Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $

Serena Federico; Gigliola Staffilani; Serena Federico; Gigliola Staffilani

doi:10.3934/mine.2022033

Mathematics in Engineering

2022, Volume 4, Issue 4: 1-23. doi: 10.3934/mine.2022033

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Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $

Serena Federico ¹,
Gigliola Staffilani ^{2
,
,}

1.
Department of Mathematics: Analysis Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Ghent, B 9000, Belgium
2.
Department of Mathematics Massachusetts Institute of Technology, 77 Massachusetts Ave, MA 02139-4307, USA

Received: 28 June 2021 Accepted: 28 August 2021 Published: 18 September 2021

In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.
- Schrödinger equations on the torus,
- variable coefficient Schrödinger equations,
- Strichartz estimates,
- periodic NLS,
- time-degenerate equations
Citation: Serena Federico, Gigliola Staffilani. Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $[J]. Mathematics in Engineering, 2022, 4(4): 1-23. doi: 10.3934/mine.2022033

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Abstract

In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.

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