Inverse problems for fractional semilinear elliptic equations

By Ru-Yu Lai, Yi-Hsuan Lin. Nonlinear Analysis, 2020

70 citations recorded by the source corpus.

This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-\Delta)^s u +a(x,u)=0$ 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 $n\geq 1$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2708313255964726ad62ad0c6d5d0b0d3555df99

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Citations in the corpus

The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “Other variants of the fractional Calderón problem include semilinear fractional (magnetic) Schrödinger equation [19, 20, 24, 25], fractional heat equation [21, 35] and fractional conductivity equation [8] (see also [6, 13] for equations arising from a nonlocal Schrödinger-type elliptic operator).”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “(i) Is reconstruction from a single measurement [15, 27] possible also in the higher order cases? (ii) Is there stability [75] in the higher order cases? (iii) Is there exponential instability [73] in the higher order cases? (iv) Is there uniqueness for the Calderón problem for fractional semilinear”