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Weak Derivatives (Revisited). Motivation for Sobolev Spaces

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Weak Derivatives (Revisited). Motivation for Sobolev Spaces Yuliya Gorb Weak Derivatives (revisited). Motivation for Sobolev Spaces Lecture 04 January 23, 2014 Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Weak Derivatives Suppose Ω ⊂ Rn is an open set Definition 1 A function f ∈ Lloc (Ω) is weakly differentiable w.r.t. xi if there exists a function 1 gi ∈ Lloc (Ω) s.t. ∂φ ∞ f dx = − gi φ dx, for all φ ∈ Cc (Ω) ∂xi Ω Ω th ∂f The function gi is called the weak i partial derivative of f , and denoted . ∂xi The weak derivative of C ∞-function coincides with the p.w. derivative. Definition n 1 Suppose that α ∈ N0 is a multi-index. A function f ∈ Lloc (Ω) has weak α α ∂ f 1 derivative D f = α ∈ Lloc (Ω) if ∂xi ∂αf ∂αφ φ dx =(−1)|α| f dx, for all φ ∈ C ∞(Ω). ∂x α ∂x α c Ω i Ω i Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Properties of Weak Derivatives Product Rule: 1 1 If f ∈ Lloc (Ω) has weak partial derivative ∂i f ∈ Lloc (Ω) and ∞ ψ ∈ C (Ω), then ψf is weakly differentiable with respect to xi and ∂i (ψf ) = (∂i ψ)f + ψ(∂i f ). Chain Rule: Let f ∈ C 1(R) s.t. f ′ ∈ L∞(R). Suppose Ω ⊂ Rn is bounded, u is weakly differentiable on Ω. Then f ◦ u is weakly differentiable on Ω and D(f ◦ u)= f ′(u)Du. Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Weak Derivatives (cont.) Theorem 1 A function f ∈ Lloc (Ω) is weakly differentiable in Ω if and only if there is a ∞ α 1 sequence {fn} of functions fn ∈ C (Ω) s.t. fn → f and ∂ fn → g inLloc (Ω) α 1 (then the weak derivative of f is given by g = ∂ f ∈ Lloc (Ω)). Lemma Define u+(x) := max{u(x), 0}, u−(x) := min{u(x), 0}. Let Ω ⊂ Rn be bounded, and u weakly differentiable on Ω. Then u+, u− and |u| are weakly differentiable on Ω and Du(x), u(x) > 0 0, u(x) ≥ 0 Du+(x)= , Du−(x)= 0, u(x) ≤ 0 −Du(x), u(x) < 0 Corollary Let u be weakly differentiable on Ω. Then Du = 0 a.e. on any set where u is constant. Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Sobolev Spaces As before, assume that Ω ⊂ Rn is an open set Definition Suppose that k ∈ N and 1 ≤ p ≤∞. The Sobolev space W k,p(Ω) consists of all locally summable functions f : Ω → R such that weak derivatives ∂αf ∈ Lp(Ω) for 0 ≤ |α|≤ k, i.e. W k,p(Ω) = {f ∈ Lp (Ω) : ∂αf ∈ Lp (Ω), |α|≤ k} The space W k,p(Ω) is equipped with the norm: 1/p |∂αf |p dx , for 1 ≤ p < ∞ f W k,p (Ω) = |α|≤k Ω (1) max sup |∂αf |, p = ∞ |α|≤k Ω ∞ k,p p Note D(Ω) := Cc (Ω) ⊂ W (Ω) ⊂ L (Ω) and f Lp (Ω) ≤ f W k,p (Ω) ′ α The norm f W k,p (Ω) := ∂ f Lp (Ω) is an equivalent norm to · W k,p (Ω) |α|≤k We also write · k,p = · W k,p (Ω) Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Motivation for Sobolev Spaces Let Ω ⊂ Rn be a C 1-domain, g ∈ C 2(Ω). Want to solve: △u(x) = 0, x ∈ Ω (2) ( u(x) = g(x), x ∈ ∂Ω Finding a C 2 solution u of (2) is equivalent to finding a minimizer of F(·)= |∇ · |2dx over class V = {v ∈ C 2(Ω) : v = g on ∂Ω} ZΩ In other words, to find u ∈ V s.t. F(u)= inf F(v) v∈V By def of inf, there exists a minimizing sequence {vj }⊂ V s.t. F(vj ) → inf F(v) ⇒ v∈V Find u ∈ V s.t. there exists a subsequence {vj′ }⊂ V with vj′ → u (in some sense) s.t. under this convergence functional F is lower semi-continuous, i.e. F( lim vj′ ) =: F(u) ≤ lim F(vj′ ) j′→∞ j′→∞ Unfortunately, this does not always hold in V ⇒ need larger space Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Motivation for Sobolev Spaces (cont.) Define an inner-product in V : (u, v)= ∇u ·∇v dx ZΩ Then under the associated norm · , {vj } is Cauchy since 2 2 vj = |∇vj | dx = F(vj ), ZΩ where F(vj ) is convergent. The natural thing to do therefore is to let V be the completion of V with respect to this norm (since V is not already complete under this norm). Note V is then a Hilbert space. Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II Motivation for Sobolev Spaces (Example) Set f := −△g, w := u − g. Then w solves △w(x) = f , x ∈ Ω (3) w(x) = 0, x ∈ ∂Ω 2 Thus, let V0 = {v ∈ C (Ω) : v =0 on ∂Ω} with completion V0 Consider bounded linear functional L on V0: L(φ)= − f φ dx. Then for any Ω φ ∈ V0: ∇w ·∇φ dx = − △wφ dx = − f φ dx = L(φ), Ω Ω Ω i.e. (w,φ)= L(φ) By Hahn-Banach Thm, L extends to a bounded linear functional L˜ on V0, and since V0 is Hilbert, by Riesz representation Thm, there exists somew ˜ ∈ V0 s.t. L˜(φ)=(˜w,φ) for all φ ∈ V0 Hence,w ˜ is a generalized solution of (3). Thus, by working in V0, we can indeed find a possible solution. The challenge is then to show that actually w˜ ∈ V0, hence, is the actual solution we search for. This depends on f and Ω However, to prove regularity results on V, we need an explicit construction for it; the abstract completion is not good enough. This leads us to define Sobolev spaces, which are those spaces are the completions we referred to above Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces Yuliya Gorb PDE II References Evans pp. 258–266 Lecture 04 Weak Derivatives (revisited). Motivation for Sobolev Spaces.
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