Toronto Math Forum

APM346-2015S => APM346--Home Assignments => HA8 => Topic started by: Chaojie Li on March 19, 2015, 09:08:28 PM

Title: Solution of Question 3
Post by: Chaojie Li 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

Title: Re: Solution of Question 3
Post by: Chaojie Li on March 19, 2015, 09:10:17 PM

This is my solution of Q3 b

Title: Re: Solution of Question 3
Post by: Victor Ivrii on March 20, 2015, 08:12:54 AM

c Is easy: if $u$ is subharmonic (or superharmonic)  and $v$ is harmonic with the same boundary value $(u-v)|_\Sigma =0$ then $u-v$ is subharmonic (or superharmonic) and then by maximum (or minimum) principle $(u-v)\le \max _\Sigma (u-v) =0$ (or $(u-v)\ge \min _\Sigma (u-v) =0$ and thus $u\le v$ ($u\ge v$ respectively)