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Weak form. of poison eq. (Ch4) (Shift Theorems (Ch4.1) (restricted form.…
Shift Theorems (Ch4.1)
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dual space \( H^{-1}:=[H_0^1(\Omega)]^* \)
or more general \( H^{-s}:=[H_0^s(\Omega)]^*, s \in \mathbb{R}^+ \)
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formulation
weak
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BLF \( A(\cdot,\cdot):V \times V \rightarrow \mathbb{R}\)
\( A(u,v)= \int_\Omega \nabla u \nabla v \,\mathrm dx + \int_{\Gamma_R}\alpha u v \,\mathrm ds\)
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Assumptions
\( \int_{\Gamma_R}\alpha \,\mathrm dx >0 \)
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LF \( f(v)= \int_\Omega fv \,\mathrm dx + \int_{\Gamma_N \cup \Gamma_R} gv \,\mathrm ds \)
weak. form \( A(u,v)=f(v), \, \forall v\in V_0\) has unique solution \( u \) (Th56) + Proof
general (Ch1.2)
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\( \Omega \) bounded and open subset of \( \mathbb{R}^d \)
\( \partial\Omega =: \Gamma = \Gamma_D +\Gamma_N + \Gamma_R \) non overlapping
Boundary conditions
\( u=u_D \) on \( \Gamma_D\)
\( \frac{ \partial u}{ \partial n}=g \) on \( \Gamma_N\)
\( \frac{ \partial u}{ \partial n} +\alpha u=g \) on \( \Gamma_R\)