Optimal regularization processes on complete Riemannian manifolds

Dave, Shantanu; Hoermann, Guenther; Kunzinger, Michael

doi:10.48550/arxiv.1003.3341

Cited by 1 publication

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“…(ii) Pick χ ∈ D(R) such that χ ≡ 1 near Z and let ũ(t) := χ(t)u(t). It then follows from [13], Prop. 3.7 that (T ε ũ − T ε u) ε is negligible on supp(ũ − u) c ⊇ Z.…”

Section: 2mentioning

confidence: 90%

“…This procedure encodes regularity and singularity features in terms of asymptotic behavior. Our approach is based on [13], to which we refer for further details.…”

Section: Regularization Of Distributions On Complete Riemannian Manif...mentioning

confidence: 99%

“…Remark 2.4. In the scalar case, an alternative proof (not relying on the above doubling-technique) of Theorem 2.2 can be found in [13], Section 3.…”

Section: 1mentioning

confidence: 99%

“…We consider regularization processes to smooth out distributions on Riemannian and globally hyperbolic Lorentzian manifolds and investigate their compatibility with the wave-equation. The Riemannian setting has been addressed in [13] and we include an introduction to this approach. For a globally hyperbolic manifold we pick a splitting of the metric as obtained by [2] which provides us with a globally defined time function and a foliation by space-like hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Regularization on Riemannian and Lorentzian Manifolds

Dave

Hörmann

Kunzinger

2012

Evolution Equations of Hyperbolic and Schrödinger Type

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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.

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“…(ii) Pick χ ∈ D(R) such that χ ≡ 1 near Z and let ũ(t) := χ(t)u(t). It then follows from [13], Prop. 3.7 that (T ε ũ − T ε u) ε is negligible on supp(ũ − u) c ⊇ Z.…”

Section: 2mentioning

confidence: 90%

“…This procedure encodes regularity and singularity features in terms of asymptotic behavior. Our approach is based on [13], to which we refer for further details.…”

Section: Regularization Of Distributions On Complete Riemannian Manif...mentioning

confidence: 99%

“…Remark 2.4. In the scalar case, an alternative proof (not relying on the above doubling-technique) of Theorem 2.2 can be found in [13], Section 3.…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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