Ω) ≥ ε0 and dist(Q, ∂Ω) ≥ ε0 for ε0:=√2 − 1. We retain notations from the previous section. Denote u the solution to (2.6) in Ω × R, u+= 1t>0u and u its extension to R3, then, using (2.9) and (2.11), one has
(3.26) u|t>0= δ 0(t − |Q − Q0|) 4π|Q − Q0| − u#, u#:= Z ∂Ω ∂νu+(P, Q0, t − |Q − P |) 4π|Q − P | dP. Let h ∈ (0, 1), χ ∈ C0∞([12, 2]), 0 ≤ χ ≤ 1, χ|[3/4,3/2] = 1 and set u
#
h(·, t) = χ(hDt)u
#(·, t) be as in (3.6) (where d = 3). In order to prove dispersive estimates for χ(hDt)u in Ω, it will be enough to obtain L∞ bounds for u#h(Q, Q0, t). We focus on the situation when Q0, Q ∈ R3\ B3(0, 1) and dist(Q0, B3(0, 1)) ≥ ε0, where ε0=
√
2 − 1 : Q0= QN(s) for some s ≥ 1 + ε0= √
2. We will prove that there exists a uniform constant C > 0, such that for all t > h and for all Q0, Q with dist(Q0, ∂Ω) ≥ ε0, dist(Q, ∂Ω) ≥ ε0, we have ku#h(Q, Q0, t)kL∞(Ω)≤ C
h2t.
Let ε > 0 and let χ0∈ C∞
0 ((−2, 2)), with χ0= 1 on [−1, 1], 0 ≤ χ0≤ 1 as in Section3.1. Let u # h,χ be given by (3.6) with the additional cut-off χ(dist(P, CQ0)/ε) in the symbol. From Proposition 3.2,
the usual dispersive bounds (1.3) hold for u#h,1−χ
0. Therefore, we now focus on u
# h,χ0.
We briefly recall the notations in Section 3.1 for d = 3: in spherical coordinates the domain is R3\ B3(0, 1) = {r ≥ 1, ϕ ∈ [0, π], ω ∈ [0, 2π]}, and, with r = 1 + x, y = π2 − ϕ, it becomes Ω = {(x, y, ω), x ≥ 0, y ∈ [−π
2, π
2], ω ∈ [0, 2π]}. When d = 3, the Fourier transform of u #
h,χ0 given in (3.10)
takes a simpler form, as the factor (3.7) equals e4π|Q−P |−iτ |Q−P |,
F (u#h,χ 0)(Q, Q0, τ ) = χ(hτ ) 4π Z P ∈S2 e−iτ |P −Q| |P − Q| χ0(dist(P, CQ0)/ε)F (∂xu +)(P, Q0, τ )dσ(P ), where, using Remark3.11, if P = (0, y, ω) we write F (∂xu+)(y, Q0, τ ) instead of F (∂xu+)(P, Q0, τ ). If Q ∈ R3\ B3(0, 1) has coordinates (r, yQ, ωQ) with r > 1 + ε0, then, for P = (1, y, ω),
|P − Q| =1 + r2− 2r cos y cos yQcos(ω − ωQ) − 2r sin y sin yQ 1/2
:= ˜ψ(y, ω, Q). Proposition 3.16. Let I0(Q, Q0, τ ) := F (u#χ0)(Q, Q0, τ )/τ , then
(3.27) I0(Q, Q0, τ ) :=τ −1 4π Z (y,ω) χ0((y − y0)/ε)e −iτ ˜ψ(y,ω,Q) ˜ ψ(y, ω, Q) F (∂xu +)(y, Q0, τ )dydω , and we have R e itτχ(hτ )τ I0(Q, Q0, τ )dτ . 1 h2t.
In the remaining of this section we prove Proposition 3.16. Let C0 = C(B(0, 1)) be the constant from Lemma2.6. We are reduced to showing that there exists a uniform constant C > 0 (independent of Q, Q0, t), such that for τ > 1 large enough, the following holds:
(3.28) |I0(Q, Q0, τ )| ≤ C
t when dist(Q, B(0, 1)) + p
s2− 1 ≤ t ≤ dist(Q, B(0, 1)) +ps2− 1 + C0. The part of the phase of (3.27) depending on y, ω equals − ˜ψ(y, ω, Q) + (y − y0)α − ψ(y0, s) and its critical point with respect to ω satisfies
∂ωψ(y, ω, Q) =˜ r cos y cos yQsin(ω − ωQ) ˜
Moreover, from Corollary 3.13, α belongs to a small neighborhood of 1 on the support of ˜χ1(α). There are two main situations: either cos yQ = 0, in which case Q0, O, Q are on the same line, or sin(ω − ωQ) = 0, when the stationary phase with respect to ω may apply.
3.2.1. Q belongs to a small conic neighborhood of the OS axis. Consider first the case cos yQ = 0, when yQ ∈ {±π
2}: if yQ = π
2, then Q and Q0 are on the half line ON : the phase ˜ψ(y, ω, Q) equals (1 + r2− 2r sin y)1/2 and the derivative of − ˜ψ(y, ω, Q) + (y − y0)α − ψ(y0, s) with respect to y is given by α + ˜r cos y
ψ(y,ω,Q) ≥ α for all y ∈ (− π 2,
π
2), hence the phase is non-stationary in y for α on the support of ˜χ1; repeated integrations by parts yield a trivial contribution O(τ−∞).
