Cubic nonlinear Schrodinger equation on three dimensional balls with radial data

We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrodinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of $X^{s,b}$ spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrodinger flow with particular initial data.