Stiffness of Harmonic Functions

Abstract

Harmonic functions cannot change rapidly. For example, if K is a compact subset of a Riemann surface R and {u} a family of harmonic functions u on R of nonconstant sign on K, then it is known that there exists a constant $q \in (0,1)$ independent of u such that ${\max _K}|u| \leqslant q{\sup _R}|u|$ for all $u \in \{ u\}$. In the present note we shall show that relations expressing such “stiffness” of harmonic functions can also be given for the Dirichlet norm and for the partial derivative with respect to the Green’s function.