Ricardo A. Sáenz

Problem 1 If $latex u$ is harmonic in the connected domain $latex \Omega$ and is not constant, then $latex u(\Omega)$ is open in $latex \R$. Suppose $latex \Omega$ is bounded and that its boundary $latex \partial\Omega$ is connected. If $latex u$ is harmonic in $latex \Omega$, then $latex u(\Omega)\subset u(\partial\Omega)$. Problem 2 A radial harmonic function on $latex \mathbb B$ is constant. A positive harmonic function on $latex \R^d$ is constant. Problem 3 Suppose $latex u(x)$ is harmonic in some domain in $latex \R^n$. Then $latex v(x) = |x|^{2-n} u\Big(\dfrac{x}{|x|^2}\Big)$ is also harmonic in a suitable domain. Problem 4 For $latex n=2$, find the Green's function for the Laplace operator on the first quadrant.  

Due October 5 Problem 1 Let $latex L^1([a,b])$ be the space of real valued continuous functions with the $latex d_1$ metric. The polynomials are dense in $latex L^1([a,b])$. Is $latex L^1([a,b])$ separable? Problem 2 Let $latex f:[a,b]\to\R$ be a continuous function such that $latex \displaystyle \int_a^b f(x) x^n dx = 0$ for all $latex n=0,1,2,\ldots$. Then $latex f(x)=0$ for all $latex x\in[a,b].$ Problem 3 If $latex X,Y$ are compact metric spaces, then the tensor space $latex \displaystyle C(X)\otimes C(Y) = \{ (x,y)\mapsto \sum_{k=1}^n f_k(x)g_k(y): f_k\in C(X), g_k\in C(Y), n\ge1\}$ is dense in $latex C(X\times Y)$. Note:  The product space $latex X\times Y$ has the metric $latex d_{X\times Y} \big( (x_1,y_1), (x_2,y_2) \big) = d_X(x_1,x_2) + d_Y(y_1,y_2).$ Problem 4 State whether the following are true: $latex \overline{A\cup B} \subset \overline{A}\cup \overline{B}$; $latex \overline{A\cup B} \supset \overline{A}\cup \overline{B}$; $latex \overline{A\cap B} \subset \ov...

Problem 1 Let $latex u_0\in L^2(0,\pi)$ be the solution of the heat equation seen in class. Then, for all $latex i,j\in\N$, $latex \displaystyle \sup_{x\in[0,\pi]}|\partial_x^i \partial_t^j u(x,t)|\to 0$ as $latex t\to\infty$. Problem 2 For any $latex u_0\in L^2(0,\pi), f\in L^2((0,\pi)\times(0,\infty))$, find a formal expression of a solution of the problem $latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = f & \text{in }(0,\pi)\times(0,\infty)\\ u(x,0) = u_0(x) & x\in (0,\pi)\\ u(0,t)=u(\pi,t)=0 & t\in (0,\infty). \end{cases}$ Problem 3 The wave operator $latex \partial_t^2 - \partial_x^2$ on $latex \R^2$ commutes with the Lorentz transformations (hyperbolic rotations) $latex \displaystyle T_\theta = \begin{pmatrix}\cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta\end{pmatrix},$ for $latex \theta\in\R$.

Due September 28 Problem 1 Let $latex f_n:[a,b]\to\R$ a monotone sequence of continuous functions which converges pointwise to the continuous function $latex f:[a,b]\to\R$. Then $latex f_n\rightrightarrows f$ on $latex [a,b]$. Problem 2 Let $latex K:[0,1]\times[0,1]\to[0,1]$ be a continuous function and define the operator $latex \mathscr L:C([0,1])\to C([0,1])$ by $latex \displaystyle \mathscr Lf(x) = \int_0^1 K(x,y) f(y) dy$. Then, the image of the closed ball $latex \bar B_1(0)$ in $latex C([0,1])$ under $latex \mathscr L$ is compact. Such operator is called a  compact operator . Let $latex w:[0,1]\to\R$ be continuous. Then the operator $latex \displaystyle \mathscr Lf(x) = \int_0^x f(t) w(t)dt$ is compact. Problem 3 Let $latex F_1\subset F_2\subset \ldots$ an increasing sequence of compact subsets of a metric space, and let $latex K \subset\subset \bigcup F_n$. Then there exists  p  such that $latex K\subset F_p$. Problem 4 Let $latex \Omega\subset\R^m$ be open and $latex f_n:\...

