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Mostrando las entradas de noviembre, 2018

### Problem set 11, PDE

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

### Homework 11, Real Analysis

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