Nonhomogeneous Boundary Value Problems of Nonlinear Schrödinger Equations in a Half Plane

Abstract

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schrödinger 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 $H^s(\mathbb{R}\times {\mathbb{R}}^{+})$ with the boundary data $h(x,t)$ in an optimal space ${\mathcal{H}}^s(0,T)$ as defined in the introduction, which is slightly weaker than the space $$H_t^{(2s+1)/4}([0,T];L_x^2(\mathbb{R}))\cap L_t^2([0,T];H_x^{s+1/2}(\mathbb{R})),$$ 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 $n>1$.

References in corpus (1)

• Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations