Strichartz estimates for Schrödinger equations with nonlinear boundary interactions
Abstract
We study a Schrödinger equation in the upper half-space with a nonlinear Neumann boundary interaction driven by the Bessel operator Ba, a>−1. The problem arises naturally as an extension formulation for a nonlocal NLS with memory and can also be interpreted as a Schrödinger evolution with a nonlinear singular source concentrated on a codimension-one interface. We first develop a complete linear theory for the associated inhomogeneous problem with nonhomogeneous Neumann data. A central ingredient is a new Duhamel representation formula that separates bulk and boundary dynamics and identifies the precise role of the boundary propagator. Using this formula, we establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure induced by the Bessel operator. The analysis reveals a basic dichotomy between the regimes a≥ 0 and −1<a<0: in the former, bulk and boundary exhibit a unified dispersive behavior, whereas in the latter the dispersive structure becomes anomalous, requiring weighted estimates and distinct functional frameworks for the bulk and boundary contributions. As an application of the linear theory, we prove well-posedness results for the nonlinear problem in the La2-mass critical and subcritical regimes. For a≥ 0, we obtain global well-posedness for sufficiently small critical data, together with local and global results in the subcritical case. In the anomalous range −1<a<0, we establish existence and uniqueness on arbitrary finite time intervals for sufficiently small critical data, as well as local well-posedness in the subcritical regime. The results provide a unified dispersive framework for Schrödinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics.
Related
- Strichartz estimates to fractional Schrödinger equations
- Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Sharp critical Strichartz estimates for a resonant hyperbolic Schrödinger flow on the 3D waveguide
- Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains
BibTeX
@misc{garofalo2026strichartz,
title = {{Strichartz estimates for Schrödinger equations with nonlinear boundary interactions}},
author = {Nicola Garofalo and Gigliola Staffilani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.23229},
eprint = {2605.23229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23229},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strichartz estimates for Schrödinger equations with nonlinear boundary interactions
AU - Nicola Garofalo
AU - Gigliola Staffilani
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2605.23229
UR - https://arxiv.org/abs/2605.23229
AB - We study a Schrödinger equation in the upper half-space with a nonlinear Neumann boundary interaction driven by the Bessel operator Ba, a>−1. The problem arises naturally as an extension formulation for a nonlocal NLS with memory and can also be interpreted as a Schrödinger evolution with a nonlinear singular source concentrated on a codimension-one interface. We first develop a complete linear theory for the associated inhomogeneous problem with nonhomogeneous Neumann data. A central ingredient is a new Duhamel representation formula that separates bulk and boundary dynamics and identifies the precise role of the boundary propagator. Using this formula, we establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure induced by the Bessel operator. The analysis reveals a basic dichotomy between the regimes a≥ 0 and −1<a<0: in the former, bulk and boundary exhibit a unified dispersive behavior, whereas in the latter the dispersive structure becomes anomalous, requiring weighted estimates and distinct functional frameworks for the bulk and boundary contributions. As an application of the linear theory, we prove well-posedness results for the nonlinear problem in the L_a^2-mass critical and subcritical regimes. For a≥ 0, we obtain global well-posedness for sufficiently small critical data, together with local and global results in the subcritical case. In the anomalous range −1<a<0, we establish existence and uniqueness on arbitrary finite time intervals for sufficiently small critical data, as well as local well-posedness in the subcritical regime. The results provide a unified dispersive framework for Schrödinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics.
ER -