### 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 ⊂ ℝ^{3} using the weighted Sobolev spaces Κ _{a}^{m}(ℙ). In particular, we show that there is no loss of Κ_{a}^{m}-regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify Κ_{a}^{m}(ℙ) with the Sobolev spaces on P associated to the metric r_{ℙ}^{-2} g_{E}, where g_{E} is the Euclidean metric and r_{ℙ}(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 ⊂ ℝ^{n} using weighted Sobolev spaces, where the weight is the distance to the boundary.

Original language | English (US) |
---|---|

Pages (from-to) | 161-206 |

Number of pages | 46 |

Journal | Documenta Mathematica |

Volume | 11 |

Issue number | 1 |

State | Published - Dec 15 2006 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Documenta Mathematica*,

*11*(1), 161-206.

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*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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sobolev spaces on lie manifolds and regularity for polyhedral domains

AU - Ammann, Bernd

AU - Ionescu, Alexandru D.

AU - Nistor, Victor

PY - 2006/12/15

Y1 - 2006/12/15

N2 - 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 rℙ-2 gE, where gE is the Euclidean metric and rℙ(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.

AB - 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 rℙ-2 gE, where gE is the Euclidean metric and rℙ(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.

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M3 - Article

AN - SCOPUS:33845468329

VL - 11

SP - 161

EP - 206

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

IS - 1

ER -