Let \(\Omega\subseteq \mathbb{R}^d\) be open and \(u\in C^2(\Omega)\). If \(u\) is subharmonic the mean value property holds, that is for suitable \(x\in \Omega\) and \(r>0\) the following inequalities holds
\[ u(x)\le \frac{1}{\lvert \partial B_r(x)\rvert} \int_{\partial B_r(x)} u \]and
\[ u(x)\le \frac{1}{\lvert B_r(x)\rvert} \int_{ B_r(x)} u. \]If \(u\) is harmonic the inequalities are identities.
Proof (sketch) Link to heading
Let \(x\in \Omega\) and \(r>0\) such that \(B_r(x)\subseteq \Omega\). Consider the function
\[\phi(\rho)=\frac{1}{\lvert \partial B_\rho(x)\rvert} \int_{\partial B_\rho(x)} u .\]This function is continuous and differentiable. Using the divergence theorem we obtain that \(\phi\) is an monoton increasing function. This implies the first inequality.
By using polar coordinates we derive the second inequality.
Remarks