Please solve this partial differential equation question.

(10 points) Solve the Laplace equation Au = 0 in the domain S := {(x, y, 2) ER3 : 1 0 be a given constant. Solve at = hum for all (27,15) 6 (0,1) x (0, +00), u(0, t) = u(1, t) = 0 for all t 2 0, u(:r:, 0) = 1 005(253) for all :r: 6 [0,1]. Solution. Boundary conditions at a: = t = 0 require 0 = 11(33 = 0, t = 0) = 1 cos 2, impossible, so no solutions. (2) (4 points) Let c > 0 be a given constant. Solve u\" = c215m for all (as, t) 6 (0,1) x (0,+oo), u(0,t) = 1.:(1, t) = 0 for all t 2 0, u(x, 0) = sinx cosa: for all a: 6 [0,1], ut(:r:,0) = cosx for all a: 6 [0,1]. Solution. Boundary conditions at a: = t = 0 require 0 = u(:c = 0, t = 0) = sin 1 cos 1, impossible, so no solutions. (3) (4 points) Let the domain be R:= {(x, y, z) ER3 : 1