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O UTLINE : k,p • definition of H (Ω), where Ω is a domain in Rn • density properties of C ∞ functions in H k,p • theorems: – interpolation – embedding – compactness • introduction to manifolds and extension of previous results • apply to DP for general elliptic operators (not just ∆) on general manifolds N OTATION : α1 αn ∂ • ∇α u = ∂x . . . ∂x∂n u 1 Here α is a multi-index with α = (α1 , . . . , αn ) ∈ Nn Pn • |α| = 1 αi the length of the multi-index. P 1/2 α 2 • k∇k uk = |α|=k |∇ u| Sobolev spaces of functions f : Rn → R Let Ω ⊆ Rn be a domain. Unless otherwise noted, we consider Ω to be open and connected. Definition: The H k,p norm of u ∈ C k Ω is: kukH k,p (Ω) k X ≡ k∇l ukLp (Ω) l=1 where kukLp (Ω) R ≡ ( Ω |u|p )1/p Definition: H k,p (Ω) ≡ C ∞ Ω k·kH k,p , the Sobolev space H k,p (Ω) k.kH k,p Definition: H0k,p (Ω) ≡ Cc∞ Ω k,p Definition: Hloc (Ω) ≡ {u : Ω → R | u|Ω1 ∈ H k,p (Ω1 ), Ω1 ( Ω} Facts: (i) H k,p (Ω) and H0k,p (Ω) are complete Banach spaces by definition k,p k,p (ii) We introduce a topology on Hloc (Ω) as follows: uk → u in Hloc if kuk − ukH k,p (Ω1 ) → 0, ∀ Ω1 ( Ω (iii) C ∞ (Ω) is dense in H k,p (Ω) Notice that (iii) =⇒ ∀u ∈ H k,p (Ω), ∃ui → u in H k,p , with ui ∈ C ∞ (Ω) Q: Is there a systematic construction of ui ? ∞ n A: Yes, R via a “mollification”. We need a function ϕ ∈ Cc (R ), with 0 ≤ ϕ ≤ 1, supp(ϕ) ⊆ B1 (0) and Rn ϕ = 1 where ϕ is radially symmetric (ie. ϕ(x) = ϕ(kxk)). We can visualize ϕ as a “bump” (see figure below). 1 2 Now, for u ∈ L1loc (Ω) and x ∈ Ω ≡ {y ∈ Ω | dist(y, ∂Ω) ≥ } (see figure above), define: Z x − y −n u (x) = ϕ u(y)dn y R The function ϕ = −n ϕ(x/) satisfies supp(ϕ ) ⊆ B (0), 0 ≤ ϕ ≤ −n and ϕ (x)dx = 1. u is the “mollification” of u. Theorem: If u ∈ L1loc (Ω) and u is as defined, then: 1) a. u ∈ C ∞ (Ω ) ∀. Here, ∇α (u ) = (∇α ϕ ) ∗ u where ∗ denotes a convolution. b. If supp(u) ⊆ K ( Ω then supp(u ) ⊆ {y ∈ Ω | dist(y, K) ≤ } for sufficiently small. (see figure below) 2) u ∈ C 0 (Ω) =⇒ u → u uniformly on Ω1 ( Ω. 3) u ∈ Lploc (Ω) =⇒ u → u in Lploc . (ku kLp (Ω1 ) ≤ kukLp (Ω1 ) ) 4) u → u a.e. Homework Prove 1) – 3). Let us examine claim 1). Z x−y u (x) = −n ϕ u(y)dn y Rn Z ∂u (x) justification ∂ x−y = −n ϕ u(y)dn y ∂x ∂x Rn Next: An alternate characterization Definition: u ∈ Lp (Ω), α a multi-index. Dα u is the “αth weak derivative of u” if Z Z (Dα u)ϕ = (−1)|α| u∇α ϕ ∀ϕ ∈ Cc∞ (Ω) Ω Ω Note 1. This is not saying that Dα u = ∇α u because u ∈ Lp (Ω) Note 2. The requirement is just that the object Dα y “behaves like” ∇α u under integration by parts. 