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Single and double layer potentials on domains with conical points. I: Straight cones. (English) Zbl 1267.47077

The paper is devoted to the analysis of single and double layer potentials in straight cones. Via the Mellin transform, the authors start by presenting a reduction process for the study of layer potentials on cones to the study of a family of layer potential operators on the basis of the cone. Thus, the invertibility of the consequent layer potential operators on cones is established by reducing to the pointwise invertibility of a family of operators on the boundary of the basis of the cone. Additionally, such invertibility turns out to hold for a range of suitable weighted Sobolev spaces. Moreover, the structure of the layer potentials in terms of operator valued Mellin convolution operators is also determined.

MSC:

47G40 Potential operators
45E05 Integral equations with kernels of Cauchy type
47A10 Spectrum, resolvent
35J25 Boundary value problems for second-order elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
58J32 Boundary value problems on manifolds
31C12 Potential theory on Riemannian manifolds and other spaces
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