By convention whenever n 2 1 and x, y € R", we write 1x1 = Vež += Ź { + ... + x; and (x, y) = X;Yi. 1. Consider 12 = {x

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By Convention Whenever N 2 1 And X Y R We Write 1x1 Vez Z X And X Y X Yi 1 Consider 12 X 1 (183.99 KiB) Viewed 60 times

By convention whenever n 2 1 and x, y € R", we write 1x1 = Vež += Ź { + ... + x; and (x, y) = X;Yi. 1. Consider 12 = {x € R2 : [x] < 1} and $2* = {x € R2 : [x] > 1}. For any x € RP \ {0}, set F(x) = x* = . For any function u : 12 - R, we define a function u* : 1* - R setting ư(x) = u(x), x + 2*. a) Show that F2(x) = and F2(x) = f are harmonic in R? \ {0} with VF1(x) = |VF2(x) = 1 1x12 and (VF1(x), VF2(x)) = 0. b) For any u € C²(2), show that u is sub-harmonic in 12 \ {0} if and only if u* € C?(92*) ) 0u is sub-harmonic in S2*. Hint: Show first the identity A(u o F) = X(VF., VF 10.ju o F + ACF )āju o F.