Sobolev spaces on Lie manifolds and regularity for polyhedral domains

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