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1

HA8 / Solution of Question 4

« on: March 19, 2015, 09:13:07 PM »

Using Newton shell theorem (see [Section 6 of Lecture 25]) prove that if Earth was a homogeneous solid ball then the gravity pull inside of it would be proportional to the distance to the center.

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HA8 / Solution of Question 3

« on: March 19, 2015, 09:08:28 PM »

1.  Using the proof of maximum principle prove the maximum principle for
    subharmonic functions and minimum principle for superharmonic
    functions.

2.  Show that minimum principle for subharmonic functions and maximum
    principle for superharmonic functions do not hold (*Hint*: construct
    counterexamples with $f=f(r)$).

3.  Prove that if $u,v,w$ are respectively harmonic, subharmonic and
    superharmonic functions in the bounded domain $\Omega$,
    coinciding on its boundary ($u|_\Sigma=v|_\Sigma=w|_\Sigma$)
    then in $w\ge u \ge v$ in $\Omega$.

SO i just post what i write on paper

3

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?