On the derivatives of harmonic functions on the boundary

Abstract:

Let U be harmonic in a closed region R, whose boundary contains a regular surface element E, with a representation $z = \phi (x,y)$. If E has bounded curvatures, and if $\phi (x,y)$ and the boundary values of U on E have continuous derivatives of order n which satisfy a Dini condition, then the partial derivatives of U of order n exist, as limits, on E, and are continuous in R at any interior point of E. Hölder conditions on the boundary values of U, or on their derivatives of order n, imply Hölder conditions on U, or the corresponding derivatives, in R, in the neighborhood of the interior points of E.