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Let I be the upper-half plane, N2 = {(x1, x2) : x2 > 0}. Consider the following Green's function: S AG = S(x y) inside 1 i OnG = 0 on 212 (here, onG denotes normal derivative at the boundary: OnG = VG- =0x2G(x, y)|x=(x1,0).) (a) Show that G = . In |x y + z Inx yR|, where yr = (y1, 42). (b) Suppose that u solves . Au = 0) inside 12 i Onu = f(x1) on 222.. Show that a necessary condition for the solution to exist is that sof(s)ds = 0. Assuming this condition is satisfied, derive the representation formula for u(x) in terms of the Green's function G. Let I be the upper-half plane, N2 = {(x1, x2) : x2 > 0}. Consider the following Green's function: S AG = S(x y) inside 1 i OnG = 0 on 212 (here, onG denotes normal derivative at the boundary: OnG = VG- =0x2G(x, y)|x=(x1,0).) (a) Show that G = . In |x y + z Inx yR|, where yr = (y1, 42). (b) Suppose that u solves . Au = 0) inside 12 i Onu = f(x1) on 222.. Show that a necessary condition for the solution to exist is that sof(s)ds = 0. Assuming this condition is satisfied, derive the representation formula for u(x) in terms of the Green's function G