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The higher-order fractional Schrödinger equation with nonlinear local perturbations: Uniqueness

Giovanni Covi, Ru-Yu Lai, Lili Yan

Abstract

We study the higher-order fractional Schrödinger equation with local nonlinear perturbations and investigate both the forward and inverse problems. We establish both the Sobolev $H^s$ and Hölder $C^s$ estimates for the well-posedness of the nonlinear problem, based on the corresponding estimates derived for the linear fractional Schrödinger equation. For the inverse problem, we show that the local nonlinear perturbations can be uniquely determined from the Dirichlet-to-Neumann map, by using the higher-order linearization and the unique continuation property of the fractional Laplace operator.

The higher-order fractional Schrödinger equation with nonlinear local perturbations: Uniqueness

Abstract

We study the higher-order fractional Schrödinger equation with local nonlinear perturbations and investigate both the forward and inverse problems. We establish both the Sobolev and Hölder estimates for the well-posedness of the nonlinear problem, based on the corresponding estimates derived for the linear fractional Schrödinger equation. For the inverse problem, we show that the local nonlinear perturbations can be uniquely determined from the Dirichlet-to-Neumann map, by using the higher-order linearization and the unique continuation property of the fractional Laplace operator.

Paper Structure

This paper contains 16 sections, 15 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Motivation and connection to the literature
  3. Outline of the article
  4. Preliminaries
  5. Function spaces
  6. The forward problem
  7. The Dirichlet-to-Neumann map
  8. Linearization
  9. Linearization
  10. Linearization of the Dirichlet-to-Neumann map
  11. Proof of the main theorem
  12. Recovering the linear coefficient
  13. An integral identity for the nonlinear coefficients
  14. Recovering the nonlinear coefficients
  15. Acknowledgments
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Unique determination) Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$. Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$. Let $\mathbf{P then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.1: Well-posedness for the nonlinear equation
  • proof
  • Definition 2.1
  • Proposition 2.3
  • ...and 18 more