For questions regarding harmonic functions.
The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as harmonic functions.
Harmonic functions appear most naturally in complex analysis and the Laplace equation is the most important PDE to study.
The Cauchy-Riemann equation together with the conjugated Cauchy-Riemann equation shows that the sum of an analytic function and an anti-analytic function is harmonic and in fact every complex harmonic function can be written as such. In particular the real/imaginary part of an analytic function is harmonic.
In any dimension, harmonic functions satisfy the following properties
Mean value property,
Maximum principle,
Harnack inequality,
Liouville's theorem.
Harmonic functions satisfy the regularity theorem for harmonic functions, which states that harmonic functions are infinitely differentiable (follows from Laplace's equation).
Please use instead the tag Laplacian if your question concerns the Laplacian as an operator.
