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

Bernd Ammann, Nadine Große and Victor Nistor

Mathematische Nachrichten, 2019, vol. 292, issue 6, 1213-1237

Abstract: Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator Δ:Hk+1(M)∩H01(M)→Hk−1(M), k∈N0, invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let ∂DM⊂∂M be an open and closed subset of the boundary of M. We say that (M,∂DM) has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance dist(x,∂DM) from a point x∈M to ∂DM⊂∂M is bounded uniformly in x (and hence, in particular, ∂DM intersects all connected components of M). For manifolds (M,∂DM) with finite width, we prove a Poincaré inequality for functions vanishing on ∂DM, thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds (M,∂DM) with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces Hs(M), s≥0.

Date: 2019
References: Add references at CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://doi.org/10.1002/mana.201700408

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:292:y:2019:i:6:p:1213-1237

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().