论文标题
分数半线性椭圆方程的逆问题
Inverse problems for fractional semilinear elliptic equations
论文作者
论文摘要
本文涉及分数半连接椭圆方程的前进和反问题$（-Δ）^s u +a（x，u）= 0 $ for $ 0 <s <1 $。对于远期问题，我们证明了该问题已得到充分的解决方案，并为小型外部数据提供了独特的解决方案。我们在这里考虑的逆问题包括两种情况。首先，我们证明，未知系数$ a（x，u）$可以从外部测量的知识（称为dirichlet to-neumann地图）中唯一确定。其次，尽管媒体中存在未知的障碍，但我们表明障碍物和系数可以从这些测量值中同时恢复。最后，我们调查了这两个分数反问题也可以通过使用单个测量方法来解决，并且对于任何维度$ n \ geq 1 $的所有结果均可解决。
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 $n\geq 1$.
