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

TL;DR

This paper investigates the invertibility and well-posedness of the Laplace-Beltrami operator on manifolds with boundary and bounded geometry, establishing conditions under which the operator is invertible and the Poisson problem is well-posed.

Contribution

It introduces the concept of finite width for manifolds with boundary and bounded geometry, proving Poincaré inequalities and invertibility of the Laplacian under these conditions.

Findings

Proves Poincaré inequality for manifolds with finite width.

Establishes invertibility of the Laplacian with mixed boundary conditions.

Shows well-posedness of the Poisson problem in higher Sobolev spaces.

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 $\Delta\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$, 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. Nach. 2001). We begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let $\partial_D M \subset \partial M$ be an open and closed subset of the boundary of $M$. We say that $(M, \partial_D M)$ has \emph{finite width} if, by definition, $M$ is a manifold with boundary and bounded geometry such that the distance $d(x, \partial_D M)$ from a point $x \in M$ to $\partial_D M \subset \partial M$ is bounded uniformly in $x$ (and hence, in particular, $\partial_D M$ intersects all connected components of $M$). For manifolds $(M, \partial_D M)$ with finite width, we prove a Poincar\'e inequality for functions vanishing on $\partial_D M$, thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincar\'e inequality then leads, as in the classical case to results on the spectrum of $\Delta$ with domain given by mixed boundary conditions, in particular, $\Delta$ is invertible for manifolds $(M, \partial_D M)$ with finite width. The bounded geometry assumption then allows us to prove the well-posedness of the Poisson problem with mixed boundary conditions in higher Sobolev spaces $H^s(M)$, $s \ge 0$.