Nonlinear Schrödinger equation in an exterior domain
Abstract
We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior Ω=ℝn∖Θ of a non-trapping obstacle Θ, with internal controls supported outside a large ball, and the corresponding boundary control problem on Ω0=BR0∖Θ with Dirichlet controls acting only on the outer sphere ∂ BR0. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces HσD(Ω), σ∈[−2,2], associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of Θ), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in H10(Ω) and in H2(Ω)∩ H10(Ω), in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.
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BibTeX
@misc{filho2026nonlinear,
title = {{Nonlinear Schrödinger equation in an exterior domain}},
author = {Roberto de A. Capistrano Filho and Lionel Rosier and Bingyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11968},
eprint = {2610.11968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11968},
note = {CC0},
}
RIS
TY - UNPB
TI - Nonlinear Schrödinger equation in an exterior domain
AU - Roberto de A. Capistrano Filho
AU - Lionel Rosier
AU - Bingyu Zhang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.11968
UR - https://arxiv.org/abs/2610.11968
AB - We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior Ω=ℝ^n∖Θ of a non-trapping obstacle Θ, with internal controls supported outside a large ball, and the corresponding boundary control problem on Ω_0=B_R_0∖Θ with Dirichlet controls acting only on the outer sphere ∂ B_R_0. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces H^σ_D(Ω), σ∈[−2,2], associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of Θ), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in H^1_0(Ω) and in H^2(Ω)∩ H^1_0(Ω), in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.
ER -