The Zakharov System on the Upper Half-Plane

In this paper we study the Zakharov system on the upper half–plane $U=\{(x,y)\in {\mathbb{R}}^2:y>0\}$ with non-homogenous boundary conditions. In particular we obtain low regularity local well–posedness using the restricted norm method of Bourgain and the Fourier–Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data. To our knowledge this is the first paper which establishes low regularity results for the 2d initial-boundary value Zakharov system.