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

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+1false(Mfalse)∩H01false(Mfalse)→Hk−1false(Mfalse), k∈double-struckN0, 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 Hsfalse(Mfalse), s≥0.

Linear parabolic equations with strong boundary degeneration

“…A Riemannian manifold with boundary is a urR manifold iff it has bounded geometry in the sense of Th. Schick [28] (also see [10], [11], [12], [17] for related definitions). Detailed proofs of these equivalences will be found in [9].…”

Analysis and boundary value problems on singular domains: An approach via bounded geometry

“…For dimension two domains M , the Poincaré inequality for (M, A, g) is equivalent to the Poincaré inequality for (M, A, g 0 ) (same proof as the conformal invariance of the Laplacian in two dimensions). The Poincaré inequality of [5] then gives: Let P = P a satisfy the strong Legendre condition with all ∇ j a bounded. Then there exists η a,f > 0 such that, for |s| < η a,f and ℓ ∈ Z + , we have an isomorphism…”

“…Let ∂ a ν be the conormal derivative associated to P , see [14]. Combining Theorem 6 with the Lax-Milgram Lemma and with the fact that the Dirichlet and Neumann boundary conditions satify the uniform Shapiro-Lopatinski regularity conditions [5,14], we obtain: Theorem 8. We assume that (M, ∂ 0 M, E, g 0 , ρ) satisfies the Hardy-Poincaré inequality.…”

“…Assume that H carries a globally defined unit normal vector field ν and let exp ⊥ (x, t) := exp M x (tν x ) be the exponential in the direction of the chosen unit normal vector. By II H we denote the second fundamental form of H. The following two definitions are from [5]. Definition 1.…”

“…Let P be a second order differential operator. We assume from now on that we are given a partition of the boundary ∂M = ∂ 0 M ⊔ ∂ 1 M into two disjoint, open subsets, as in [5], and order i differential boundary conditions B i on ∂ i M . See, for example, [7,20] for general results on boundary value problems on smooth domains, [11,18,23,22] for the case of non-smooth domains, and [4,14] for more general boundary conditions involving projections.…”