A general theory of hypersurface potentials
The author investigates hypersurface potentials with general kernels. Particularly, he proves \[ \begin{aligned} \lim_{t\to +0} \int_{\mathbb{R}^{n-1}} \varphi (x) K(x, t)dx &= \gamma \varphi (0)+ \int_{\mathbb{R}^{n-1}} \varphi (x) K(x, 0) dx,\\ \lim _{\substack{ x\to x_0\\ x\in \nu_{x_0} }}\;\int_\Sigma \varphi (y) {\partial \over {\partial \nu_{x_0}}} h(x- y) d\sigma_y &= \gamma (x_0) \varphi (x_0)+ \int_\Sigma \varphi (y) {\partial \over {\partial \nu_{x_0}}} h(x_0- y) d\sigma_y, \end{aligned} \] 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.
hypersurface potentials with general kernels
scientific article; zbMATH DE number 878441
scientific article