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 \(\PageIndex{1}\): 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 \(z_0 = x_0 + iy_0\) then

\[u(x_0, y_0) = \dfrac{1}{2\pi} \int_{0}^{2\pi} u(z_0 + re^{i \theta})\ d\theta \nonumber \]

Proof

Let \(f = u + iv\) be an analytic function with \(u\) as its real part. The mean value property for \(f\) says

\[\begin{array} {rcl} {u(x_0, y_0) + iv(x_0, y_0) = f(z_0)} & = & {\dfrac{1}{2\pi} \int_{0}^{2\pi} f(z_0 + re^{i \theta})\ d\theta} \\ {} & = & {\dfrac{1}{2\pi} \int_{0}^{2\pi} u(z_0 + re^{i \theta} + iv (z_0 + re^{i \theta}) \ d\theta} \end{array} \nonumber \]

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

Theorem \(\PageIndex{2}\): Maximum Principle

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

Suppose \(z_0\) is in \(A\). If \(u\) has a relative maximum or minimum at \(z_0\) then \(u\) is constant on a disk centered at \(z_0\).
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.