Problem 1 Classify the following second-order PDE. $latex \displaystyle \sum_{i=1}^n \partial^2_{x_i x_i}u + \sum_{1\le i<j\le n} \partial^2_{x_i x_j}u = 0$ $latex \displaystyle \sum_{1\le i<j \le n} \partial^2_{x_i x_j}u = 0$ Problem 2 Use energy methods to discuss the uniqueness of the problem $latex \begin{cases} \Delta u - u^3 = f & \text{in }\Omega\\u = \phi & \text{on }\partial\Omega.\end{cases}$ Problem 3 Let $latex \Omega$ be a bounded $latex C^1$-domain in $latex \R^n$ and  u be a $latex C^2$-function in $latex \bar\Omega\times[0,T]$ satisfying $latex \begin{cases} u_t - \Delta u = f & \text{in }\Omega\times(0,\infty)\\ u(\cdot, 0) = u_0 & \text{in }\Omega\\ u=0 & \text{on }\partial\Omega\times(0,\infty).\end{cases}$ Then $latex \displaystyle \sup_{0\le t \le T} \int_\Omega |\nabla u(\cdot,t)|^2 dx + \int_0^T\int_\Omega |\partial_t u|^2 dx dt \le C\Big( \int_\Omega |\nabla u_0|^2 dx + \int_0^T\int_\Omega |f|^2 dxdt \Big),$ where  C is a positive...

Due September 21 Problem 1 Let  X  be a compact space and $latex f:X\to Y$ a continuous bijection. Then $latex f^{-1}:Y\to X$ is continuous. Give an example of a continuous bijection $latex f:X\to Y$, for a noncompact X , whose inverse is not continuous. Problem 2 Let  X be a discrete metric space. Any function on  X is uniformly continuos. Is  X always compact? If not, give necessary and sufficient conditions for a discrete space to be compact. Problem 3 Let X be a  subspace of  Y ,  i.e. a subset of the metric space  Y with the same metric. We say  X is closed in  Y if  X contains all its limits: if $latex x_n\in X$ is a sequence that converges to $latex x_0\in Y$, then $latex x_0\in X$. If  Y is complete, then  X is complete if and only if  X is closed in  Y . Problem 4 Every bounded set in $latex \R^l$ is totally bounded. Every closed and bounded set in $latex \R^l$ is compact.

Due September 14 Problem 1 If the measurable $latex f_n\searrow f\ge 0$ with $latex \int f_1 < \infty$, then $latex \int f_n \to \int f.$ Explain the condition $latex \int f_1 < \infty$. Problem 2 There exists a positive continuous $latex f \in L^1(\R)$ such that $latex \limsup_{|x|\to\infty} f(x) = \infty.$ If $latex f\in L^1(\R)$ is uniformly continuous, then $latex \lim_{|x|\to\infty}f(x) = 0.$ Problem 3 If $latex f\in L^1(\R)$ and $latex F(x) = \int_{-\infty}^x f$. Then  F is uniformly continuous. Problem 4 Let $latex f:D\to\R$ be uniformly continuous, with $latex D\subset\R$. If $latex x_0$ is a limit point of D , then  f has limit at $latex x_0$. f has a continuous extension to $latex \bar D$, the closure of  D.

Problem 1 Let $latex a = (a_1, \ldots, a_n), b, f, u_0$ continuous functions with $latex |a|\le 1/\kappa$, and $latex u\in C^1(\bar\R^{n+1}_+)$ a solution to the IVP $latex \displaystyle \partial_t u + \sum_{i=1}^n a_i \partial_{x_i}u + bu = f \qquad \text{in  }\R^{n+1}_+,$ $latex u(x,0) = u_0(x) \qquad \text{on } \R^n.$ Then, for any $latex P = (X,T)\in\R^{n+1}_+$, $latex \displaystyle \sup_{C_\kappa}(P)|e^{-\alpha t}u| \le \sup_{\partial_{-}C_\kappa(P)}|u_0| + \frac{1}{\alpha + \inf_{C_\kappa(P)}b} \sup_{C_\kappa(P)}|e^{-\alpha t}f|,$ where $latex \alpha>0$ is a constant such that $latex \displaystyle \alpha + \inf_{C_\kappa(P)}b > 0.$ Problem 2 Let $latex a = (a_1, \ldots,a_n), b, f, u_0$ be $latex C^1$ functions with $latex |a|\le 1/\kappa$, and $latex u$ a $latex C^2(\bar\R^{n+1}_+)$ solution of the IVP of the previous problem. Then, for any $latex P=(X,T)\in\R^{n+1}_+$, $latex \displaystyle ||u||_{C^1(C_\kappa(P))} \le C \big( ||u_0||_{C^1(\partial_-C_\kappa(P))} + ||f||_{C...