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« **on:** March 19, 2015, 09:06:43 PM »
Using the proof of mean value theorem prove that if $\Delta u\ge 0$ in $B(y,r)$ then

1. $u(y)$ does not exceed the mean value of $u$ over the sphere $S(y,r)$ bounding this ball:

\begin{equation}

u(y)\le \frac{1}{\sigma_n r^{n-1}}\int_{S(y,r)} u\,dS.

\label{equ-H8.2}

\end{equation}

2. $u(y)$ does not exceed the mean value of $u$ over this ball $B(y,r)$:

\begin{equation}

u(y)\le \frac{1}{\omega_n r^n}\int_{B(y,r)} u\,dV.

\label{equ-H8.3} \end{equation}

3. Formulate similar statements for functions satisfying $\Delta u\le 0$ (in the next problem we refer to them as (a)' and (b)').

**Definition**

a. Functions having property (a) (or (b) does not matter) of the previous problem are called *subharmoni*c

b. Functions having property (a)' (or (b)' does not matter) are called *superharmonic*.

Well, so 9 pm is good to post solution right?