Let yQ= −π2, in which case Q ∈ OS, Q = QS(r). For yQ= −π2 the phase ˜ψ(y, ω, Q) is independent of ω and has a simpler form ˜ψ(y, ω, Q) = (1 + r2+ 2r sin y)1/2 = ψ(−y, r), where ψ has been defined in (3.9). The second derivative of ψ(−y, r) vanishes only when −y = yc = arcsin1r. Let −y = yc+ ˜z: we first prove that for |˜z| ≥ π4, the phase of I0is non-stationary with respect to y for α close to 1. Lemma 3.17. Let ˜χ1 be as in Corollary3.13, supported in (3/4, 5/4). Let χ2(α) ∈ C∞ be a smooth cut-off, 0 ≤ χ2(α) ≤ 1, supported for 1 − α2≤ 121 and such that 1 − χ2 is supported for 1 − α2≥ 161.
(1) For α on the support of χ2 and for |y + arcsin1r| ≥ π
4, the phase of I0 is non-stationary with respect to y and the contribution of the integral (3.27) for |y + arcsin1r| ≥ π
4 is O(τ −∞). (2) For α on the support of ˜χ1(1 − χ2), the phase of I0 has non-degenerate critical points with
respect to y, α but if ε < 1/24, |y − y0| > 2ε (hence, y 6∈ suppχ0((· − y0)/ε)). The contribution of the integral (3.27) with additional cut-off (1 − χ2(α)) is O(τ−∞).
Proof. From the discussion above, it follows that for yQ = −π2, the phase ˜ψ(y, ω, Q) is independent of ω. We denote it ψ(−y, r) = (1 + r2+ 2r sin y)1/2: it has an unique degenerate critical point of order exactly two, yc = arcsin1r. Critical points with respect to y satisfy α = ∂yψ(−y, r) = ψ(−y,r)r cos y which translates into (r sin y+1)ψ2(−y,r)2 = 1 − α
2. Let α on the support of χ
2. Let −y = yc + ˜z, yc = arcsin1r, and assume first ˜z ≤ −π4, then, as y ∈ (−π/2, π/2) and 0 < arcsin1r ≤ π/4, we have (−˜z) ∈ (π/4, 3π/4). We claim that in this case we have 10(1 + r sin y)2 > ψ(−y, r)2, in which case the critical point α satisfies 101 < (r sin y+1)ψ2(−y,r)2 = 1 − α
2, so it doesn’t belong to the support of χ2. As ψ(−y, r)2= (1 + r sin y)2+ r2cos2y, the last inequality is equivalent to 9(1 + r sin y)2> r2cos2y. As 1 + r sin y = 1 − r sinarcsin1r+ ˜z=√r2− 1 sin(−˜z) + 1 − cos(˜z) > 0, the last inequality reads as
3pr2− 1 sin(−˜z) + 1 − cos(˜z)> r cos(arcsin1 r+ ˜z) =
p
r2− 1 cos(˜z) + sin(˜z). As (−˜z) ∈ (π/4, 3π/4), then sin(−˜z) > cos ˜z and the last inequality holds true (for all r ≥ 1+ε0=
√ 2). Therefore on the support of χ2where |1 − α2| < 121, the phase is non-stationary with respect to y, and we obtain a O(τ−∞) contribution. Let now ˜z ≥ π4: as y ∈ (−π2,π2), we must have arcsin1r ≤ π
2− ˜z < π 4 which further implies r ≥√2 and π4 < ˜z ≤ π2 − arcsin1
r. As sin ˜z ≥ √ 2 2 , we compute |1 + r sin y| = −1 + r sin(arcsin1 r + ˜z) = −1 + cos ˜z + p r2− 1 sin ˜z. The function g(˜z) :=√r2− 1 sin ˜z + cos ˜z − 1 is increasing on (π
4, π 2) as g 0(˜z) = r cos(arcsin1 r+ ˜z) > 0, hence g(˜z) ≥ g(π 4) = √ 2 2 ( √
r2− 1 + 1) − 1. As r ≥√2, the same arguments allow to obtain, again, 10|1 + r sin y| ≥ ψ(−y, r), so the phase is non-stationary with respect to y for α on the support of χ2. Let now α ∈ supp( ˜χ1(1 − χ2)), then α is close to 1 but such that 1 − α2 > 161. At yQ = −π2, the phase of I0 equals − ˜ψ(−y, r) + (y − y0)α − ψ(y0, s) +23(−ζ0(α))3/2, where the last term comes from the factor A+(τ2/3ζ0(α))−1 of (3.19). Using (3.13) yields 2
3(− ˜ζ) 3/2(ρ) = 2 3 √ 2(ρ−1)ρ1/23/2 1 + O(ρ − 1) and for ζ0(α) = ζ(0, α) = α2/3ζ(˜ 1 α), 2 3(−ζ0(α)) 3/2=2 3(−α 2/3ζ(1/α))˜ 3/2= 2 3 √ 2(1 − α)3/2(1 + O(1 − α)), hence the phase of I0 is stationary with respect to α when y − y0+
√
2√1 − α(1 + O(1 − α)) = 0. As |y − y0| ≤ 2ε on the support of χ0, we must have √2√1 − α ≤ 4ε there. As 14 ≤ √1 − α2 on
