6.5: Maximum Principle and Mean Value Property
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These are similar to the corresponding properties of analytic functions. Indeed, we deduce them from those corresponding properties.
If u is a harmonic function then u satisfies the mean value property. That is, suppose u is harmonic on and inside a circle of radius r centered at z0=x0+iy0 then
u(x0,y0)=12π∫2π0u(z0+reiθ) dθ
- Proof
-
Let f=u+iv be an analytic function with u as its real part. The mean value property for f says
u(x0,y0)+iv(x0,y0)=f(z0)=12π∫2π0f(z0+reiθ) dθ=12π∫2π0u(z0+reiθ+iv(z0+reiθ) dθ
Looking at the real parts of this equation proves the theorem.
Suppose u(x,y) is harmonic on a open region A.
- Suppose z0 is in A. If u has a relative maximum or minimum at z0 then u is constant on a disk centered at z0.
- If A is bounded and connected and u is continuous on the boundary of A then the absolute maximum and absolute minimum of u occur on the boundary.
- Proof
-
The proof for maxima is identical to the one for the maximum modulus principle. The proof for minima comes by looking at the maxima of −u.
For analytic functions we only talked about maxima because we had to use the modulus in order to have real values. Since |−f|=|f| we couldn’t use the trick of turning minima into maxima by using a minus sign.