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HA8 / Re: Solution of Question 2
« on: March 19, 2015, 09:07:29 PM »
part c
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18
HA8 / Solution of Question 2
« 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 subharmonic
b. Functions having property (a)' (or (b)' does not matter) are called superharmonic.
Well, so 9 pm is good to post solution right?
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 subharmonic
b. Functions having property (a)' (or (b)' does not matter) are called superharmonic.
Well, so 9 pm is good to post solution right?
Pages: 1 [2]