Abstract
We study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichletboundary conditions. Special care is devoted to the space where the boundary data belong. When $\Omega$ isthe complement of a non-trapping obstacle, well-posednessfor boundary data of optimal regularity is obtained by transposition arguments. If $\Omega^c$ is convex,a local smoothing property(similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As anapplication local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.
Highlights
The purpose of this article is to study the initial boundary value problem (IBVP) i∂tu + ∆Du = f,(x, t) ∈ Ω × [0, T ], u|t=0 = u0, x ∈ Ω, u|Σ = g, (x, t) ∈ Σ := ∂Ω × [0, T ], (IBVP)where f may be a forcing term or a nonlinearity depending on u, typically behaving like a power of u
The homogeneous BVP for the Schrodinger equation in non trivial geometrical settings has received a lot of interest over the last years
Ivanovici [12] proved that the full range of Strichartz estimates holds for the homogeneous BVP posed outside a strictly convex set, and obtained with Planchon [13] the well-posedness of the energy critical quintic Schrodinger equation on general non-trapping domains
Summary
The purpose of this article is to study the initial boundary value problem (IBVP). where f may be a forcing term or a nonlinearity depending on u (but not its gradient), typically behaving like a power of u. The first results that were not consequences of semi-groups arguments were obtained in dimension 2 for u0 ∈ H01 (see Brezis-Gallouet [5], Tsutsumi [29]) which is precisely the level of regularity where the semi-groups arguments do not work anymore They received a number of significant extensions, until the work of Burq-Gerard-Tzvetkov [6] who obtained the first results (to our knowledge) of global well-posedness for large dimensions and data when the equation is posed on the complement of a compact “non trapping” obstacle. From this little computation, it appears that the boundary data g should belong to some anisotropic Sobolev space of the kind H(s+1/2)/2(Rt, L2(∂Ω)) ∩ L2(Rt, Hs+1/2(∂Ω)). Consistant numerology of the boundary data g ∈ L2([0, T ], H1) ∩ H1/2([0, T ], L2) ( a slight loss will be necessary for the nonlinear problem, but not for the linear one)
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