Marius Mitrea, University of Missouri-Columbia, Department of Mathematics, Columbia, MO 65211, e-mail: marius@math.missouri.edu; Victor Nistor, Pennsylvania State University, Math. Dept., University Park, PA 16802, e-mail: nistor@math.psu.edu
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 \cite{FJL1} and Kral-Wedland \cite{KralWendland}. 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."
Keywords: layer potentials, manifolds with cylindrical ends, Dirichlet problem
Classification (MSC 2000): 31C12, 58J05, 58J40, 35J05
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