Sobolev Spaces (Ch3)

Basics (Ch3.1)

Equivalent norms on $H^1$ & sub-spaces (Ch3.4)

Trace (Ch3.3)

Sobolev spaces $ W^k_p(\Omega) $ (Ch3.2)

smooth funct. with compact supp $ \mathcal{D}(\Omega):=C^\infty_0 (\Omega)$ (P25)

support $ supp\{u\} $ (P25)

locally integrable function space $ L^{loc}_1(\Omega) $(P26)

classical diff. operator $ D^\alpha $, with multi-index $ \alpha $ (P25)

LFs on $ \mathcal{D}(\Omega) $ are $ \mathcal{D}' $ the Distributions ($ L_2 $ inner product with $ u \in C(\Omega)$)

integration by parts (eq3.2)

Lipschitz boundary (Def 42)

Motivation: apply Lax-Milgram on 2nd order var problems in $ H^1 $ therefore we have to prove coercivity

Derivatives

Embedding $ H^k \rightarrow H^l $ for $ k>l $ is compact (Th51)

Bramble Hilbert Lemma (Th54)
$ U $ HS, $ L:H^k \rightarrow U $ cont. lin. operator with $ Lq=0 , q \in P^{k-1}$
$ \Rightarrow \left\Vert Lv \right\Vert_U \leq \left\Vert v \right\Vert_{H^k} $

Friedrichs inequality (Th52)
$ \left\Vert v\right\Vert _{L_{2}} \preceq \left\Vert \nabla v\right\Vert _{L_{2}}, \forall v\in V_D=\{v \in H^1(\Omega):tr_{\Gamma{_D}}v=0 \} $

Sobolevs embedding theorem (Th55)

Application

Tartar + Proof = 3 equivalent norms (Th50)

Trace Space $ H^{1/2} $ (Ch3.3.1)

is BS (Th40)

closed subspace $ H^1_0(\Omega) $ (P30)

2 alternative definitions, which are equivalent (under moderate restrictions) (P27) (Th41, Th43)

Poincare (Th53)
$ \left\Vert v\right\Vert _{H^1(\Omega)}^2 \leq \left\Vert \nabla v\right\Vert _{L_{2}}^2 + \left(\int_{\Omega}vdx\right)^{2} $

Derivation in 1D (P28)

norms (P27)

compact $ \Leftrightarrow $ bounded

$ H^k(\Omega) := W^k_2(\Omega) $ is HS

if $ \Omega $ is bounded domain: $ supp\{u\} $ is compact in $ \Omega \Leftrightarrow u$ vanishes on $ \partial\Omega $

generalized derivative $ D^\alpha_g u $ of $u \in \mathcal{D}' $(Def37)

$ \Rightarrow $ compact embedding $id: H^1 \rightarrow H^0=L_2 $

weak is more general than classical, but more restrictive than generalized

is BS and existence of corresponding function in $ H^1 $ + Proof (Le49) = Trace Theorem?

weak derivative $ D^\alpha_w u $ of $u \in L^{loc}_1(\Omega)$(Def38)

Hilbert space interpolation (P35)

Application: $ V=H^1, W=L_2, \left\Vert v\right\Vert_A = \left\Vert \nabla v\right\Vert_{L_{2}} $

trace space onto one edge $ E $ = interpolation space $ H^{1/2}(E) $

space & norm (P34 & eq3.4)

well defined & cont. trace operator $ tr:H^1((0,h)) \rightarrow \mathbb{R} $ (Th44)

well defined & cont. trace operator $ tr:H^1(\Omega) \rightarrow L_2(\partial\Omega) $ (Th45) Trace Theorem?

integration by parts in $ H^1 $ (P31)

$ \Rightarrow \left\Vert u\right\Vert _{H^1(\Omega)}^2 \simeq \left\Vert \nabla v\right\Vert _{L_{2}(\Omega)}^2 + \left\Vert v\right\Vert _{L_{2}(\omega)}$, with $ \omega \subset \Omega, \left\vert \omega \right\vert > 0 $ in $ \mathbb{R}^d $

Sobolev spaces over sub-domains (P31)

$ \Rightarrow \left\Vert u\right\Vert _{H^1(\Omega)}^2 \simeq \left\Vert \nabla v\right\Vert _{L_{2}(\Omega)}^2 + \left\Vert v\right\Vert _{L_{2}(\gamma)}$, with $ \gamma \subset \partial\Omega, \left\vert \gamma \right\vert > 0 $ in $ \mathbb{R}^{d-1} $

Extension operators (P32)

interpolation space $ V_s $

interpolation norm

goal: explicit form of the trace-norm

conditions for $ u \in H^1(\Omega)$(Th46)

Construction of sub-domains $ \Omega_i $ and interfaces $ \gamma_{ij} $ in $ \Omega $ (P31)

extension operator (eq3.3)
$ (Eu)(\widehat{x})=u(x) \forall x \in \cup S_i $
$ (Eu)(x)=u(x) \forall x \in \Omega $

Extension $ E_0u(\widehat{x})=(1-\eta) u(x) $ is extension from $ H^1(\Omega) $ to $ H^1_0(\widetilde{\Omega}) $ and its bounded (Th48) ($E_0$ vanishes at $ \partial \widetilde{\Omega} $ )

$ \Rightarrow $ finite element space is sub-space of $ H^1 $

Extension from $ H^1_0({\Omega}) $ onto larger domains trivial, by zero (P33)

Construction (P32)

ext. operator $ E:H^1(\Omega) \rightarrow H^1(\widetilde{\Omega}) $ is well defined and bounded (Th47)