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Mathematics LibreTexts

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.

Theorem 6.5.1: Mean Value Property

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)=12π2π0u(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)=12π2π0f(z0+reiθ) dθ=12π2π0u(z0+reiθ+iv(z0+reiθ) dθ

Looking at the real parts of this equation proves the theorem.

Theorem 6.5.2: Maximum Principle

Suppose u(x,y) is harmonic on a open region A.

  1. 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.
  2. 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.

Note

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.


This page titled 6.5: Maximum Principle and Mean Value Property is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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