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Optimal constants in nontrapping resolvent estimates and applications in numerical analysis
- Source :
- Pure Appl. Analysis 2 (2020) 157-202
- Publication Year :
- 2018
-
Abstract
- We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ times the length of the longest generalized bicharacteristic of $|\xi|_g^2-1$ remaining in the support of $\chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.<br />Comment: 40 pages
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Numerical Analysis
Subjects
Details
- Database :
- arXiv
- Journal :
- Pure Appl. Analysis 2 (2020) 157-202
- Publication Type :
- Report
- Accession number :
- edsarx.1810.13426
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2140/paa.2020.2.157