Shift Theorems (Ch4.1)

formulation

restricted form. of Poisson eq. (P42)

H20H10L2H1H2

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 \( V_D=\left\{u \in V:tr_{\Gamma_D}u=u_D\right\}\)