Maximum principle for harmonic functions

Statement

Let $U\subseteq\mathbb R^n$ be connected and let $u:U\to\mathbb R$ be harmonic. If $u$ attains a maximum or a minimum at an interior point, then $u$ is constant on $U$ .

Boundary form

If $U$ is bounded, $u$ is continuous on $\overline U$ , and harmonic on $U$ , then

\max_{\overline U}u=\max_{\partial U}u,\qquad \min_{\overline U}u=\min_{\partial U}u.

Proof mechanism

The mean-value property says that an interior value is the average of nearby values. If it is already maximal, every nearby value must be equal to it. Connectedness propagates this local constancy throughout the domain.

Use

Applying the boundary form to the difference of two solutions proves uniqueness for the Dirichlet problem. It also bounds derivatives or auxiliary harmonic functions by their boundary values.

References

Lawrence C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010. Publisher record.