Let \(u \in C^0(\Omega)\) and \(\Omega\subseteq \mathbb{R}^d\) is an open set.
We call \(u\) is subharmonic if for every \(B_r(x)\subseteq \Omega\)
\[u(x) \leq \int_{\partial B(x,r)} u \, d\sigma.\]The function \(u\) is superharmonic if it pointwise larger then the mean-value and harmonic if it is both subharmonic and superharmonic.
Remarks
- If \(u \in C^2\), subharmonic definitions are equivalent due to (0x68032fd6) .
- If \(u_1\) and \(u_2\) are subharmonic then \(\max(u_1, u_2)\) is subharmonic (apply the definition to see this).