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+%
+% sobolev.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Sobolev-Raum}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Vektorrraum $W$}
+Funktionen $f\colon \Omega\to\mathbb{C}$
+\begin{itemize}
+\item
+$f\in L^2(\Omega)$
+\item
+$\nabla f\in L^2(\Omega)$
+\item
+homogene Randbedingungen:
+$f_{|\partial \Omega}=0$
+\end{itemize}
+\end{block}
+\begin{block}{Skalarprodukt}
+\begin{align*}
+\langle f,g\rangle_W
+&=
+\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)
+\\
+&\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)
+\\
+&=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Vollständigkeit}
+\dots
+\end{block}
+\begin{block}{Anwendung}
+``Ein Hilbertraum für jedes partielle Differentialgleichungsproblem''
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup