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

Abstract

This paper discusses the initial-boundary-value problems (IBVPs) 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 H s (0, T ) as defined in the introduction, which is slightly weaker than the spacethe 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 estimates for this operator. The results presented in the paper hold also for the IBVP posed in a half space R n × R + with any n > 1.2010 Mathematics Subject Classification. Primary 35Q55.It turns out, however, that the space W s (0, T ) is not the optimal choice. Indeed, as the trace of the solution v of the linear Schrödinger equation in R 2 ,most likely does not belong to the space W s (0, T ) (see the discussion in Section 2), we will show that the solution v possesses the following trace property (see Lemma 2.1 in Section 2):then the trace of the solution v ∈ C(R; H s (R 2 )), v b (x, t) = v(x, 0, t), belongs to the space H s (R 2 ) := ß w ∈ L 2 (R 2 ) : Ä 1 + |λ| + |ξ| 2 | ä s/2 λ + |ξ| 2 1/4 F[w](λ, ξ) ∈ L 2 (R 2 ) ™

Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications

“…The main issue is thus to prove estimate (3.9) : it was obtained very recently by [23] with }u} L p W s,q in the left hand side for bounded time intervals. While the core of the L p t L q estimate does not require significant modifications we include a full proof for comfort of the reader.…”

“…The idea in [23] is to use a T T˚argument similar to the classical one for the Schrödinger equation, namely if we set p ψ " 2ηp gpξ,´|ξ| 2`η2 q1 ηě0 then (3.10) reads…”

“…We point out however that our (uniform) Kreiss-Lopatinskii condition derived was quite restrictive, and in particular forbid the Neuman boundary condition, a limitation which is lifted here. On the issue of Strichartz estimates, Y.Ran, S.M.Sun and B.Y.Zhang considered in [23] the IBVP (1.1) on a half space with nonhomogeneous Dirichlet boundary conditions. They derived explicit solution formulas in the spirit of their work on the Korteweg de Vries equation with J.Bona [8], and managed to use them to obtain local in time Strichartz estimates without loss of derivatives.…”

“…Note however that in the appendix we provide a construction showing that it is less accurate for evanescent waves (solutions that exist only for BVPs and remain localized near the boundary). Although not stated explicitly in [23], we might roughly summarize their linear results as follows:…”

Inhomogeneous Dirichlet boundary value problem for nonlinear Schrödinger equations in the upper half-space

“…There are some papers discussing the inhomogeneous initial boundary value problem for multidimensional nonlinear Schrödinger equations. Among them, we refer [1,2,27,28,31]. These papers are devoted to study fundamental questions of local existence and uniqueness of solutions in Sobolev space H s , s > 0, by applying classical tools such as Strichartz and energy estimates.…”