《H_space_v2》.pdf

Chapter 7 The Sobolev Spaces Hs Note that H k (Ω) = W k,2 (Ω). 7.1 The Sobolev Spaces H 1 Definition. We say f ∈ H 1 (Ω) if f ∈ L2 (Ω) and it’s distributional gradient ∇f ∈ L2 (Ω). H1 (Ω) is a Hilbert space with norm 1/2 f H 1 (Ω) = |f |2 + |∇f |2 Ω Ω and natural inner product f, g 1 = fg + ∂ f∂ g H (Ω) i i Ω Ω i in H 1 . The term Ω |∇f |2 dx is called the kinetic energy of f . Fourier representation of H 1 (Rn ) norm implies L2 (Rn ) space with different measure by Fourier charac- terization later. Lemma. H 1 (Ω) is complete. Proof. Take {fn } Cauchy in H 1 (Ω) and show fn → f ∈ H 1 (Ω). The proof is based on H¨older’s inequality. Assume {fm } is Cauchy in H 1 (Ω) and, by the completeness 2 2 2 2 of L , there is a f ∈ L (Ω) such that fm → f in L (Ω), and there is a vector b ∈ L such that ∇fm → b in L2 (Ω). ∞ Now we need to show that b = ∇f in D (Ω). Note that for all φ ∈ Cc (Ω) ( φ ∈ D(Ω) ), we have ∇φ(x)f (x) dx = lim ∇φ(x)fm (x) dx. m→∞ Ω With Cauchy-Schwarz inequality we have ∇φ(x) f (x) − fm (x) dx ≤ ∇φL2 (Ω) f − fm L2 (Ω) → 0 Ω since φ ∈ D(