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).

Post Tagged with 

Comments & Responses

Leave a Reply