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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