Ricardo A. Sáenz

Problem 1 Identify $latex \R^2$ with $latex \C$. The 2-dimensional zonal harmonics are then given by $latex Z_0^{e^{i\theta}}(e^{i\varphi}) = \dfrac{1}{2\pi},\qquad Z_k^{i\theta}(e^{i\varphi}) = \dfrac{1}{\pi}\cos k(\varphi - \theta),\quad k>0.$ Verify explicitly the properties of zonal harmonics seen in class. 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 expansio...

Due November 2 Problem 1 If $latex A\subset X$ is connected, then so is its closure $latex \bar A$. If A is connected and $latex A \subset B \subset \bar A$, then so is B . If $latex A\subset \R^n$ is convex, then $latex \bar A$ is convex. Problem 2 If $latex f:[0,1]\to[0,1]$ is continuous, then there exists $latex c\in[0,1]$ such that $latex f(c) = c$. If $latex f:\mathbb S^1\to\R$ is continuous, then there exists $latex \xi\in\mathbb S^1$ such that $latex f(\xi) = f(-\xi)$. Problem 3 State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cap B$ is path connected. If $latex A, B\subset\R^n$ are convex, then $latex A\cap B$ is convex. Problem 4 Let $latex A\cap B\not=\emptyset$ in some metric space. State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cup B$ is path connected. If $latex A,B\subset\R^n$ are convex, then $latex A\cup B$ is convex.  

Problem 1 Let $latex u\in C^2$ be a solution of $latex \begin{cases} \Delta u = 0 & \text{in } \R^n\setminus B_R\\ u=0 & \text{on } \partial B_R.\end{cases}$ Then $latex u\equiv 0$ if $latex \displaystyle \lim_{|x|\to\infty}\frac{u(x)}{\log |x|} = 0,\quad n=2;\qquad \lim_{|x|\to\infty} u(x) = 0,\quad n=3.$ Problem 2 Let $latex \Omega\subset\R^n$ be a bounded $latex C^1$-domain that satisfies the inner sphere condition, and $latex f\in C(\Omega)$ bounded. If $latex u\in C^2(\Omega)\cap C^1(\bar \Omega)$ is a solution of $latex \begin{cases} \Delta u = f & \text{in } \Omega\\ u=0 & \text{on } \partial\Omega,\end{cases}$ then $latex \displaystyle \sup_{\partial\Omega}\Big|\frac{\partial u}{\partial\nu}\Big| \le C \sup_\Omega|f|,$ where $latex C>0$ depends only on $latex n,\Omega$. Problem 3 A polynomial $latex p\in\mathscr P_k$ if and only if $latex \sum x_j \partial_j p = kp.$ Let $latex p\in\mathscr P_k$, and consider its orthogonal projection $latex |x|^2q$ onto ...

Due October 19 Problem 1 Let  X be a complete metric space. The countable intersection of dense $latex G_\delta$ sets in  X is a dense $latex G_\delta$ set in  X . If a set and its complement are dense subsets of  X , at most one can be $latex G_\delta$. If X doesn't have isolated points, a countable dense subset of  X cannot be $latex G_\delta$. Problem 2 Let  X be a complete metric space. If $latex O\subset X$ is open, then  O is a metric subspace of the second category. If $latex \{F_n\}$ are closed subsets of  X with $latex X = \bigcup_n F_n$, then $latex \bigcup_n \text{int}(F_n)$ is dense in  X . Problem 3 Let  X be a complete metric space. The set of points of discontinuity of the characteristic function $latex \chi_O$ of an open set  O is a nowhere dense subset of  X . Given open sets $latex \{O_n\}$, there exists $latex x\in X$ such that $latex \chi_{O_n}$ is continuous at  x for each  n . Problem 4 Let $latex d(x) = d(x,\Z)$ denote the dista...

Problem 1 Find the Green's function for the Laplace equation in the upper half-space $latex \R^{n+1}_+ = \{x_{n+1}>0\}$ and then derive a formal integral representation for a solution of the Dirichlet problem $latex \begin{cases} \Delta u = 0 &\text{in } \R^{n+1}_+\\ u = \phi & \text{on }\{x_{n+1}=0\}.\end{cases}$ Problem 2 Use the Poisson integration formula to prove the Harnack inequality $latex \displaystyle\Big(\frac{R}{R+r}\Big)^{n-2} \frac{R-r}{R+r} u(x_0) \le u(x) \le \Big(\frac{R}{R-r}\Big)^{n-2} \frac{R+r}{R-r}u(x_0),$ for a harmonic function  u in $latex B_R(x_0)\subset\R^n$, where $latex r=|x-x_0|<R$. Problem 3 Use Problem 2 to give another proof of Liuville's theorem: a harmonic function in $latex \R^n$ bounded from above, or from below, is constant.

Due October 12 Problem 1 Let  X be a metric space. We say that a collection $latex \{F_\alpha\}$ of subsets of  X has the finite intersection property (FIP) if any finite subcollection $latex \{F_{\alpha_1}, F_{\alpha_2},\ldots,F_{\alpha_k}\}$ of them has nonempty intersection: $latex F_{\alpha_1}\cap F_{\alpha_2}\cap\ldots\cap F_{\alpha_k}\not=\emptyset.$ X is compact if and only if the intersection of any collection of closed sets that has the FIP is nonempty. Give an example of a decreasing sequence of nonempty closed sets in a metric space with empty intersection. Problem 2 The closed ball $latex \bar B_r(x_0) = \{ x\in X: d(x,x_0)\le r\}$ is a closed set in  X . Is $latex \bar B_r(x_0) = \overline{B_r(x_0)}$ in every metric space? Problem 3 If $latex f:X\to Y$ is continuous, its  graph  $latex G=\{(x,f(x)): x\in X\}$ is closed in $latex X\times Y$. Problem 4 If X is a metric space, then $latex \diam \bar A = \diam A$ for any $latex A\subset X$.