Topics
- Introduction to the course.
- Review of Chapters 1 and 2.
Readings
- Make sure you know the content of those two chapters.
Topics
- Review of Chapters 3 and 4.
- Topology induced by a family of seminorms.
- Nets.
Readings
- Make sure you know the content of those two chapters.
- Section 5.7
Topics
- Review of Chapter 5.
- Topology induced by a family of seminorms.
- Nets.
Readings
Topics
- Duality between $L^p_{loc}$ and $L^q_c$.
Readings
Jan 27
To be rescheduled
Topics
Readings
Topics
Readings
Topics
- The $W^{k,\infty}$ spaces.
- Complete proof of the extension theorem.
- The last item of the proof of the existence of the trace map.
Readings
Topics
Readings
Topics
Readings
Homework #1
Denote by $Lip(\Omega)$ the set of continuous functions on $\Omega$ that are globally Lipschitz. Namely, if $f\in Lip(\Omega)$, there is a $L>0$ such that $|f(x)-f(y)|\leq L\|x-y\|$
for every $x,y\in\Omega$.
- Prove that $\|f\|_{Lip}=\|f\|_{C^0}+\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$ is a norm on $Lip([0,1])$ and show that, with this norm, $Lip([0,1])$ is a Banach space.
- Prove that $W^{1,\infty}([0,1])=Lip([0,1])$.
- Let $f\in L^p(\mathbb R^2_+)$. Prove that $\|f\|_{L^p(\mathbb R\times(0,\varepsilon))}\to0$ as $\varepsilon\to0^+$ without using Poincare' inequality.
- Let $\eta\in C^\infty([0,\infty))$ be such that $\eta=0$ in $[0,1]$, $\eta=1$ in $[2,\infty)$ and $\eta\in(0,1)$ in $(1,2)$.
Set $\eta_\varepsilon(x)=\eta(x/\varepsilon)$. Show that there is a constant $c>0$ such that $|\eta'_\varepsilon(x)||\leq c/\varepsilon$ for $x\in(\varepsilon,2\varepsilon)$.
- Verify using Theorem 6.2.6 that $f(x)=0$ for $\in[-1,0)$ and $f(x)=1$ for $\in(0,1]$ does not belong to $W^{1,p}([-1,1])$ for $p\in[1,\infty)$ while $g(x)=x+1/2$ for $x\in(-1/2,0)$,
$g(x)=1/2-x$ for $x\in(0,1/2)$ and $g(x)=0$ otherwise does belong to $W^{1,p}([-1,1])$.