Be sure that this examination has 3 pages. Let c(u) be a smooth function satisfying both c (u) < 0 and c(u) < 0 for all u > 0. Consider laplace"s equation for u = u(r, ) in the wedge 0 r r, 0 , where (r, ) are polar coordinates: urr+ 1 r2 u = 0 , u(r, ) = 0 , 0 , u(r, ) = 1 , u bounded as r 0 . For what values of the wedge-angle is the partial derivative ur bounded as r 0? (remark: you can still nd even if you are unable to solve part (i)). Consider a sphere of radius a with azimuthal symmetry and with surface potential u(a, ) = f ( ). Then, the solution u(r, ) to the axisymmetric laplace"s equation, de ned both inside and outside the sphere, satis es. 0 < < , r2 sin u(a, ) = f ( ) ,
