Mean value properties of solutions to the Helmholtz and modified Helmholtz equations

Abstract: Mean value properties of solutions to the $m$-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given; it involves the Euler--Poisson--Darboux equation. Despite the similar form of these properties for both equations, their consequences distinguish essentially. The restricted mean value property for harmonic functions is amended so that a function, satisfying it in a bounded domain of a special class, solves the modified Helmholtz equation in this domain.