This is Advance Partial Differential Equation question
By convention whenever n 2 1 and x, y e R”, we write |x= + ... + x} and (x, y) = { Σχ Xiyi c) Assume that u € C?(22") n C°(D*) is harmonic such that lim«|--00 og u(x) = 0. Show that 4. = limxl- u(x) exists with U. S" 1 21 u(cos(2), sin(O))de. Hint: You may use the removable singularity theorem. d) Show that there is no harmonic function u € C?(12") n C°(22") with lim x= u(x) = 0) and u(w) = 1 for all w with |wl = 1.
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