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, well-known 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.
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Acknowledgments
We would like to thank Bernd Ammann for several useful discussions. We also gladly acknowledge the hospitality of the SFB 1085 Higher Invariants at the Faculty of Mathematics at the University of Regensburg where parts of this article was written. We are also very grateful to Alexander Engel and Mirela Kohr for carefully reading our paper and for making several very useful comments.
N.G. has been partially supported by SPP 2026 (Geometry at infinity), funded by the DFG. V.N. has been partially supported by ANR-14-CE25-0012-01
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Große, N., Nistor, V. Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry. Potential Anal 53, 407–447 (2020). https://doi.org/10.1007/s11118-019-09774-y
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DOI: https://doi.org/10.1007/s11118-019-09774-y