Some extensions of the arithmetic-mean theory of Robin's integral equation for bodies with vertices (Q1111122)

The Robin's equation in \(E^ 2\) \(\sigma (A)=1/\pi \oint_{C}\sigma (A')\cos x/| r| ds'-1/2\pi (\partial U/\partial \nu)_ A\) is considered, where C is a convex closed contour with a finite number of vertices, A, A'\(\in C\), \(r=A'A\), x is the angle between r and the outward normal \(\nu\) to C at A. The existence and uniqueness of its solution is established by a kind of iteration relating to arithmetic-mean \(\Omega_ z\) of a function \(\phi\) (s) which is a linear operator defined by \(\Omega_ z\phi (s)=\phi (s)+a\) for \(s\in [z,c-z]\) and \(=a\) for \(s\in [0,z)\cup (c-z,c]\) where \(a=1/c(\int^{z}_{0}+\int^{c}_{c-z})\phi (s)ds\) (c is the arc-length of C). The results are also extended in \(E^ 3\). This work is a continuation of an earlier paper of the author [ibid. 3, 1-19 (1981; Zbl 0462.45004)].