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The double layer potential operator over polyhedral domains. I: Solvability in weighted Sobolev spaces. (English) Zbl 0749.31002

Let \(\Omega\) be a simply connected bounded domain in \(\mathbb{R}^ 3\), then the harmonic double layer potential operator on \(\partial\Omega\) is defined by \[ Ku(x)=\int_{\partial\Omega}(x-y)\cdot n_ y u(y) d\sigma(y) / 2\pi| x-y|^ 3,\quad x\in\partial\Omega, \] where \(d\sigma\) is the surface measure on \(\partial\Omega\), and \(n\) is the outward pointing normal vector to \(\partial\Omega\). The author considers the integral equation \((\lambda-K)u=f\), when \(\Omega\) is a bounded polyhedral domain in \(\mathbb{R}\) and \(\lambda\) (\(|\lambda|\leq 1\)) is a complex constant. In particular by applying Mellin transform techniques directly to the integral equation he studies the mapping properties and proves the invertibility of \(\lambda-K\) in weighted Sobolev spaces. The paper also contains some results on properties of the Mellin transform itself.

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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