Harmonic Mean Value Property

Worked Example

Let's consider the nonharmonic quadratic function u(x, y) = x² + y².

Its Laplacian is Δu = ∂²u/∂x² + ∂²u/∂y² = 2 + 2 = 4. Since the Laplacian is not zero, the function is not harmonic.

Let's evaluate its circle mean at the center (1, 0) with a radius of 1. Using polar coordinates centered at (1, 0), x = 1 + r cos(θ) and y = r sin(θ). For r = 1, x = 1 + cos(θ) and y = sin(θ).

1. Substitute x and y: u(1 + cos(θ), sin(θ)) = (1 + cos(θ))² + sin²(θ).
2. Expand: 1 + 2cos(θ) + cos²(θ) + sin²(θ).
3. Simplify using cos²(θ) + sin²(θ) = 1: 2 + 2cos(θ).
4. Integrate from 0 to 2π and divide by 2π to find the mean: The integral of cos(θ) is 0, so the average is simply 2.

The value at the center is u(1, 0) = 1² + 0² = 1. The expected difference between the circle mean (2) and the center value (1) is 1. Since they are not equal, this function does not satisfy the mean value property.

Experiment

Use the interactive module to explore this property yourself.

1. Select Field: Nonharmonic quadratic.
2. Set the Center x to 1 and Center y to 0.
3. Set the Radius to 1.
4. Observe that the Center value is 1, the Circle mean is 2, and their Mean difference is 1, with a non-zero Laplacian of 4.

Limitations

Note that when the radius is 0, the circle degenerates to a point. In this trivial case, the "circle mean" is simply the value at the center point itself. Therefore, the mean value property holds trivially for any continuous function at r=0, regardless of whether it is harmonic or not. A single agreement at a zero radius is not a valid test of harmonicity. The property must hold for all circles within the domain.

References

• Axler, S., Bourdon, P., & Ramey, W. (2001). Harmonic Function Theory. https://www.axler.net/HFT.pdf

Harmonic Function Theory is a comprehensive text detailing the properties of harmonic functions, including proofs and generalizations of the mean value property.