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Mostrando las entradas de septiembre, 2018

Problem set 7, PDE

Problem 1 If u is harmonic in the connected domain Ω and is not constant, then u(Ω) is open in \R. Suppose Ω is bounded and that its boundary Ω is connected. If u is harmonic in Ω, then u(Ω)u(Ω). Problem 2 A radial harmonic function on B is constant. A positive harmonic function on \Rd is constant. Problem 3 Suppose u(x) is harmonic in some domain in \Rn. Then v(x)=|x|2nu(x|x|2) 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 7, Real Analysis

Due October 5 Problem 1 Let L1([a,b]) be the space of real valued continuous functions with the d1 metric. The polynomials are dense in L1([a,b]). Is L1([a,b]) separable? Problem 2 Let f:[a,b]\R be a continuous function such that baf(x)xndx=0 for all n=0,1,2,. Then f(x)=0 for all x[a,b]. Problem 3 If X,Y are compact metric spaces, then the tensor space C(X)C(Y)={(x,y)nk=1fk(x)gk(y):fkC(X),gkC(Y),n1} is dense in C(X×Y). Note:  The product space X×Y has the metric dX×Y((x1,y1),(x2,y2))=dX(x1,x2)+dY(y1,y2). Problem 4 State whether the following are true: ¯AB¯A¯B; ¯AB¯A¯B; $latex \overline{A\cap B} \subset \ov...

Problem set 6, PDE

Problem 1 Let u0L2(0,π) be the solution of the heat equation seen in class. Then, for all i,j\N, sup as t\to\infty. Problem 2 For any 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 \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 \partial_t^2 - \partial_x^2 on \R^2 commutes with the Lorentz transformations (hyperbolic rotations) \displaystyle T_\theta = \begin{pmatrix}\cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta\end{pmatrix}, for \theta\in\R.

Homework 6, Real Analysis

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

Problem set 5, PDE

Problem 1 Classify the following second-order PDE. \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 \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 \begin{cases} \Delta u - u^3 = f & \text{in }\Omega\\u = \phi & \text{on }\partial\Omega.\end{cases} Problem 3 Let \Omega be a bounded C^1-domain in \R^n and  u be a C^2-function in \bar\Omega\times[0,T] satisfying \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 \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...

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...