Inverse problems for fractional semilinear elliptic equations

Ru Yu Lai, Yi Hsuan Lin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation (−Δ)su+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.

Original languageEnglish (US)
Article number112699
JournalNonlinear Analysis, Theory, Methods and Applications
Volume216
DOIs
StatePublished - Mar 2022

Bibliographical note

Funding Information:
R.-Y. Lai is partially supported by the NSF, United States grant DMS-1714490 . Y.-H. Lin is partially supported by the Ministry of Science and Technology Taiwan , under the Columbus Program: MOST-109-2636-M-009-006 .

Publisher Copyright:
© 2021 Elsevier Ltd

Keywords

  • Calderón problem
  • Dirichlet-to-Neumann map
  • Fractional Laplacian
  • Higher order linearization
  • Maximum principle
  • Runge approximation
  • Semilinear elliptic equations
  • Single measurement

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