Global uniqueness for the fractional semilinear schrödinger equation

Ru Yu Lai; Yi-Hsuan Lin

doi:10.1090/proc/14319

Global uniqueness for the fractional semilinear schrödinger equation

Ru Yu Lai, Yi-Hsuan Lin

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation (-Δ)su + q(x, u) = 0 with s ε (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide an L ^∞ estimate for this nonlocal equation under appropriate regularity assumptions.

Original language	English
Pages (from-to)	1189-1199
Number of pages	11
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	3
DOIs	https://doi.org/10.1090/proc/14319
State	Published - 1 Mar 2019

Keywords

Calderón’s problem
Fractional schrödinger equation
Maximum principle
Nonlocal
Partial data
Semilinear

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title = "Global uniqueness for the fractional semilinear schr{\"o}dinger equation",

abstract = " We study global uniqueness in an inverse problem for the fractional semilinear Schr{\"o}dinger equation (-Δ)su + q(x, u) = 0 with s ε (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide an L ∞ estimate for this nonlocal equation under appropriate regularity assumptions. ",

keywords = "Calder{\'o}n{\textquoteright}s problem, Fractional schr{\"o}dinger equation, Maximum principle, Nonlocal, Partial data, Semilinear",

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AB - We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation (-Δ)su + q(x, u) = 0 with s ε (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide an L ∞ estimate for this nonlocal equation under appropriate regularity assumptions.

KW - Calderón’s problem

KW - Fractional schrödinger equation

KW - Maximum principle

KW - Nonlocal

KW - Partial data

KW - Semilinear

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