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1). 2 We define H0m (G) to be the closure in H m (G) of C0∞ (G). Generally, H0m (G) is a proper subspace of H m (G). Note that for any f ∈ H m (G) we have (∂ α f, ϕ)L2 (G) = (−1)|α| (f, D α ϕ)L2 (G) , |α| ≤ m , ϕ ∈ C0∞ (G) . We can extend this result by continuity to obtain the generalized integrationby-parts formula (∂ α f, g)L2 (G) = (−1)|α| (f, ∂ α g)L2 (G) , f ∈ H m (G) , g ∈ H0m (G) , |α| ≤ m . This formula suggests that H0m (G) consists of functions in H m (G) which vanish on ∂G together with their derivatives through order m − 1.

It is an easy consequence of the Lebesgue dominated convergence theorem that θj u → u in L2 (Rn+ ) and for each k, 1 ≤ k ≤ n − 1, that ∂k (θj u) = θj (∂k u) → ∂k u in L2 (Rn+ ) as j → ∞. Similarly, θj (∂n u) → ∂n u and we have ∂n (θj u) = θj (∂n u) + θj u, so we need only to show that θj u → 0 in L2 (Rn+ ) as j → ∞. Since γ0 (u) = 0 we have u(x , s) = 0s ∂n u(x , t) dt for x ∈ Rn−1 and s ≥ 0. From this follows the estimate |u(x , s)|2 ≤ s s 0 |∂n u(x , t)|2 dt . Thus, we obtain for each x ∈ Rn−1 ∞ 0 |θj (s)u(x , s)|2 ds ≤ 2/j 0 ≤ 8j (2j)2 s 2/j 0 s 0 s 0 |∂n u(x , t)|2 dt ds |∂n u(x , t)|2 dt ds .

Then we have βj ∈ C0∞ (Rn ), βj has support in Gj , βj (x) ≥ 0, x ∈ Rn and CHAPTER II. DISTRIBUTIONS AND SOBOLEV SPACES 44 ¯ That is, {βj : 0 ≤ j ≤ N } is {βj (x) : 0 ≤ j ≤ N } = 1 for each x ∈ G. ¯ a partition-of-unity subordinate to the open cover {Gj : 0 ≤ j ≤ N } of G and {βj : 1 ≤ j ≤ N } is a partition-of-unity subordinate to the open cover {Gj : 1 ≤ j ≤ N } of ∂G. Suppose we are given a u ∈ H m (G). Then we have u = N j=0 {βj u} on m G and we can show that each pointwise product βj u is in H (G ∩ Gj ) with support in Gj .

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