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Download PDF by Schrijver A.: A Course in Combinatorial Optimization

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4) =p d |um − ϑu|p dϑ dx dϑ u(um − ϑu)|um − ϑu|p−2 dϑ dx 0 1 →p u(u − ϑu)|u − ϑu|p−2 dϑ dx = |u|p dx , 0 where ... dx denotes integration over IRn ; that is, IRn |u|p dx + IRn |vm |p dx → 1 . 5) E(um ) = E(vm + u) 1 |∇u|2 + 2∇u∇vm + |∇vm |2 = 2 IRn + a(x) |u|2 + 2uvm + |vm |2 dx = E(u) + E(vm ) + IRn ∇u∇vm + a(x)uvm dx, and the last term converges to zero by weak convergence vm = um − u H 1,2 (IRn ). 0 in Moreover, for any ε > 0, letting Ωε = {x ∈ IRn ; |a(x) − a∞ | ≥ ε} ⊂⊂ IRn , since vm → 0 locally in L2 , the integral IRn (a(x) − a∞ )|vm |2 dx ≤ε IRn |vm |2 + sup |a(x)| IRn |vm |2 dx Ωε ≤ cε + o(1) .

For u ∈ C0∞ (IRn ), and t ≥ 0 let Ω(t) = {x ∈ IRn ; |u(x)| > t} . Then ∞ |u(x)| = χΩ(t) (x) dt 0 for almost every x ∈ Ω, and hence by Minkowsky’s inequality u L q1 ∞ ≤ χΩ(t) 0 = ∞ L q1 Ln Ω(t) 0 ∞ ≤ K(n) dt 1/q1 dt P Ω(t); IRn dt . 8) follows. 8) the general case p ≥ 1 can be derived by applying H¨ older’s np n ∞ = sq1 , where s = np−p ≥ 1. Then for u ∈ C inequality. Denote q = n−p 0 (IR ) n−p we can estimate 44 Chapter I. 9) ≤ sK(n) u D1,p . 7 Bounded domains. 9) is never achieved on any domain Ω different from IRn ; in particular, it is never achieved on a bounded domain.

Hence ϕ weakly in H01,p (Ω) by the Arz´ela-Ascoli theorem we may assume that ϕm and uniformly in Ω. ) Thus (∇um − ∇u)∇ϕm dx = Ω (∇um − ∇u)(∇ϕm − ∇ϕ) dx + o(1) Ω (∇um − ∇ul )(∇ϕm − ∇ϕ) dx + o(1) = lim l→∞ Ω (fm − fl )(ϕm − ϕ) dx + o(1) = lim l→∞ Ω ≤ 2 sup fl l∈IN where o(1) → 0 as m → ∞. L1 ϕm − ϕ L∞ + o(1) = o(1), 3. Compensated Compactness 33 It follows that ∇um → ∇u in Lq0 (Ω) for some q0 ≥ 1. But then, by H¨ older’s inequality, for any q < 2 we have ∇um − ∇u where 1 q =γ+ Lq ≤ ∇um − ∇u γ L1 ∇um − ∇u 1−γ L2 → 0, 1−γ .

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A Course in Combinatorial Optimization by Schrijver A.

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