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Mostrando las entradas de septiembre, 2018

### Problem set 7, PDE

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

### Homework 5, Real Analysis

Due September 21 Problem 1 Let  X  be a compact space and $f:X\to Y$ a continuous bijection. Then $f^{-1}:Y\to X$ is continuous. Give an example of a continuous bijection $f:X\to Y$, 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 $x_n\in X$ is a sequence that converges to $x_0\in Y$, then $x_0\in X$. If  Y is complete, then  X is complete if and only if  X is closed in  Y . Problem 4 Every bounded set in $\R^l$ is totally bounded. Every closed and bounded set in $\R^l$ is compact.

### Homework 4, Real Analysis

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

### Problem set 4, PDE

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