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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-03 18:51:36 +0200 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-03 18:51:36 +0200 |
commit | 680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d (patch) | |
tree | aff901d861b84c62e8eb2b7ff9981ae13a3f5baa /vorlesungen/slides/2/hilbertraum/sobolev.tex | |
parent | add slides (diff) | |
download | SeminarMatrizen-680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d.tar.gz SeminarMatrizen-680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d.zip |
phases
Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/sobolev.tex')
-rw-r--r-- | vorlesungen/slides/2/hilbertraum/sobolev.tex | 23 |
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} |