Sobolev spaces on lie manifolds and regularity for polyhedral domains

Bernd Ammann, Alexandru D. Ionescu, Victor Nistor

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains ℙ ⊂ ℝ3 using the weighted Sobolev spaces Κ am(ℙ). In particular, we show that there is no loss of Κam-regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify Κam(ℙ) with the Sobolev spaces on P associated to the metric r-2 gE, where gE is the Euclidean metric and r(x) is a smoothing of the Euclidean distance from x to the set of singular points of ℙ. A suitable compactification of the interior of ℙ then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded domain with boundary O ⊂ ℝn using weighted Sobolev spaces, where the weight is the distance to the boundary.

Original languageEnglish (US)
Pages (from-to)161-206
Number of pages46
JournalDocumenta Mathematica
Volume11
Issue number1
StatePublished - Dec 15 2006

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Sobolev Spaces
Regularity
Weighted Sobolev Spaces
Compactification
Elliptic Regularity
Metric
Subset
Pseudodifferential Operators
Elliptic Systems
Euclidean Distance
Singular Point
Well-posedness
Submanifolds
Differential operator
Smoothing
Bounded Domain
Euclidean
Vector Field
Lie Algebra
Interior

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Ammann, B., Ionescu, A. D., & Nistor, V. (2006). Sobolev spaces on lie manifolds and regularity for polyhedral domains. Documenta Mathematica, 11(1), 161-206.
Ammann, Bernd ; Ionescu, Alexandru D. ; Nistor, Victor. / Sobolev spaces on lie manifolds and regularity for polyhedral domains. In: Documenta Mathematica. 2006 ; Vol. 11, No. 1. pp. 161-206.
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Ammann, B, Ionescu, AD & Nistor, V 2006, 'Sobolev spaces on lie manifolds and regularity for polyhedral domains', Documenta Mathematica, vol. 11, no. 1, pp. 161-206.

Sobolev spaces on lie manifolds and regularity for polyhedral domains. / Ammann, Bernd; Ionescu, Alexandru D.; Nistor, Victor.

In: Documenta Mathematica, Vol. 11, No. 1, 15.12.2006, p. 161-206.

Research output: Contribution to journalArticle

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Ammann B, Ionescu AD, Nistor V. Sobolev spaces on lie manifolds and regularity for polyhedral domains. Documenta Mathematica. 2006 Dec 15;11(1):161-206.