Nonlinear Neumann boundary value problem for NLS in the half-space
A talk in the Oberseminar Analysis series by
Shun Tsuhara
| Abstract: | We study the initial-boundary value problem for the nonlinear Schr\"odinger equation with a nonlinear Neumann boundary condition in the half-space.
In the one-dimensional case, Batal--\"Ozsar\i \, (2016) and Hayashi--Ogawa--Sato (2025) established the well-posedness of the problem in $H^1$ and $L^2$, respectively.
However, higher-dimensional cases have not been thoroughly investigated.
As far as the speaker is aware, the existence of $L^2$-solutions in higher dimensions has been partially studied by Ogawa--Sato--T. (2024), while other regularity classes remain open.
In this talk, we consider the existence of $H^1$-solutions for the problem in the higher-dimensional half-space.
Our approach is based on a new representation formula for solutions to the linear problem introduced by Audiard (2019).
Using this representation, we establish new boundary Strichartz estimates for the linear solution, including estimates with respect to both time and spatial derivatives.
These estimates allow us to establish $H^1$ well-posedness via the contraction mapping principle. Zoom Link: https://uni-bielefeld.zoom-x.de/j/68790273760?pwd=2aOMmlyA59zaLFbVbonREYcGAbJYRN.1 |