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

2016/09/18 by Ran, Yu, Sun, Shu-Ming, Zhang, Bing-Yu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.05418

Abstract

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schrödinger equations posed in a half plane ℝ × ℝ⁺ with nonhomogeneous Dirichlet boundary conditions. For any given s ≥ 0, if the initial data φ(x, y) are in Sobolev space H^s(ℝ× ℝ⁺) with the boundary data h ( x, t) in an optimal space \cal H^s(0,T) as defined in the introduction, which is slightly weaker than the space H^(2s+1)/4_t ([0, T]; L_x²(ℝ ) ) ∩ L²_t ( [ 0, T]; H^{s+ 1/2} _x ( ℝ ) ), the local well-posedness of the IBVP in C ( [0, T] ; H^s ( ℝ× 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 ℝⁿ× ℝ⁺ with any n>1.

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