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 language||English (US)|
|Journal||Nonlinear Analysis, Theory, Methods and Applications|
|State||Published - Mar 2022|
Bibliographical noteFunding 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 .
© 2021 Elsevier Ltd
- Calderón problem
- Dirichlet-to-Neumann map
- Fractional Laplacian
- Higher order linearization
- Maximum principle
- Runge approximation
- Semilinear elliptic equations
- Single measurement