Inverse problems for fractional semilinear elliptic equations. (March 2022)

• Record Type: Journal Article
• Authors: Lai, Ru-Yu; Lin, Yi-Hsuan
• Is Part Of: Nonlinear analysis. Volume 216(2022)
• Publication Date: 2022-03
• Subjects: Calderón problem -- Dirichlet-to-Neumann map -- Semilinear elliptic equations -- Fractional Laplacian -- Higher order linearization -- Maximum principle -- Runge approximation -- Single measurement; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112699
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation ( − Δ ) s u + a ( x, u ) = 0 for 0 < s < 1 . For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient a ( x, u ) can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient can be recovered concurrently from these measurements. Finally, we investigate that these two fractional inverse problems can also be solved by using a single measurement, and all results hold for any dimension n ≥ 1 .