Suppose \(\Omega\subseteq \mathbb{R}^d\) is a open. A function \(u\in C^2(\Omega)\) is harmonic if the Laplacian of \(u\) vanishes, i.e. \(\Delta u=0\).
Remarks
- A harmonic function satisfies the Laplaces equation .
New harmonic functions from old Link to heading
Regularity Link to heading
- Harmonic functions are smooth (Weyl’s theorem ).