Problem 8.20

Let be harmonic in some region and assume , . Define for the functions

We derived in 8.6.2 the monotonicity formula

(a) Prove that

(b) Show

(c) Define the frequency function

and derive Almgren’s monotonicity formula: .
(d) Demonstrate next that and consequently

for and . This is an estimate from below on how fast a nonconstant harmonic function must grow near a point where it vanishes.

Solution

(a)

Since for all , we have Adding from 1 to , integrating it on , noting that , we have

which is the second “=”.

Note that

and

which implies the first “=”.

(b)

From the definition of and monotonicity formula, it sufficient to prove

Note that denotes the derivative of with respect to the radial variable .

From (a), we have

where denotes outer normal vector, which completes the proof.

(c)

Clearly, utilizing (b) and (a), we have

(d)

From (a) and (c), it is easy to check . Furthermore, we have , which implies