Ricardo A. Sáenz

Problem 1 Find solutions of the following initial-value problems in $latex \R^2$. $latex \partial_y u + (1+x^2)\partial_x u - u = 0; u(x,0) = \tan x$ $latex \partial_y u + \partial_x u = u^2; u(x,0) = h(x)$ Problem 2 Find a smooth function $latex a(x,y)$ in $latex \R^2$ such that the equation $latex \partial_y u + a(x,y) \partial_x u = 0$ does not have global solutions for any Cauchy data $latex \{y=0\}$. Problem 3 Let $latex \alpha\in\R$ and $latex h(x)$ a continuous function in $latex \R$, and consider que problem $latex y\partial_x u + x \partial_y u = \alpha u; \quad u(x,0) = h(x).$ Find all points where $latex \{y=0\}$ is characteristic. What is the compatibility condition on  h on those points? Find the solution of the initial-value problem away of the point in (1). What is the domain of this solution? For the cases $latex \alpha=1, h(x) = x$ and $latex \alpha=3, h(x) = x$, check whether this solution can be extended over the characteristic points. Find the chara...

Due Septembre 7 Problem 1 If $latex E, F\subset\R$ are a closed and a compact disjoint sets, then $latex \text{dist}(E,F)>0$. Problem 2 If $latex E\subset\R$ is closed, then it is measurable. Follow the next steps. Prove that it is sufficient to assume that  E is compact, and thus $latex |E|_* < \infty.$ Given $latex \e>0$, choose an open $latex U\supset E$ with $latex |U|_* < |E|_* + \e.$ Prove that we can write $latex U\setminus E = \bigcup I_j$, where the $latex I_j$ are disjoint open intervals. If $latex I,J$ are disjoint open intervals, then $latex |I\cup J|_* = |I| + |J|.$ For each  N , $latex |U|_* \ge |E|_* + \sum_{j=1}^N |I_j|$. Conclude $latex |U\setminus E|_* < \e$. Problem 3 Find a sequence of measurable sets $latex E_1 \supset E_2 \supset ...$ such that, for $latex E = \bigcap E_j$, $latex |E| \not= \lim |E_j|$. Problem 4 For $latex E\subset\R$, let $latex U_n=\{x\in\R: \text{dist}(x,E) < 1/n\}$. If  E is compact, then $latex |E| = \lim |U_n...

Problem 1 If, for $latex k=1, 2, \ldots, N$, $latex y_k(x_1, \ldots, x_{n-1},t) = u(x_1 + \ldots + x_{n-1},t)$, then $latex Y = (y_k)$ solves the system $latex \displaystyle \partial_t y_k = \frac{Mr}{r - (x_1 + \ldots + x_{n-1}) - (y_1 + \ldots + y_N)} \Big( \sum_{i=1}^{n-1}\sum_{j=1}^N \partial_{x_i}y_j + 1 \Big)$ with $latex y_k(x,0)=0$ if and only if $latex u(s,t)$ solves the equation $latex \displaystyle \partial_t u = \frac{Mr}{r - s - Nu} \big( N(n-1)\partial_su + 1\big)$ with $latex u(x,0)=0$. Problem 2 The function $latex \displaystyle u(s,t) = \frac{r - s - \sqrt{(r-s)^2 - 2MNnrt}}{Nn}$ solves the equation of the previous problem near $latex (0,0)$. Problem 3 Find a solution as a power series expansion of the initial-value problem $latex \displaystyle \begin{cases} u_{tt} - u_{xx} - u=0,\quad (x,t)\in\R\times(0,\infty)\\ u(x,0) = x,\quad \partial_t u(x,0) = -x. \end{cases}$ Identify this solution.

Due August 31 Problem 1 Let $latex f_n:[a,b]\to\R$ be Riemann-integrable and $latex f_n\rightrightarrows f$. f is Riemann-integrable on $latex [a,b]$. ( Hint:  Given $latex \e > 0$, find a partition $latex \mathscr P$ of $latex [a,b]$ such that $latex U(f,\mathscr P) - L(f,\mathscr P) < \e,$ where $latex U(f,\mathscr P), L(f,\mathscr P)$ are the upper and lower sums of  f  with respect to $latex \mathscr P$, respectively.) $latex \displaystyle \int_a^b f_n \to \int_a^b f$. Problem 2 Consider the functions $latex \displaystyle f_n(x) = \frac{x}{1 + nx^2}.$ Then $latex f_n\rightrightarrows 0$, but $latex f_n'(0)\not\to 0$. Problem 3 Let $latex f\in C^k(\mathbb S)$, a  k -continuously differentiable periodic function, with period $latex 2\pi$, and let $latex a_n$ be its  n th Fourier coefficient. There exists $latex C>0$ such that $latex |a_n| \le \dfrac{C}{|n|^k}$. The series $latex \sum a_n e^{inx}$ converges uniformly if $latex k\ge 2$. Problem 4 Let  X be a metr...