On Mean Values of Solutions to Differential Equations

The methods developed in recent decades for identifying and obtaining mean values formulas for solutions of differential equations are described. The results obtained using these methods are presented. Mean values formulas for the Laplace–Beltrami operator in non-Euclidean spaces, for the wave equations in Euclidean space, on a sphere, in Lobachevskii space, two-point mean values formulas for elliptic equations are presented. The method of accompanying distributions is described, which makes it possible to obtain new mean values formulas. The mean value formula for a hyperbolic operator factorizing into linear factors of the first order is presented. A consequence of this formula is the ‘‘inclusions-exclusions’’ formula for polynomials.