On the derivatives of harmonic functions on the boundary
Oliver Dimon Kellogg · Transactions of the American Mathematical Society · 1931
Let U be harmonic in a closed region R , whose boundary contains a regular surface element E , with a representation z = ϕ ( x , y ) z = \phi (x,y) . If E has bounded curvatures, and if ϕ ( x , y ) \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 .