From 9e7524c25a0ba5a643fbb7555d01311f69aa603e Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 3 Jun 2021 17:18:58 +0200
Subject: add slides

---
 vorlesungen/slides/2/hilbertraum/sobolev.tex | 48 ++++++++++++++++++++++++++++
 1 file changed, 48 insertions(+)
 create mode 100644 vorlesungen/slides/2/hilbertraum/sobolev.tex

(limited to 'vorlesungen/slides/2/hilbertraum/sobolev.tex')

diff --git a/vorlesungen/slides/2/hilbertraum/sobolev.tex b/vorlesungen/slides/2/hilbertraum/sobolev.tex
new file mode 100644
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+++ b/vorlesungen/slides/2/hilbertraum/sobolev.tex
@@ -0,0 +1,48 @@
+%
+% 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
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