L¹-elliptic regularity and H=W on the whole L^p-scale on arbitrary manifolds

2014/05/12 by Davide Guidetti, Guidetti, Davide, Batu Güneysu +3
Mathematics · Medicine · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Soft tissue tumor case studies #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1405.2654

27 pages

arxiv created 2014/05/12 · openalex publication_date 2014/05/12 · arxiv updated 2014/05/13 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We define abstract Sobolev type spaces on L^p-scales, p∈ [1,∞), on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families \mathfrakP of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalized ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers-Serrin\rqs theorem on the whole L^p-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local elliptic regularity result in L¹ on the Besov scale, which shows that the above generalized ellipticity condition is satisfied on the whole L^p-scale, if some differential operator from \mathfrakP that has a sufficiently high (but not necessarily the highest) order is elliptic.

Citations

• Geometric Wave Equations

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