Ricardo A. Sáenz

Due date: December 7 Problem 1 Let $latex f_n(x) = a_n x^2 + b_n x + c_n$ be a sequence of quadratic polynomials such that $latex \displaystyle \int_0^1 |f_n(x)| dx \to 0.$ Then the coefficient sequences $latex a_n, b_n, c_n$ all converge to zero. Problem 2 For $latex r\in\mathbb Z_+$, let $latex \mathscr P_r$ be the space of polynomials of degree at most $latex r$. If $latex f_n\in\mathscr P_r$ converge uniformly to f  in [0,1], then $latex f\in\mathscr P_r$. The polynomials $latex \displaystyle f_n(x) = 1 + \frac{1}{2}x + \frac{1}{2^2}x^2 + \ldots + \frac{1}{2^n} x^n$ converge uniformly on [0,1], but their limit is not a polynomial function. Problem 3 Let $latex \mathscr H$ be the subspace of functions $latex f\in C([0,1])$ that satisfy $latex f(1 - x) = f(x)$. Then $latex \mathscr H$ is an infinite dimensional closed subspace of $latex C([0,1])$. Problem 4 Let $latex \mathscr I: C([0,1])\to C([0,1])$ be the operator given by $latex \disp...

Due November 30 Problem 1 Let  Y be a finite dimensional subspace of the Hilbert space  X . Then  Y is closed in  X . Problem 2 For each $latex n\in\Z$, let $latex e_n(x) = e^{2\pi i nx}$. Then $latex \displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$   Problem 3 Let  Y be a close subspace of the Hilbert space  X and $latex T:X\to Y$ the orthogonal projection onto  Y , $latex Tx = \text{Proj}_Y x.$ T is continuous. Problem 4 Let  Y be a closed subspace of the Hilbert space  X , and let $latex Y^\perp = \{ x\in X: x\perp Y\}.$ $latex Y^\perp$ is a closed subspace of  X . $latex X\cong Y\oplus Y^\perp$.

Problem 1 Let $latex \Omega\subset\R^n$ be a bounded domain and $latex u_0\in C(\bar\Omega)$. Suppose that $latex u\in C^{2,1}(\Omega\times(0,\infty))\cap C(\bar\Omega\times[0,\infty))$ is a solution of $latex \begin{cases} \partial_t u - \Delta u=0 & \text{in }\Omega\times(0,\infty)\\ u(\cdot,0) = u_0 & \text{on } \Omega \\ u=0 & \text{on } \partial\Omega\times(0,\infty).\end{cases}$ Then there exist constants $latex \mu, C>0$, depending only on $latex n,\Omega$, such that $latex \displaystyle \sup_\Omega |u(\cdot,t)| \le C e^{-\mu t} \sup_\Omega |u_0|$ for any $latex t>0$. Problem 2 Let $latex \Omega\subset\R^n$ be a bounded domain, $latex c\in C(\bar\Omega\times[0,T])$ with $latex c\ge -c_0$ for a constant $latex c_0\ge 0$, and $latex u_0\in C(\Omega)$ nonnegative. Suppose $latex u\in C^{2,1}(\Omega\times(0,T])\cap C(\bar\Omega\times[0,T])$ is a solution of $latex \begin{cases} \partial_t u - \Delta u + cu = -u^2 & \text{in }\Omega\times(0,T]\\ u(\cdot,0) = u_0...

Due November 23 Problem 1 Let  X be a metric space, $latex \mathcal C_X$ the set of its nonempty compact subsets and $latex d_H$ the Hausdorff metric. If  X is discrete, then $latex (\mathcal C_X,d_H)$ is discrete. Let $latex A\subset X$ be a finite set of isolated points of  X . Then  A is an isolated point in $latex \mathcal C_X$. Problem 2 Let $latex X=C([-1,1])$ with the inner product $latex \displaystyle (f,g) = \int_{-1}^1 f\bar g.$ Apply the Gram-Schmidt process to the sequence $latex f_n(x) = x^n$ to obtain the orthonormal polynomials $latex p_0, p_1, p_2, p_3$, such that each $latex p_n$ is of degree  n . These are the first  Legendre polynomials . Problem 3 If $latex ||\cdot||_1, ||\cdot||_2$ are norms induced by inner products in $latex \R^l$, then they are equivalent: there exist constants $latex c,C>0$ such that $latex c||x||_1 \le ||x||_2 \le C||x||_1$ for all $latex x\in\R^l$. ( Hint: use the Gram-Schmidt process to construct orthonrmal bases for each inner ...

