Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann map

We study the semiclassical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [Vodev, Commun. Math. Phys. 2015;336(3):1141-1166;...

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Bibliographic Details
Published in:Communications in partial differential equations 2019-05, Vol.44 (5), p.367-396
Main Author: Vodev, Georgi
Format: Article
Language:English
Subjects:
Analysis of PDEs
Asymptotic properties
Dirichlet problem
Dirichlet-to-Neumann map
Eigenvalues
glancing region
Mathematics
parametrix
Refraction
Riemann manifold
Smooth boundaries
transmission eigenvalues
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Summary:We study the semiclassical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [Vodev, Commun. Math. Phys. 2015;336(3):1141-1166; Vodev, Asymptotic Anal. 2018;106:147-168]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in (Vodev, Asymptotic Anal. 2018;106:147-168) in the isotropic case when the restrictions of the refraction indices on the boundary coincide.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2018.1547746