3 Note 3. Many functions have weak derivatives. R If u ∈ C k (Ω), then ∇α u exists ∀α such that |α| ≤ k. Integrating by parts, Ω ∇α uϕ = R (−1)|α| u∇α ϕ. Thus, for u ∈ C k (Ω), ∇α u = Dα u and the weak derivative generalizes the ordinary derivative. Claim: Suppose u ∈ C 0 (Ω) is a Lipschitz function (ie. ∃L with |u(x) − u(y)| ≤ L|x − y| ∀x, y ∈ Ω. L is called the Lipschitz constant.) Then u has a weak derivative, and Du = ∇u whenever it exists. Homework Prove the claim. Also, what about when ∇u does not exist? What is Du? Now, the alternate characterization. Definition: W k,p (Ω) = {u ∈ Lploc (Ω) | Dα u exists for |α| ≤ k, and Dα u ∈ Lp (Ω)} k,p Definition: Wloc (Ω) = {u ∈ Lploc (Ω) | u|Ω1 ∈ W k,p (Ω1 ) ∀Ω1 ( Ω} Aside: Why do we have H k,p and W k,p spaces? H k,p has nice density properties while W k,p has nice analytic features — u ∈ W k,p have weak derivatives, W k,p is more general than C k so that W k,p is good for PDEs. Goal: Prove that if Ω is a “nice” domain, then H k,p (Ω) = W k,p (Ω). Proposition: W k,p (Ω) is a Banach space {also reflexive (for p > 1) and separable (for p < ∞). This is a consequence of the fact that Lp is reflexive if p > 1 and separable if p < ∞.} Proof: Show that W k,p (Ω) is complete. Suppose that ui ∈ W k,p (Ω) is Cauchy in the W k,p norm Pk kui kW k,p = l=0 kDl ui kLp . Then Dα ui are all Cauchy in Lp . Thus, there is a subsequence ui0 such that Dα ui0 → v α in Lp for some v α ∈ Lp (Ω). We have ui0 → v ∈ Lp and we need to show v α = Dα v. Choose ϕ ∈ Cc∞ (Ω). Then, Z Z v α ϕ = lim Dα ui ϕ Z |α| = lim(−1) u i ∇α ϕ Z = (−1)|α| v∇α ϕ Since this holds ∀ϕ, v α = Dα v Note that the previous calculation says that if u ∈ H k,p (Ω) then there exists a sequence ui ∈ C ∞ (Ω) such that ui → u in H k,p (Ω) =⇒ ∇α ui are Cauchy in W k,p norm. Thus, the H k,p (Ω) and W k,p (Ω) norms are the same on C ∞ functions and v α = Dα u so that H k,p (Ω) ⊆ W k,p (Ω). k,p k,p Proposition: Hloc (Ω) = Wloc (Ω) k,p Proof: ⊆ is already done. We look at ⊇ by taking u ∈ Wloc (Ω). Use mollification. Let Ω1 ( Ω, take u|Ω1 and extend it by setting it equal to zero outside of Ω1 . Call this extension ū : Ω1 → R. Mollify ū: Z x − y −n n d y, ū (x) = ū(y)ϕ where < 0 < dist(Ω1 , ∂Ω)/2 (Aside: Ω1 ( supp(ū ) ( Ω) Fact: ū → ū in Lp on K ( supp(ū0 ). Homework Fact: ∇α ū → Dα ū in Lp on K ( supp(ū0 ). In particular, for K = Ω1 and 0 ≤ |α| ≤ k, ∇α ū |Ω1 Lp → (Dα ū)|Ω1 = (Dα u)|Ω1 Thus, ū → u on Ω1 , ū ∈ C ∞ (Ω). That is, u ∈ H k,p (Ω1 ).