## Monotone formula of harmonic functions

Suppose , and is a harmonic function is

For , we can define

,

In general is called the frequency of . If is a homogeneous harmonic polynomial of degree , then

Suppose , where is a spherical harmonic polynomial functions on , then ,

.

is a non-decreasing function in

Since , then . From the Green formula

We will prove that for .

By the Green formula

where is the j-th component of outer normal vector of .

Again apply green formula

Plug (3) into (2), we get

Sum over j and i, noticing is haromic,

So

Plugging (1) and (4) to , we get

since the cauchy inequality.

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