Spherical mean

Definition

Consider an open set U in the Euclidean space Rⁿ and a continuous function u defined on U with real or complex values. Let x be a point in U and r > 0 be such that the closed ball B(x, r) of center x and radius r is contained in U. The spherical mean over the sphere of radius r centered at x is defined as

1 ω n − 1 ( r ) ∫ ∂ B ( x , r ) u ( y ) d S ( y )

where ∂B(x, r) is the (n−1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and ω_n−1(r) is the "surface area" of this (n−1)-sphere.

Equivalently, the spherical mean is given by

1 ω n − 1 ∫ ∥ y ∥ = 1 u ( x + r y ) d S ( y )

where ω_n−1 is the area of the (n−1)-sphere of radius 1.

The spherical mean is often denoted as

∫ ∂ B ( x , r ) − u ( y ) d S ( y ) .

The spherical mean is also defined for Riemannian manifolds in a natural manner.

Properties and uses

From the continuity of u it follows that the function

is continuous, and its limit as r → 0 is u ( x ) .

Spherical means are used in finding the solution of the wave equation u t t = c 2 Δ u for t > 0 with prescribed boundary conditions at t = 0.

If U is an open set in R n and u is a C² function defined on U , then u is harmonic if and only if for all x in U and all r > 0 such that the closed ball B ( x , r ) is contained in U one has

This result can be used to prove the maximum principle for harmonic functions.