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Boundary values of multidimensional integrals of potential type. (English. Russian original) Zbl 0853.31005

Russ. Acad. Sci., Dokl., Math. 47, No. 1, 104-107 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 328, No. 4, 431-433 (1993).
This note enunciates a generalization of the jump formula for the normal derivative of a single layer potential [cf. O. D. Kellogg, Foundations of potential theory (Springer-Verlag, Berlin) (1929; JFM 55.0282.01, Reprint Dover 1953; Springer 1967; Zbl 0152.31301)] and M. E. Taylor [Partial differential equations. II (Appl. Math. Sci. 116, Springer-Verlag, Berlin) (1996)]. The jump formula is given with respect to an oriented hypersurface \(G\) in \(\mathbb{R}^n\) for operators of the form \[ Au (z)= \int_G {{\Omega (z, y-z)} \over {|y-z |^{n-1}}} u(y) d_y G, \qquad z\in \mathbb{R}^n \setminus G, \] where \(\Omega (z, \zeta)\) is homogeneous of degree zero in \(\zeta\in \mathbb{R}^n\) and \(u\) is a function on \(G\).
Given a point \(z_0\in G\) which does not belong to the boundary of \(G\), the jump coefficients for \(Au\) at \(z_0\) are expressed only in terms of \(\Omega\) and the vector normal to \(G\) at the point \(z_0\); this expression should be compared with Theorem 4.28, p. 30 in [W. J. Trjitzinsky, Acta Math. 84, 1-128 (1950; Zbl 0037.19503)].

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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