A characterisation of harmonic functions by quadrature identities of annular domains and related results

Abstract

A new characterization of harmonic functions is obtained. It is based on quadrature identities involving mean values over annular domains and over concentric spheres lying within these domains or on their boundaries. The analogous result with a logarithmic weight in the volume means is conjectured. The similar characterization is derived for real-valued solutions of the modified Helmholtz equation (panharmonic functions), whereas for solutions of the Helmholtz equation (metaharmonic functions) the analogous result is just outlined. Weighted mean value identities are also obtained for solutions of these equations in two-dimensions. The dependence of coefficients in these identities on a logarithmic weight is analysed and characterizations of these functions are conjectured.