Spherical mean

In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.

Definition

Consider an open set $$U in the Euclidean space $$mathbb R^n 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

: $$frac{1}{omega_{n-1}(r)}intlimits_{partial B(x, r)} ! u(y) ,dS(y)

where $$partial B(x, r) is the ("n"−1)-sphere forming the boundary of $$B(x, r) and $$omega_{n-1}(r) is the "surface area" of this $$(n-1)-sphere.

Equivalently, the spherical mean is given by

: $$frac{1}{omega_{n-1intlimits_{|y|=1} ! u(x+ry) ,dS(y)

where $$omega_{n-1} is the area of the $$(n-1)-sphere of radius 1.

The spherical mean is often denoted as

: $$intlimits_{partial B(x, r)}!!!!!!!!!!!-, u(y) ,dS(y).

Properties and uses

* From the continuity of $$u it follows that the function

::$$r o intlimits_{partial B(x, r)}!!!!!!!!!!!-, u(y) ,dS(y)

:is continuous, and its limit as $$r o 0 is $$u(x).

* Spherical means are used in finding the solution of the wave equation $$u_{tt}=c^2Delta u for $$t>0 with prescribed boundary conditions at $$t=0.

* If $$U is an open set in $$mathbb R^n and $$u is a "C"² function defined on $$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

::$$u(x)=intlimits_{partial B(x, r)}!!!!!!!!!!!-, u(y) ,dS(y).

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

References

*cite book
last = Evans
first = Lawrence C.
title = Partial differential equations
publisher = American Mathematical Society
date = 1998
pages =
isbn = 0821807722

*cite book
last = Sabelfeld
first = K. K.
coauthors = Shalimova, I. A.
title = Spherical means for PDEs
publisher = VSP
date = 1997
pages =
isbn = 9067642118

External links

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