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diff --git a/vorlesungen/slides/2/hilbertraum/laplace.tex b/vorlesungen/slides/2/hilbertraum/laplace.tex new file mode 100644 index 0000000..8f6b196 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/laplace.tex @@ -0,0 +1,66 @@ +% +% laplace.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Höhere Dimension} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Problem} +Gegeben: $\Omega\subset\mathbb{R}^n$ ein Gebiet +\\ +Gesucht: Lösungen von $\Delta u=0$ mit $u_{|\partial\Omega}=0$ +\end{block} +\uncover<2->{% +\begin{block}{Funktionen} +Hilbertraum $H$ der Funktionen $f:\overline{\Omega}\to\mathbb{C}$ +mit $f_{|\partial\Omega}=0$ +\end{block}} +\uncover<3->{% +\begin{block}{Skalarprodukt} +\[ +\langle f,g\rangle += +\int_{\Omega} \overline{f}(x) g(x)\,d\mu(x) +\] +\end{block}} +\uncover<4->{% +\begin{block}{Laplace-Operator} +\[ +\Delta \psi = \operatorname{div}\operatorname{grad}\psi +\] +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<5->{% +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,\Delta g\rangle +&\uncover<6->{= +\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)} +\\ +&\uncover<7->{= +\int_{\partial\Omega} +\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)} +\\ +&\uncover<7->{\qquad +- +\int_{\Omega} +\operatorname{grad}\overline{f}(x)\cdot \operatorname{grad}g(x) +\,d\mu(x)} +\\ +&\uncover<8->{=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)} +\\ +&\uncover<9->{= +\langle \Delta f,g\rangle} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |