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Geometric Regularization on Riemannian and Lorentzian Manifolds

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Evolution Equations of Hyperbolic and Schrödinger Type

Part of the book series: Progress in Mathematics ((PM,volume 301))

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Abstract

We investigate regularizations of distributional sections of vector bundles by means of nets of smooth sections that preserve the main regularity properties of the original distributions (singular support, wavefront set, Sobolev regularity). The underlying regularization mechanism is based on functional calculus of elliptic operators with finite speed of propagation with respect to a complete Riemannian metric. As an application we consider the interplay between the wave equation on a Lorentzian manifold and corresponding Riemannian regularizations, and under additional regularity assumptions we derive bounds on the rate of convergence of their commutator. We also show that the restriction to underlying space-like foliations behaves well with respect to these regularizations.

Mathematics Subject Classification. Primary 58J37; Secondary 46F30, 46T30, 35A27, 53C50.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Wien, Austria

    Shantanu Dave, Günther Hörmann & Michael Kunzinger

Authors
  1. Shantanu Dave
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  2. Günther Hörmann
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  3. Michael Kunzinger
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Corresponding author

Correspondence to Shantanu Dave .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

  3. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

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Dave, S., Hörmann, G., Kunzinger, M. (2012). Geometric Regularization on Riemannian and Lorentzian Manifolds. In: Ruzhansky, M., Sugimoto, M., Wirth, J. (eds) Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol 301. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0454-7_5

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