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Chapter 6 Question 3

(1/1)

Andrew Hardy:
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:
Andrew see this:
http://forum.math.toronto.edu/index.php?topic=1066.0

Andrew Hardy:
JX, I dont follow Professor Ivrii's response. Do you mind elaborating?

Victor Ivrii:
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.