论文标题
无限领域的分数Calderón问题和庞加莱的不平等现象
Fractional Calderón problems and Poincaré inequalities on unbounded domains
论文作者
论文摘要
我们将许多最近的唯一性结果推广到分数Calderón问题上，以涵盖所有非空外部域的案例。我们工作的亮点是唯一性和局部数据逆问题的唯一性的表征是在整个欧几里得空间中支撑的电导率的一个方向上的域上的分数电导率方程式的表征，并衰减了无限度的恒定背景电导率。我们将Ghosh，Salo和Uhlmann的分数Calderón问题的唯一性证明概括为一般的抽象环境，以利用其论点的全部力量。这使我们可以观察到许多反问题的唯一性结果对于低阶分数laplacian的较高局部扰动也有唯一性结果。我们给出具体的示例模型来说明这些好奇的情况，并证明对域上任何一个方向界限的任何顺序的分数laplacians证明了庞加莱的不平等。我们在这些一般环境中建立runge近似结果，在界定集合的情况下也改善规律性假设，并证明一般的外部确定结果。在另一个伴侣工作中构建了对部分数据的反向电导率问题的唯一性的反示例。
We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.
