\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\DeclareMathOperator{\Ker}{{\rm Ker}}
\begin{document}
\doclabel{Math 617: Homework 8}
\docdate{Due: March 21, 2012}
\begin{aproblems}

\hproblem D\&M 3.19

\hproblem  Suppose $f\in L^1_{\rm loc}(\Reals)$ and $\phi$ is a continuously 
differentiable function with compact support.  Define
$$
(f * \phi)(x) = \int_\Reals \phi(x-y) f(y)\; dy;
$$
this is the {\bf convolution} of $\phi$ and $f$.
Show that $f*\phi$ is differentiable and
$$
(f * \phi)'(x) = \int_\Reals \phi'(x-y)  f(y)\; dy.
$$
Conclude that if $\phi$ is an infinitely differentiable function
with compact support, then $f*\phi$ is infinitely differentiable.
\Hint Use the Dominated Convergence Theorem.

\hproblem Suppose $\phi$ is a continuously differentiable
function on $[0,1]$ such that $\phi(0)=0$.  Show that
there is a constant $c>0$ such that
$$
\int_0^1 \phi'(x)^2\; dx \ge c \int_0^1 \phi(x)^2\; dx.
$$

\hproblem We define $H^1_0((0,1))$ to be the
closure of $\mathcal{D}((0,1))$ in the $H^1$ norm.
Show that the bilinear form 
$$
\varphi(u,v) = \int_0^1 u'(x)v'(x)\; dx
$$
is coercive on $H^1_0([0,1])$.

\hproblem Let $f\in L^2([0,1])$. Show that there is
a unique function $u\in H^1_0([0,1])$ such that
$$
\int_0^1 u' \psi' = \int_0^1 f\psi
$$
for all $\psi\in \mathcal{D}((0,1))$.
You may assume that $H^1((0,1))$ is complete.

\hproblem  There exists an infinitely differentiable function $\phi:\Reals\ra\Reals$
such that $\phi\ge 0$, the support of $\phi$ is contained in $[-1,1]$,
and $\int_\Reals \phi = 1$.  You do not need to prove this.  Instead,
prove:
\begin{subproblems}
  \item For each $n\in\Nats$, let $\phi_n(x)=n\phi(nx)$.
  Show that $\phi_n$ all the aforementioned properties that $\phi$ has,
  except that the support of $\phi_n$ in contained in $[-1/n,1/n]$.
  \item Suppose $g\in L^\infty(\Reals)$.  Let $g_n=\phi_n*g$.  Show $||g_n||_\infty \le ||g||_\infty$.
  \item Suppose $g\in L^1_{\rm loc}(\Reals)$ and is continuous at some $x\in \Reals$.  Show that $g_n(x)\ra g(x)$.  Hint: $g(x)=\int_\Reals \phi_n(x-y)g(x)\;dy$.
\end{subproblems}
\end{aproblems}
\end{document}