I am studying Partial Differential Equations currently at my university. I unfortunately haven't been able to take a course on functional analysis prior to this course and I have been having a hard time understanding some of the concepts. I was hoping someone would be kind enough to clarify some of my misunderstandings. The course follows the book Evans aswell.
- What does it mean for a function to belong to a particular space such as $W^{k,p}(U)$ or $L^p(U)$?
I understand that either space has a norm. Namely, for a function $f\in L^p(U)$, $$||f||_{L^p(U)}=\left(\int_U|f|^p\, dx\right)^{1/p}$$ and for a function $u\in W^{k,p}(U)$, with multiindex $\alpha$, \begin{equation*} ||u||_{W^{k,p}(U)}:=\begin{cases} \left( \sum_{|\alpha|\leq k} \int_U |D^\alpha u|^p\, dx\right)^{\frac{1}{p}}, \ \ \ &if \ 1\leq p<\infty\\ \sum_{|\alpha|\leq k} \mathrm{ess}\sup_{U}(|D^\alpha u|), &if \ p=\infty. \end{cases} \end{equation*} If these norms exist and are finite does this imply that the functions belong in the respective spaces? If possible, I was hoping someone could give be an example of this $W^{k,p}(U)$ norm for some given $k$. Would I be correct in saying that for $k=3$ for example, \begin{equation*} ||u||_{W^{3,p}(U)}:=\begin{cases} \left( \sum_{|\alpha|\leq 3} \int_U |D^\alpha u|^p\, dx\right)^{\frac{1}{p}}, \ \ \ &if \ 1\leq p<\infty\\ \sum_{|\alpha|\leq 3} \mathrm{ess}\sup_{U}(|D^\alpha u|), &if \ p=\infty \end{cases} =\begin{cases} \left( \int_U |u|^p+|Du|^p+|D^2u|^p+|D^3u|^p\, dx\right)^{\frac{1}{p}}, \ \ \ &if \ 1\leq p<\infty\\ \mathrm{ess}\sup_{U}(|u|^p)+\mathrm{ess}\sup_{U}(|Du|^p)+\mathrm{ess}\sup_{U}(|D^2u|^p)+\mathrm{ess}\sup_{U}(|D^3u|^p), &if \ p=\infty. \end{cases} \end{equation*} Please forgive me if this is completely incorrect. I feel as though there may be a gap in my understanding.
- Why is it that we write $W^{k,2}(U)=H^k(U)$?
Is this because the norm on $W^{k,2}(U)$ is equivalent to the norm on $H^k(U)$?
- This probably links in with the above questions, but why, from this can we deduce that $$||u||^2_{H^1_0(U)}=||u||^2_{L^2(U)}+||Du||^2_{L^2(U)}$$ and additionally, $$||u||^2_{H^2_0(U)}=||u||^2_{L^2(U)}+||Du||^2_{L^2(U)}+||D^2u||^2_{L^2(U)}.$$ Where $H^k_0$ denotes the closure of $H^k$.
I hope that someone can help me better understand some of these concepts. Any help will be appreciated.
(1) Yes, although they might not be "functions" in the traditional sense.
(2) This is just notation.
(3) Once again, these are just definitions. If you were given a slightly different definition for the norm, then this one will be equivalent (likely through the equivalence of $\ell^1$ and $\ell^2$ vector norms). Also, $H_0^k$ doesn't denote the closure of $H^k$. Rather, it denotes the closure of $C_c^\infty$ in the $H^k$ norm (or, equivalently, the subspace of $H^k$ whose elements have trace-zero).