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-rw-r--r--vorlesungen/slides/2/hilbertraum/sobolev.tex23
1 files changed, 13 insertions, 10 deletions
diff --git a/vorlesungen/slides/2/hilbertraum/sobolev.tex b/vorlesungen/slides/2/hilbertraum/sobolev.tex
index 425c263..828d34d 100644
--- a/vorlesungen/slides/2/hilbertraum/sobolev.tex
+++ b/vorlesungen/slides/2/hilbertraum/sobolev.tex
@@ -14,34 +14,37 @@
\begin{block}{Vektorrraum $W$}
Funktionen $f\colon \Omega\to\mathbb{C}$
\begin{itemize}
-\item
+\item<2->
$f\in L^2(\Omega)$
-\item
+\item<3->
$\nabla f\in L^2(\Omega)$
-\item
+\item<4->
homogene Randbedingungen:
$f_{|\partial \Omega}=0$
\end{itemize}
\end{block}
+\uncover<5->{%
\begin{block}{Skalarprodukt}
\begin{align*}
\langle f,g\rangle_W
-&=
-\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)
+&\uncover<6->{=
+\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)}
\\
-&\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)
+&\uncover<7->{\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)}
\\
-&=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}
+&\uncover<8->{=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}}
\end{align*}
-\end{block}
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
+\uncover<9->{%
\begin{block}{Vollständigkeit}
\dots
-\end{block}
+\end{block}}
+\uncover<10->{%
\begin{block}{Anwendung}
``Ein Hilbertraum für jedes partielle Differentialgleichungsproblem''
-\end{block}
+\end{block}}
\end{column}
\end{columns}
\end{frame}