Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Abstract

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P, C) is equivalent to the uniform regularity of the natural family (Px, Cx) of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincaré inequality of (Ammann-Große-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, wellknown theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition. 12 4. Variational boundary conditions and regularity 13 4.

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

“…In particular, we can assume that for all , where as independent of w . The main point of this proof (exploited in greater generality in ) is that also satisfies elliptic regularity. This is because it is strongly elliptic, since this property is preserved by limits in which the strong ellipticity constant is bounded from below and strongly elliptic operators with Dirichlet boundary conditions satisfy elliptic regularity (see the references above).…”

“…If and , then we say that f is the “classical” and that . These elementary but tedious details pertain more to analysis than geometry, so we skip the simple proofs (the reader can see more details in a more general setting in ), which however follow a different line than the one indicated here. A more substantial remark is that strongly elliptic operators with Neumann boundary conditions still satisfy elliptic regularity, so the proof of Lemma extends to the case of Neumann boundary conditions.…”

“…Some of them were included in the first version of the associated ArXiv preprint, 1611.00281.v1, which had a slightly different title and will not be published in that form. The results of that preprint that were not already included in this paper, as well as others, will be included in two forthcoming papers .…”

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

“…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.…”

“…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. We assume that ρ 2 P , ρB 1 , and B 0 have coefficients in W ∞,∞ (g).…”