The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

We consider the initial-boundary value problem of the nonlinear Schr & ouml;dinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in L2(R+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2({\mathbb {R}}<^>2_+)$$\end{document} and Hs(R+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s({\mathbb {R}}<^>2_+)$$\end{document}, we consider the time local well-posedness of the problem in L2(R+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2({\mathbb {R}}<^>2_+)$$\end{document} and Hs(R+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s({\mathbb {R}}<^>2_+)$$\end{document}.