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Chapter 6 Question 3

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**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.

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