Author Topic: Chapter 6 Question 3  (Read 871 times)

Andrew Hardy

  • Full Member
  • ***
  • Posts: 34
  • Karma: 10
    • View Profile
Chapter 6 Question 3
« on: March 11, 2018, 06:37:39 PM »
The question asks about a 2D Laplace Dirichlet Problem on a circle.  From Strauss Chapter 6.3, this question derives Poisson's formula of the form $$ u(r, \theta ) = (a^2-r^2) \int_{0}^{2\pi} \frac {h(\phi)} {a^2-2ar\cos(\theta - \phi) +r^2} \frac {d\phi}{2\pi} $$

Strauss mentions that we require finite solutions. This is analogous to max $ u < \infty $ but then how do we arrive at a solution to Part 1 of this question? What am I missing?

Jingxuan Zhang

  • Elder Member
  • *****
  • Posts: 106
  • Karma: 20
    • View Profile
Re: Chapter 6 Question 3
« Reply #1 on: March 11, 2018, 07:50:48 PM »

Andrew Hardy

  • Full Member
  • ***
  • Posts: 34
  • Karma: 10
    • View Profile
Re: Chapter 6 Question 3
« Reply #2 on: March 11, 2018, 09:40:52 PM »
JX, I dont follow Professor Ivrii's response. Do you mind elaborating?

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2452
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Chapter 6 Question 3
« Reply #3 on: March 12, 2018, 03:15:21 AM »
equation $\Delta u=0$ should be satisfied in the disk: $\{r<a\}$ rather than in the disk without its center $\{0<r<a\}$. There is a maximum principle for Laplace, which means that $u$ must be bounded at $0$,
\begin{equation}
\max |u|<\infty.\tag{*}
\end{equation}
This eliminates $r^\nu$ with $\nu<0$ and $\ln(r)$.

Actually there is much stronger statement: $u$ is $C^\infty$ and even real-analytic (could be decomposed into Taylor series by $x,y$ of non-zero radius of convergence). This eliminates $r^\nu$ with positive but non-integer $\nu$.

However, if we consider a sector $\{r<a, \theta_1<\theta <\theta_2\}$ we need to request (*) explicitly. Ditto in unbounded domains.