当前位置: X-MOL 学术Inverse Probl. Imaging › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
Inverse Problems and Imaging ( IF 1.2 ) Pub Date : 2021-01-27 , DOI: 10.3934/ipi.2021009
Giovanni Covi , Keijo Mönkkönen , Jesse Railo

We prove a unique continuation property for the fractional Laplacian $ (-\Delta)^s $ when $ s \in (-n/2, \infty)\setminus \mathbb{Z} $ where $ n\geq 1 $. In addition, we study Poincaré-type inequalities for the operator $ (-\Delta)^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.

中文翻译:

高阶分数拉普拉斯算子的独特连续性和Poincaré不等式及其在反问题中的应用

当$ s \ in(-n / 2,\ infty)\ setminus \ mathbb {Z} $其中$ n \ geq 1 $时,我们证明了分数拉普拉斯$(-\ Delta)^ s $的唯一连续性。另外,我们研究了当s \ geq 0 $时,算子$(-\ Delta)^ s $的庞加莱型不等式。我们应用结果表明,一个人可以从与高阶分数磁性Schrödinger方程相关的Dirichlet-Neumann映射中唯一地恢复到一个规范,并具有一定的电势和磁势。我们还研究了具有奇异电势的高阶分数阶Schrödinger方程。在这两种情况下,我们都获得了方程的Runge近似属性。此外,我们证明了在低规则性下$ d $平面Radon变换的部分数据问题的唯一性结果。我们的工作为更一般的算子扩展了反问题中的一些最新结果。
更新日期:2021-01-27
down
wechat
bug