Nonhomogeneous Boundary Value Problems of Nonlinear Schr\"odinger Equations in a Half Plane

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schr"odinger equations posed in a half plane $\mathbb{R}\times {\mathbb{R}}^{+}$ with nonhomogeneous Dirichlet boundary conditions. For any given $s\ge 0$, if the initial data $\phi (x,y)$ are in Sobolev space $H^s($\mathbb{R}$\times $\mathbb{R}$^+) $withtheboundarydata$ h ( x, t) $inanoptimalspace${\cal H}^s(0,T)$ as defined in the introduction, which is slightly weaker than the space $$H_t^{(2s+1)\mathrm{/}4}([0,T];L_x^2(\mathbb{R}))\cap L_t^2([0,T];H_x^{s+1\mathrm{/}2}(\mathbb{R})),$$ the local well-posedness of the IBVP in $ C ( [0, T] ; H^s ( $\mathbb{R}$\times $\mathbb{R}$^+ ) )$ is proved. The global well-posedness is also discussed for $s=1$. The main idea of the proof is to derive a boundary integral operator for the corresponding nonhomogeneous boundary condition and obtain the Strichartz’s estimates for this operator. The results presented in the paper hold for the IBVP posed in a half space ${\mathbb{R}}^n\times {\mathbb{R}}^{+}$ with any $n>1$.