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Sobolev spaces on Lie manifolds and regularity for polyhedral domains

Bernd Ammann, Alexandru D. Ionescu, Victor Nistor

TL;DR

The paper develops a robust Sobolev- and elliptic-regularity framework on Lie manifolds, unifying analysis on non-smooth domains via a Lie-structure-at-infinity. It defines regular open subsets as the natural domains for analysis, proves trace and extension theorems, and establishes a regularity theory for strongly elliptic problems in weighted Kondratiev spaces, notably for polyhedral domains in 3D by conformal blow-up and desingularization. A global tubular neighborhood theorem for tame Lie submanifolds and a pseudodifferential calculus Ψ_V(M0) adapted to the Lie structure at infinity are introduced, together with mapping properties and elliptic regularity results in this calculus. The results yield no loss of regularity for strongly elliptic systems with smooth coefficients in weighted Sobolev spaces on polyhedral domains and provide a versatile toolbox for further geometric-analytic problems on Lie manifolds and their regular open subsets.

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 $\PP \subset \RR^3$ using the weighted Sobolev spaces $\Kond{m}a(\PP)$. In particular, we show that there is no loss of $\Kond{m}a$--regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify $\Kond{m}a(\PP)$ with the Sobolev spaces on $\PP$ associated to the metric $r_{\PP}^{-2} g_E$, where $g_E$ is the Euclidean metric and $r_{\PP}(x)$ is a smoothing of the Euclidean distance from $x$ to the set of singular points of $\PP$. A suitable compactification of the interior of $\PP$ 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 $\maO \subset \RR^n$ using weighted Sobolev spaces, where the weight is the distance to the boundary.

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

TL;DR

The paper develops a robust Sobolev- and elliptic-regularity framework on Lie manifolds, unifying analysis on non-smooth domains via a Lie-structure-at-infinity. It defines regular open subsets as the natural domains for analysis, proves trace and extension theorems, and establishes a regularity theory for strongly elliptic problems in weighted Kondratiev spaces, notably for polyhedral domains in 3D by conformal blow-up and desingularization. A global tubular neighborhood theorem for tame Lie submanifolds and a pseudodifferential calculus Ψ_V(M0) adapted to the Lie structure at infinity are introduced, together with mapping properties and elliptic regularity results in this calculus. The results yield no loss of regularity for strongly elliptic systems with smooth coefficients in weighted Sobolev spaces on polyhedral domains and provide a versatile toolbox for further geometric-analytic problems on Lie manifolds and their regular open subsets.

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 using the weighted Sobolev spaces . In particular, we show that there is no loss of --regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify with the Sobolev spaces on associated to the metric , where is the Euclidean metric and is a smoothing of the Euclidean distance from to the set of singular points of . A suitable compactification of the interior of 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 using weighted Sobolev spaces, where the weight is the distance to the boundary.

Paper Structure

This paper contains 22 sections, 36 theorems, 102 equations, 1 figure.

Table of Contents

  1. Lie manifolds
  2. Definition of Lie manifolds
  3. Riemannian metric
  4. Examples
  5. $\mathcal{V}$-differential operators
  6. Regular open sets
  7. Curvilinear polygonal domains
  8. Submanifolds
  9. General submanifolds
  10. Second fundamental form
  11. Tame submanifolds
  12. Sobolev spaces
  13. Definition of Sobolev spaces using vector fields and connections
  14. Definition of Sobolev spaces using partitions of unity
  15. Sobolev spaces on regular open subsets
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\mathbb P \subset \mathbb R^3$ be a polyhedral domain and $P$ be a strongly elliptic, second order differential operator with coefficients in $\mathcal{C}^\infty(\overline{\mathbb P})$. Let $u \in {\mathcal{K}}^{1}_{a+1}(\mathbb P)$, $u = 0$ on $\partial \mathbb P$, $a\in\mathbb R$. If $Pu \in The same result holds for strongly elliptic systems.

Figures (1)

  • Figure 1: A regular open set $\Omega$. Note that the interior of $\partial_\infty \overline{\Omega}$ is contained in $\Omega$, but the true boundary $\partial_{\rm top} \Omega =\partial_{\rm top} \overline\Omega$ is not contained in $\Omega$

Theorems & Definitions (80)

  • Theorem 1
  • Example 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 70 more