Existence and Hölder continuity of derivatives of single layer \(\Phi\)- potentials (Q792509)

In the n dimensional space \({\mathbb{R}}^ n\), let S be a compact portion of a k dimensional lipschitzian surface, \(\Phi\) a continuously differentiable function in \({\mathbb{R}}^ n\backslash \{0\}\), \(\sigma\) a measure on S. The single layer \(\Phi\) potential of \(\sigma\) is defined by \(V(x)=\int \Phi(x-y)d\sigma(y).\) V is continuously differentiable outside of S but in general not on S. The following problems are investigated: (i) Existence of non tangential limits of derivatives of V; (ii) Hölder continuity of derivatives of V on non tangential sets; (iii) Existence of derivatives of V on S; (iv) Hölder continuity of derivatives of V on S. In the classical case of the newtonian kernel \(\Phi\) and when \(\sigma\) is of the form fds many results of this kind were already known. These results are extended to the general case under suitable restrictions on \(\Phi\) and \(\sigma\).