Problem set 10, PDE

Problem 1

Identify $\R^2$ with $\C$.

1. The 2-dimensional zonal harmonics are then given by
$Z_0^{e^{i\theta}}(e^{i\varphi}) = \dfrac{1}{2\pi},\qquad Z_k^{i\theta}(e^{i\varphi}) = \dfrac{1}{\pi}\cos k(\varphi - \theta),\quad k>0.$
2. Verify explicitly the properties of zonal harmonics seen in class.
3. The polynomial $F_k$ given by $F_k(x\cdot y) = Z_k^x(y)$ is equal to $\dfrac{1}{\pi}T_k$ for $k>0$, where $T_k$ is the Chebyshev polynomial given by $R_k(\cos\theta) = \cos k\theta$.

Problem 2

The Gegenbauer polynomials $C_k^\lambda$ are given by the generating function

$\displaystyle \sum_{k=0}^\infty C_k^\lambda(t) r^k = (1 - 2rt + r^2)^{-\lambda}.$

The polynomials $F_k$, for $n\ge 3$, are given by

$\displaystyle F_k(t) = \frac{n-2+2k}{\omega_n (n-2)}C_k^{(n-2)/2}(t).$

(Hint: Apply the operator $D=1 + \dfrac{r}{\lambda}\dfrac{d}{dr}$ to the generating function above, and consider the expansion of the Poisson kernel in the $F_k$.)

Problem 3

Let $\Omega\subset\R^n$ be a bounded domain and f a bounded function in $C^\alpha(\Omega)$ for some $0 < \alpha < 1,$ i.e. there exists some $C>0$ such that

$|f(x) - f(y)| \le C |x-y|^\alpha, \qquad x,y\in\Omega.$

Then the Newtonian potential $w_f\in C^2(\Omega)$, $\Delta w_f = f$ in $\Omega$, and the second derivatives of $w_f$ are in $C^\alpha(\Omega)$.

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