NONHOMOGENEOUS BOUNDARY VALUE PROBLEMS OF NONLINEAR SCHRODINGER EQUATIONS IN A HALF PLANE

This paper studies the initial- boundary- value problems (IBVPs) of nonlinear Schrodinger equations posed in a half plane R x R+ with nonhomogeneous Dirichlet boundary conditions. For any given s >= 0, if the initial data phi(x,y) are in Sobolev space H-s(R x R+) with the boundary data h (x, t) in an optimal space Hs (0,T) as de fi ned in the introduction, the local wellposedness of the IBVP in C([0,T]; H-s (R x R+)) is proved under necessary compatibility conditions on phi and h. The global well- posedness is also discussed for s = 1. The main idea of the proof for the local well- posedness is to derive a boundary integral operator for the corresponding nonhomogeneous boundary condition and obtain the Strichartz estimates for this operator. The space H-s (0,T) for h (x,t) is consistent with the temporal trace result that can be derived for the solutions of pure initial- value problems on R-2