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High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation

Published online by Cambridge University Press:  20 November 2018

Fernando Cardoso  and
Georgi Vodev
Fernando Cardoso
Affiliation:
Universidade Federal de Pernambuco Departamento de Matemàtica CEP. 50540-740 Recife-Pe Brazil, e-mail: fernando@dmat.ufpe.br
Georgi Vodev
Affiliation:
Université de Nantes Département de Mathématiques UMR 6629 du CNRS 2, rue de la Houssiniére BP 92208 44072 Nantes Cedex 03 France, e-mail: vodev@math.univ-nantes.fr
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Abstract

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We prove an uniform Hölder continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation.

Keywords

35B3735J1547F05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 4 , 01 December 2004 , pp. 504 - 514
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Burq, N., Décroissance de l’énergie locale de l’équation des ondes pour le probl`eme extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998), 1–29.Google Scholar
[2] Burq, N., Lower bounds for shape resonances widths of long-range Schrödinger operators. Amer. J. Math. 124 (2002), 677–735.Google Scholar
[3] Cardoso, F. and Vodev, G., Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II. Ann. Henri Poincaré, 3 (2002), 673–691.Google Scholar
[4] Vasy, A. and Zworski, M., Semiclassical estimates in asymptotically Euclidean scattering. Comm. Math. Phys. 212 (2000), 205–217.Google Scholar
[5] Vodev, G., Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds with cusps. Comm. Partial Differential Equations 27 (2002), 1437–1465.Google Scholar