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Resolvent estimates and local decay of waves on conic manifolds
- Publication Year :
- 2012
-
Abstract
- 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.<br />Comment: Incorporates appendix from previous version into body of text
- Subjects :
- Mathematics - Analysis of PDEs
Mathematical Physics
35B34, 58J47, 58J50
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1209.4883
- Document Type :
- Working Paper