Problem 1 $latex \displaystyle \int_{-\infty}^\infty e^{-\pi x^2} e^{-2\pi i x\xi} dx = e^{-\pi\xi^2}$ for any $latex \xi\in\R$.  Hint:  For each N,  calculate the contour integral $latex \displaystyle \int_\gamma e^{-\pi (z+i\xi)^2} dz$ over the contour $latex \gamma$ around the rectangle with vertices $latex -N, N, N-i\xi, -N-i\xi$. Take $latex N\to\infty$. Problem 2 $latex \displaystyle \int_{\R^n} e^{-\pi|x|^2} e^{-2\pi i x\cdot \xi} dx = e^{-\pi|\xi|^2}$ for every $latex \xi\in\R^n$. Problem 3 $latex K(x,t) = t^{-n/2} e^{-|x|^2/4t}$ satisfies the heat equation. $latex G(x,t) = (1 - 4\alpha t)^{-n/2} e^{\alpha|x|^2/(1-4\alpha t)}$, for any $latex \alpha>0$, also satisfies the heat equation. Problem 4 Let $latex u_0:[0,\infty)\to\R$ be bounded and continuous, with $latex u_0(0)=0$. Find an integral representation for the solution of the problem $latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = 0 & x>0, t>0 \\ u(x,0) = u_0(x) & x>0 \\ u(0,t) =...

Due November 16 Problem 1 If $latex A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix}$, then the map $latex x\mapsto Ax$ is a contraction in $latex \R^2$. Problem 2 Let $latex F(x,t) = \dfrac{tx}{x^2+1}$. Then, for all $latex t\in\R$, $latex |F(x,t) - F(y,t)| \le |t||x-y|$. Problem 3 The function $latex f(x) = \sqrt x$ on $latex [0,\infty)$ is uniformly continuous but not Lipschitz. Problem 4 Consider the operator $latex \Phi:C([-1,1])\to C([-1,1])$ given by $latex \displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,$ for any $latex x(t)\in C([-1,1])$. Starting from the constant function $latex x_0(s)=1$, verify explicitly that the  n th iteration of $latex x_{n+1} = \Phi(x_n)$ is the  n th Taylor polynomial of $latex t\mapsto e^{t^2}$ around $latex t=0$.

In all problems, $latex \Omega\subset\R^n$ is a $latex C^1$ bounded domain, and derivatives are understood as weak derivatives. Problem 1 The space $latex H^1(\Omega)$ is a Hilbert space with respect to the inner product $latex \displaystyle \langle u, v \rangle_{H^1} = \int_\Omega \big( u\cdot v + \nabla u \cdot \nabla v \big).$ Problem 2 The $latex H_0^1$ and $latex H^1$ norms are equivalent in the space $latex H_0^1(\Omega)$. Problem 3 The restriction $latex u\mapsto u|_{\mathbb \partial\Omega}$ is bounded from $latex H^1(\Omega)$ into $latex L^2(\partial\Omega)$. ( Hint: Extend the normal field $latex \nu$ from $latex \partial\Omega$ to $latex \bar\Omega$ and use the divergence theorem, together with the Cauchy inequality.) Problem 4 Consider, in the disk $latex \mathbb D$, the functions $latex u_k(r,\theta) = r \cos k\theta$, $latex k\in\N.$ The $latex u_k$ are orthogonal in $latex H^1(\mathbb D)$, with respect to the $latex H^1$ product. ( Hint:  Use integration in polar coord...

Due November 9 Problem 1 The fixed points of a continuous $latex f:\mathbb B^n\to\mathbb B^n$ might not be interior. Problem 2 The Brouwer fixed point theorem is false for the open ball. Problem 3 Let $latex K\subset\R^n$ be compact and convex, and $latex f:K\to K$ continuous. Then  f has a fixed point. Problem 4 Let $latex K\subset\R^n$ be compact and convex with $latex C^1$ boundary, $latex x_0\in K$, and $latex b:K\setminus\{x_0\}\to\partial K$ given by the intersection point of the line from $latex x_0$ to $latex x$, on the side of $latex x$. Then $latex b\in C^1$. Note that, if $latex x_0$ is an interior point of $latex K$, then $latex b$ is a retraction from $latex K\setminus\{x_0\}$ onto $latex \partial K$.  

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 $latex A\subset X$.    

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.