Harmonic function - Knowlpedia

Definition

Let $U\subseteq\mathbb R^n$ be open. A $C^2$ function $u:U\to\mathbb R$ is harmonic if

\Delta u=\sum_{j=1}^n\partial_{x_j}^2u=0

on $U$ .

Mean-value characterization

A continuous function is harmonic exactly when every closed ball $\overline{B_r(x)}\subset U$ satisfies

u(x)=\frac1{|B_r|}\int_{B_r(x)}u(y)\,dy.

Equivalently, one may use averages over boundary spheres.

Because the mean-value equality gives both sub-mean inequalities, $u$ is harmonic exactly when both $u$ and $-u$ are subharmonic.

Complex-analytic source

The real and imaginary parts of a holomorphic function are harmonic. Conversely, on a simply connected planar domain, every real-valued harmonic function is locally the real part of a holomorphic function.

Consequences

Harmonic functions are smooth and real analytic. The harmonic maximum principle gives uniqueness for the Dirichlet problem and controls Poisson extensions.

References

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