TY - JOUR

T1 - Boundary value problems and layer potentials on manifolds with cylindrical ends

AU - Mitrea, Marius

AU - Nistor, Victor

N1 - Funding 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/.

PY - 2007/12

Y1 - 2007/12

N2 - 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."

AB - 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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U2 - 10.1007/s10587-007-0118-9

DO - 10.1007/s10587-007-0118-9

M3 - Article

AN - SCOPUS:36448967472

SN - 0011-4642

VL - 57

SP - 1151

EP - 1197

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

IS - 4

ER -