still more changes pdes

Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex

@@ -78,7 +78,7 @@
     $$
   \end{proof}
   \begin{theorem}
-    Let $\Omega\subseteq \RR^d$ be a set and $1<p<\infty$. Then, if $f_n\rightharpoonup f$ in $L^p(\Omega)$, then: $$\norm{f}_p\leq \liminf_{n\to\infty}{\norm{f_n}_p}$$ In addition, if $\displaystyle\norm{f}_p=\lim_{n\to\infty}{\norm{f_n}_p}$, then $f_n\to f$ in $L^p(\Omega)$.
+    Let $\Omega\subseteq \RR^d$ be a set and $1<p<\infty$. If $f_n\rightharpoonup f$ in $L^p(\Omega)$, then: $$\norm{f}_p\leq \liminf_{n\to\infty}{\norm{f_n}_p}$$ In addition, if $\displaystyle\norm{f}_p=\lim_{n\to\infty}{\norm{f_n}_p}$, then $f_n\to f$ in $L^p(\Omega)$.
   \end{theorem}
   \begin{proof}
     The first point is \mcref{ATFAPDE:lower_semicontinuity_thm} in the case $E = L^p(\Omega)$. We only prove the second point for $p=2$. If $\displaystyle\norm{f}_2=\lim_{n\to\infty}{\norm{f_n}_2}$, then:
@@ -309,7 +309,7 @@
   \begin{theorem}
     Let $\Omega\subseteq \RR^d$ be a bounded domain with $\mathcal{C}^k$ boundary. Then, $\forall m\leq k$, if $1\leq p<\frac{d}{m}$,
     % now add the conclusions of the gagliardo theorem
-    there is an embedding $W^{m,p}(\Omega)\hookrightarrow L^q(\Omega)$, where $\displaystyle\frac{1}{q}=\frac{1}{p}-\frac{m}{d}$. If $p>\frac{d}{m}$, then $W^{m,p}(\Omega)\hookrightarrow \mathcal{C}^{k-m,\theta}(\Omega)$, where $\theta=m-\frac{d}{p}-\ell$ and $\ell:=\left\lfloor m-\frac{d}{p}\right\rfloor$.
+    there is an embedding $W^{m,p}(\Omega)\hookrightarrow L^q(\Omega)$, where $\displaystyle\frac{1}{q}=\frac{1}{p}-\frac{m}{d}$. If $p>\frac{d}{m}$, then $W^{m,p}(\Omega)\hookrightarrow \mathcal{C}^{\ell,\theta} (\overline{\Omega})$, where $\ell:=\left\lfloor m-\frac{d}{p}\right\rfloor$ and $\theta:=m-\frac{d}{p}-\ell\in [0,1)$.
   \end{theorem}
   \begin{theorem}[Reillich-Kondrachov's compactness theorem]\label{ATFAPDE:reillich_kondrachov_compactness}
     Let $\Omega\subseteq \RR^d$ be a bounded domain with $\mathcal{C}^k$ boundary. Then, $\forall m\leq k$ we have: