Shift Theorems (Ch4.1)
formulation
restricted form. of Poisson eq. (P42)
H20⊂H10⊂L2⊂H−1⊂H−2
dual space \( H^{-1}:=[H_0^1(\Omega)]^* \)
or more general \( H^{-s}:=[H_0^s(\Omega)]^*, s \in \mathbb{R}^+ \)
Shift Theorem (Th57)
\( H^{1+s} \) regularity P(43)
weak
Shift non-convex domains to solve w/ weighted Sobolev spaces (Th58)
general (Ch1.2)
derivation (w/ boundary conditions) (P6)
BLF \( A(\cdot,\cdot):V \times V \rightarrow \mathbb{R}\)
\( A(u,v)= \int_\Omega \nabla u \nabla v \,\mathrm dx + \int_{\Gamma_R}\alpha u v \,\mathrm ds\)
functions & (their) spaces (P41)
Assumptions
\( -\Delta u =f \) in \( \Omega \)
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
\( \Delta:=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}} \)
\( \Omega \) 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 \( \Gamma_R\)
\( \int_{\Gamma_R}\alpha \,\mathrm dx >0 \)
\( \left\vert \Gamma_D \right\vert > 0 \)
\( g \in L_2 (\Gamma_N \cup \Gamma_R )\)
\( f \in L_2 (\Omega)\)
\( u_D \in H^{1/2}(\Gamma_D)\)
closed subspace \( V_0=\left\{v:tr_{\Gamma_D}v=0\right\}\)
HS \( V:=H^1(\Omega)\)
\( \alpha \in L_{\infty}(\Gamma_D), \alpha>0 \)
linear manifold \( V_D=\left\{u \in V:tr_{\Gamma_D}u=u_D\right\}\)