Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStabilisation of Waves on Product Manifolds by Boundary Strips
166
0
0.0
(
0
)
Authors
Ruoyu P. T. Wang
Ask ChatGPT about the research
No Arabic abstract
We show that a transversely geometrically controlling boundary damping strip is sufficient but not necessary for $t^{-1/2}$-decay of waves on product manifolds. We give a general scheme to turn resolvent estimates for impedance problems on cross-sections to wave decay on product manifolds.
rate research
Read More
We consider manifolds with conic singularites that are isometric to $mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a very weak Huygens principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr{o}dinger equation.
We investigate the dispersive properties of solutions to the Schrodinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrodinger flow on each eigenspace of the link manifold satisfies a weighted $L^1to L^infty$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schrodinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral measure. In even dimensions, we prove a similar estimate, but with a loss of $t^{1/2}$ compared to the sharp Euclidean estimate.
We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schrodinger equation, because of the strong indefiniteness of the energy.
We prove that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the perturbed biharmonic operator $Delta_g^2+q$ on a conformally transversally anisotropic Riemannian manifold of dimension $ge 3$ with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [51]. In particular, our result is applicable and new in the case of smooth bounded domains in the $3$-dimensional Euclidean space as well as in the case of $3$-dimensional admissible manifolds.
We show that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the Schrodinger operator $-Delta_g+q$ on a conformally transversally anisotropic manifold of dimension $geq 3$, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of Nachman-Street to our setting. We also identify the main space introduced by Nachman-Street with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet-to-Neumann map.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
