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Boundary value problems and layer potentials on manifolds with cylindrical ends

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Abstract

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis [10] and Kral-Wedland [18]. We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.”

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri-Columbia, Columbia, MO, 65211

    Marius Mitrea

  2. Math. Dept., Pennsylvania State University, University Park, PA, 16802

    Victor Nistor

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  1. Marius Mitrea
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  2. Victor Nistor
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Correspondence to Marius Mitrea.

Additional information

Mitrea was partially supported by NSF Grant DMS-0139801 and a UMC Research Board Grant. Nistor was partially supported by NSF Grants DMS-0209497 and DMS-0200808. Manuscripts available from http://www.math.missouri.edu/marius and http://www.math.psu.edu/nistor/.

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Mitrea, M., Nistor, V. Boundary value problems and layer potentials on manifolds with cylindrical ends. Czech Math J 57, 1151–1197 (2007). https://doi.org/10.1007/s10587-007-0118-9

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  • DOI: https://doi.org/10.1007/s10587-007-0118-9

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