To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator $M$ along the trapped ray between the two obstacles. Using this method Ikawa and G{é}rard established the existence of parallel rows of poles in a strip $Im z\leq c$ as $Re z$ tends to infinity. Assuming that the boundaries are analytic and the eigenvalues of Poincar{é} map are non-resonant we use the Birkhoff normal form for $M$ to improve this result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.
We prove the existence of a resonance free region in scattering by a strictly convex obstacle with the Robin boundary condition. More precisely, we show that the scattering resonances lie below a cubic curve which is the same as in the case of the Neumann boundary condition. This generalizes earlier results on cubic poles free regions obtained for the Dirichlet boundary condition.
We prove Strichartz estimates without loss for the Schrödinger equation and the wave equation outside finitely many strictly convex obstacles verifying Ikawa's condition, extending the approach we introduced previously for the two convex case.
We prove global Strichartz estimates without loss for the wave equation outside two strictly convex obstacles, following the roadmap introduced in [Lafontaine, 2017] for the Schrödinger equation. Moreover, we show a first step toward the large data scattering for the critical non linear equation associated to this geometrical setting, and prove the scattering for a class of non trapping obstacles close to the two convex framework.
We consider the defocusing energy-critical nonlinear Schrödinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.
We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3D.
We prove global Strichartz estimates without loss outside two strictly convex obstacles, combining arguments from M.Ikawa (1982,1988) with more recent ones inspired by N.Burq, C.Guillarmou, and A. Hassell (2010) and O. Ivanovici (2010). This is to be contrasted with the loss in the smoothing estimate for the Schrödinger equation with Dirichlet boundary conditions in the presence of a trapped geodesic.
We study the distribution of resonances for smooth strictly convex obstacles under general boundary conditions. We show that under a pinched curvature condition for the boundary of the obstacle, the resonances are separated into cubic bands and the distribution in each bands satisfies Weyl's law.
We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the sec…
We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument…
We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on th…
This paper concerns spectral clusters of the Neumann Laplacian on compact Riemannian manifolds with strictly geodesically concave boundary. We prove an inequality which controls the $L^p$ norms of spectral clusters.
We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.
In this paper, we are interested in the problem of scattering by strictly convex obstacles in the plane. We provide an upper bound for the number $N(r,γ)$ of resonances in the box $\{r \le \Re(λ) \le r + 1$; $\Im(λ) \ge - γ\}$. It was proved in a work of S. Nonnenmacher, J. Sj{ö}strand and M. Zworski (2014) that $N (r,γ) = O_γ(r^{d_H})$ where $2d_H + 1$ is the Hausdorff dimension of the trapped set of the billiard flow. In this article, we provide an improved upper bound in the band $0 \le γ< γ_{cl} /2$, where $γ_{cl}$ is the classical decay rate of the flow. This improved Weyl upper bound is …
For scattering off a smooth, strictly convex obstacle $Ω\subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S \subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| \mathrm{Vol}(\partial Ω)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. U…
We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.
Global existence and scattering for the nonlinear defocusing Schrödinger equation in 2 dimensions are known for domains exterior to star-shaped obstacles and for nonlinearities that grow at least as the quintic power. In this paper, we extend the global existence result for all non-trapping obstacles and for nonlinearities with power strictly greater than quartic. For such nonlinearities, we also prove scattering for a class of so-called almost star-shaped obstacles.
We prove the scattering for a defocusing nonlinear Schrödinger equation with a sum of two repulsive potentials with strictly convex level surfaces, thus providing a scattering result in a trapped setting similar to the exterior of two strictly convex obstacles.
In this paper, we consider the Schrödinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obs…
This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly conv…
We consider the two-dimensional high-frequency plane wave scattering problem in the exterior of a finite collection of disjoint, compact, smooth, strictly convex obstacles with Neumann boundary conditions. Using integral equation formulations, we determine the Hörmander classes and derive high-frequency asymptotic expansions of the total fields corresponding to multiple scattering iterations on the boundaries of the scattering obstacles. These asymptotic expansions are used to obtain sharp wavenumber dependent estimates on the derivatives of multiple scattering total fields which, in turn, all…
We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) $H^1$ norm of the solution is bounded by the $L^2$ norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacle…
