By convention whenever n 2 1 and x, y e R", we write n |x = + + x2 and (x, y) = xiyi i=1 = = : a 1. Consider 12 = {x € R

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By Convention Whenever N 2 1 And X Y E R We Write N X X2 And X Y Xiyi I 1 A 1 Consider 12 X R 1 (205.31 KiB) Viewed 44 times

By convention whenever n 2 1 and x, y e R", we write n |x = + + x2 and (x, y) = xiyi i=1 = = : a 1. Consider 12 = {x € R2 : [x] < 1} and N* = {x € R2 : [x] > 11. For any x € RP \ {0}, set F(x) = x* = . For any function u : 1 → R, we define a function u* : 1* → R setting u*(x) = u(x*), x + 2*. a) Show that F1(x) = 1 and F2(x) = it are harmonic in RP \ {0} with |VF1(x)| = |VF2(x) = 1 |x12 and (VF1(x), VF2(x)) = 0. b) For any u € C²(12), show that u is sub-harmonic in 2 \ {0} if and only if u* e C?(92*) is sub-harmonic in 2*. Hint: Show first the identity Δ(uo F) = Xxvf, VF;)@ju o F + (F;)ą ju o F. , , ο + Σ. j