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 $latex \partial K$.