By convention whenever n 2 1 and x, y e R”, we write |x| = Vx} + + x; and (x,y) = {xivi . i=1 u(x) c) Assume that u € C?

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By Convention Whenever N 2 1 And X Y E R We Write X Vx X And X Y Xivi I 1 U X C Assume That U C 1 (160.54 KiB) Viewed 33 times

By convention whenever n 2 1 and x, y e R”, we write |x| = Vx} + + x; and (x,y) = {xivi . i=1 u(x) c) Assume that u € C?(92%) n C°(22") is harmonic such that limx1400 = 0. Show that log |x| Uo0 = limx140 U(x) exists with = U.. = 1 207 u(cos(2), sin(O))de. Hint: You may use the removable singularity theorem. d) Show that there is no harmonic function u € C²(12) n C°(0*) with lim x1-00 U(x) = 0) and u(w) = 1 for all w with \w] = 1.