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Problem set 10, PDE

Problem 1

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

  1. The 2-dimensional zonal harmonics are then given by
    $latex Z_0^{e^{i\theta}}(e^{i\varphi}) = \dfrac{1}{2\pi},\qquad Z_k^{i\theta}(e^{i\varphi}) = \dfrac{1}{\pi}\cos k(\varphi - \theta),\quad k>0.$

  2. Verify explicitly the properties of zonal harmonics seen in class.

  3. The polynomial $latex F_k$ given by $latex F_k(x\cdot y) = Z_k^x(y)$ is equal to $latex \dfrac{1}{\pi}T_k$ for $latex k>0$, where $latex T_k$ is the Chebyshev polynomial given by $latex R_k(\cos\theta) = \cos k\theta$.

Problem 2

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

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

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

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

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

Problem 3

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

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

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