LAUSR

Mean value properties of harmonic functions (solutions of the Laplace equation; see [1], p. 25, about the origin of the term ‘harmonic’) are well known as well as various versions of assertions converse to these properties; see the survey articles [2] and [3]. On the other hand, analogous properties of solutions to several other simple partial differential equations are studied less thoroughly and, it may be said, fragmentary. Only recently the mean value property for balls was obtained (see [4]) for solutions to the m-dimensional Helmholtz equation: ∇u+ λu = 0, λ ∈ R \ {0}. (1.1)