Boundary value problems and layer potentials on manifolds with cylindrical ends

Abstract

Abstract. We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the noncompactness of the boundary, which prevents us from using the standard characterization of Fredholm and compact (pseudo-)differential operators between Sobolev spaces. Our approach, which involves the study of layer potentials depending on a parameter on compact manifolds as an intermediate step, yields the invertibility of the relevant boundary integral operators in the global, noncompact setting, which is rather unexpected. As an application, we prove the well-posedness of the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichletto-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are "almost translation invariant at infinity," a calculus that is closely related to Melrose's b-calculus [29,28], which we study in this paper. The proof of the convergence of the layer potentials and of the existence of the Dirichlet-to-Neumann map are based on a good understanding of resolvents of elliptic operators that are translation invariant at infinity.

Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

“…These estimates follow from the results in [35], see Section 2. The same estimates can be used for domains with cusps.…”

“…The proofs use some estimates on the Dirichlet problem in Sobolev spaces with weights [17,25,35]. These estimates follow from the results in [35], see Section 2.…”

Pseudodifferential operators on manifolds with a Lie structure at infinity

“…An earlier important motivation for the construction of these algebras was the method of layer potentials for boundary value problems and questions in analysis on locally symmetric spaces. See for example [4], [5], [6], [8], [18], [19], [24], [32].…”

Integral Equations of Diffraction Problems with Unbounded Smooth Obstacles

“…The diffraction (transmission) problems for Helmholtz and Maxwell equations on smooth bounded obstacles are classical problems of Mathematical Physics (see for instance [5,[16][17][18]27,37], and references cited there). There is an extensive literature devoted to diffraction problems on specific unbounded obstacles (see for instance [6,8,9,11,[13][14][15]20,30,[32][33][34]36,39,58], and references given there).…”