论文标题
独特的延续属性和庞加莱的不平等，用于高级分数拉普拉斯人，施加了反问题
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
论文作者
论文摘要
当$ s \ in（-n/2，\ infty）\ setMinus \ mathbb {z} $时，我们证明了分数laplacian $（-Δ）^s $的唯一延续属性。此外，当$ s \ geq 0 $时，我们研究了操作员$（-Δ）^s $的庞加莱型不平等。我们应用结果表明，从与高阶分数磁性schrödinger方程相关的dirichlet到neumann地图上，可以唯一恢复，最多可恢复量规，电势和磁性电位。我们还研究具有单数电势的高阶分数schrödinger方程。在这两种情况下，我们都获得了方程式的Runge近似属性。此外，我们证明了$ d $ - 平面ra ra的部分数据问题的唯一性结果。我们的工作扩展了一些最新的总体运营商反向问题的结果。
We prove a unique continuation property for the fractional Laplacian $(-Δ)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincaré-type inequalities for the operator $(-Δ)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.
