On the non-homogeneous boundary value problem for Schrödinger equations

Abstract

In this paper we study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When Ω is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If Ω c is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.

Well-posedness for nonlinear Schrödinger equations with boundary forces in low dimensions by Strichartz estimates

“…Observe that there is a loss of 1 6p + derivatives in the estimate (3) compared to the estimate (2).…”

“…[8][9][10][11]20] study one-dimensional NLS with boundary forces on such a domain. There are also some results on related equations such as Ginzburg-Landau equations with boundary forces; see for example [2,6,7,15]. In high dimensions (d ≥ 3), well-posedness for NLS on bounded domains, even with homogeneous boundary datum h ≡ 0, has not been studied well yet.…”

Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

“…(The precise conditions which are sufficient to assure the existence, uniqueness and well-posedness of such a solution, until a certain time T > 0, are non-trivial [7].) This problem has applications, for example, in ionospheric modification experiments [8].…”

Complex Ginzburg–Landau equations with dynamic boundary conditions

“…Our proofs for strong solutions will use Sobolev embeddings. Therefore, we will have some restriction on p. In [4], it is proven that the defocusing cubic NLS with dynamic boundary conditions is globally well-posed for N = 2. Here, we improve this result in the context of the CGLE by proving the well-posedness of global solutions for dimensions N ≤ 3.…”

“…There are only a few results on the CGLE under nonhomogeneous boundary conditions ( [1], [2], [17]- [18], [43]). The NLS subject to inhomogenenous or nonlinear boundary conditions, which can be considered a limiting case of the CGLE, took much more attention in recent years; see for example [4], [5], [6], [15], [16], [24], [25], [26], [30], [31], [38]- [41], and [47].…”