Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

Abstract

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving [11].

Substructuring Methods in Nonlinear Function Spaces

“…We restrict our attention to k = 1, p = 2. The following is a special case of a result proved in [5]. We denote by D(Ω, E) the smooth sections in (E, π, Ω) and by D(Ω, E| Γ ) the smooth sections of the bundle restriction on Γ .…”

Boundary value problems for noncompact boundaries of Spincmanifolds and spectral estimates

“…In particular, the restriction operator will have a bounded linear right inversethat is called extension operator E. For more details on the definition of bounded geometry on manifolds with boundary see [26]. For the equivalence of all those different definitions of Sobolev-norms involved here and the corresponding theorems for submanifolds (not necessarily hypersurfaces) see [14].…”

“…Those will be Fermi coordinates and there will be a adapted synchronous trivialization of E. This will allow that we can use the trace theorem on R n on the individual charts to obtain the trace theorem on (M, Σ). In the following, we restrict to trace theorems for Sobolev spaces over L 2 , for more general domains as Sobolev spaces over L p or Triebel-Lizorkin spaces see [14]. Before we define Sobolev spaces for sections of E, we introduce Fermi coordinates adapted to the boundary and a corresponding synchronous trivialization of the vector bundle: …”

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

“…The proof of Theorem is based on a Poincaré inequality on M for functions vanishing on , under the assumption that the pair has finite width, and on local regularity results (see Theorems and ). The higher regularity results is obtained using a description of Sobolev spaces on manifolds with bounded geometry using partitions of unity and is valid without the finite width assumption (we only need that M is with boundary and bounded geometry for our regularity result to be true). We also need the classical regularity of the Dirichlet and Neumann problems for strongly elliptic operators on smooth domains.…”

“…In particular, we introduce and study submanifolds with bounded geometry and we devise a method to construct manifolds with boundary and bounded geometry using a gluing procedure, see Corollary . In Subsection 2.4 we recall the definitions of basic coordinate charts on manifolds with boundary and bounded geometry and use them to define the Sobolev spaces, as for example in . The Poincaré inequality for functions vanishing on and its proof, together with some geometric preliminaries can be found in Section 3.…”