Weighted Hardy inequality with higher dimensional ... - CAPDE

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Weighted Hardy inequality with higher dimensional ... - CAPDE

and from which we deduce that(2.13) ∇α · ∇ log(˜δ) = 1˜δ ∇α · ∇˜δ = O(˜δ − 3 2 ).By Lemma 2.1 we have thatα∆ log(˜δ) = α N − k − 2˜δ 2(1 + O(˜δ)).Taking back the above estimate together with (2.13) and (2.12) in (2.9), we get(2.14) ∆ log(w) = α N − k − 2˜δ 2 (1 + O(˜δ)) + O(| log(˜δ)|˜δ − 3 2 ).We also haveand thus∇(log(w)) = ∇(α log(˜δ)) = α ∇˜δ˜δ+ log(˜δ)∇α|∇(log(w))| 2 = α2 2α log(˜δ)+ ∇˜δ · ∇α + | log(˜δ)|˜δ 2 ˜δ2 |∇α| 2 = α2˜δ + O(| log(˜δ)|˜δ − 3 2 2 ).Putting this together with (2.14) in (2.8), we conclude that(2.15)Now we define the function∆ww = α N − k − 2 + α2˜δ 2 ˜δ + O(| log(˜δ)| ˜δ − 3 2 2 ).v(x) := d(x) w(x),where we recall that d is the distance function to the boundary of U. It is clear that(2.16) ∆v = w∆d + d∆w + 2∇d · ∇w.Notice that∇w = w ∇ log(w) = w(log(˜δ)∇α + α ∇˜δ˜δ)and so(2.17) ∇d · ∇w = wRecall 2. of Lemma 2.1 that we rewrite(2.18) ∇d · ∇˜δ = d˜δ .(log(˜δ)∇d · ∇α + α˜δ )∇d · ∇˜δ .10

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