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Zbl 0855.31004
Cialdea, Alberto
A general theory of hypersurface potentials. (English)
[J] Ann. Mat. Pura Appl., IV. Ser. 168, 37-61 (1995). ISSN 0373-3114; ISSN 1618-1891/e

The author investigates hypersurface potentials with general kernels. Particularly, he proves $$\align \lim_{t\to +0} \int_{\bbfR^{n-1}} \varphi (x) K(x, t)dx &= \gamma \varphi (0)+ \int_{\bbfR^{n-1}} \varphi (x) K(x, 0) dx,\\ \lim \Sb x\to x_0\\ x\in \nu_{x_0} \endSb\ \int_\Sigma \varphi (y) {\partial \over {\partial \nu_{x_0}}} h(x- y) d\sigma_y &= \gamma (x_0) \varphi (x_0)+ \int_\Sigma \varphi (y) {\partial \over {\partial \nu_{x_0}}} h(x_0- y) d\sigma_y, \endalign$$ where $\varphi$ belongs to the Hölder's class, Lyapunov's manifold $\Sigma$ is the boundary of a bounded domain, the values $\gamma$ and $\gamma (x_0)$ are defined by the kernels, $\nu_{x_0}$ is the inner normal at the point $x_0\in \Sigma$, $K(x, t)$ and $h(x)$ are kernels of special classes. Potentials of functions of the class $L_p$ and potentials of measures are considered as well.
[A.F.Grishin (Khar'kov)]

MSC 2000:: *31B25 Boundary behavior of harmonic functions (higher-dim.)
35B15 Almost periodic solutions of PDE

Keywords: singular integral; homogeneous kernel; Lyapunov's manifold; hypersurface potentials with general kernels; potentials of measures

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Zentralblatt MATH Berlin [Germany]