Local and global well-posedness for the nonlinear Schr\"{o}dinger equation with nonhomogeneous boundary conditions
In this paper, we study the initial-boundary value problem for the nonlinear Schr\"{o}dinger equation in $\mathbb{R}^{n}_{+}$ \begin{equation*} i\partial_{t}u+\Delta u+\lambda |u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^{s}(\mathbb{R}^{n}_{+})$ with $s\in[0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in[1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^{s}(\mathbb{R}_{+})$ is extended to $H^{s}(\mathbb{R}^{n}_{+})$. Additionally, we obtain global solutions in the lower regularity setting $s\in[0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^{2}(\mathbb{R}^{n}_{+})$ a priori estimates.