arXiv 1403.22552014-03math.AP

用复几何光学解从柯西数据恢复势函数

Recovering a Potential from Cauchy Data via Complex Geometrical Optics Solutions

Hoai-Minh Nguyen, Daniel Spirn

摘要：

本文研究了从Dirichlet到Neumann映射中恢复$d \geq 3$的$\mathbb{R}^d$中的域中的潜在$q$的问题.这个问题是通过刘维变换的逆卡尔德龙电导率问题。由Haberman和Tataru [11]以及Nachman和Lavine [17]的工作可知，一阶导数的电导率类和$W^{2,d/2}$电导率类分别具有唯一性。Haberman和Tataru的证明基于最初由Sylvester和Uhlmann [22]提出的复几何光学（CGO）解的构造，在Bourgain [2]引入的函数空间中。在Ferreira等人的工作中，第二个结果的证明。[10]基于通过Carleman估计构造CGO解。本文的主要目标是了解是否是一种方法，这是基于建设CGO解决方案的精神，西尔维斯特和乌尔曼，只涉及标准的索伯列夫空间可以用来获得这些结果。事实上，我们能够得到一个新的证明的唯一性的卡尔德龙问题的1）一个略有不同的类在[11]中，和2）类的$W^{2,d/2}$电导率。声明1）的证明是基于CGO解的新估计和一些与[11]相同的平均估计。声明2）的证明一方面基于Kenig等人[14]的广义Sobolev不等式，另一方面仅涉及CGO解的标准估计[22]。我们还能够证明3）$W^{s, 3/s}$（$\supsetneqq W^{2, 3/2}$）类电导率在三维中具有$3/2 < s < 2$的势的唯一性。据我们所知，声明3）是新的。

英文摘要：

This paper is devoted to the problem of recovering a potential $q$ in a domain in $\mathbb{R}^d$ for $d \geq 3$ from the Dirichlet to Neumann map. This problem is related to the inverse Calderón conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of $W^{2,d/2}$ conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvester and Uhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of the second result, in the work of Ferreira et al. [10], is based on the construction of CGO solutions via Carleman estimates. The main goal of the paper is to understand whether or not an approach which is based on the construction of CGO solutions in the spirit of Sylvester and Uhlmann and involves only standard Sobolev spaces can be used to obtain these results. In fact, we are able to obtain a new proof of uniqueness for the Calderón problem for 1) a slightly different class as the one in [11], and for 2) the class of $W^{2,d/2}$ conductivities. The proof of statement 1) is based on a new estimate for CGO solutions and some averaging estimates in the same spirit as in [11]. The proof of statement 2) is on the one hand based on a generalized Sobolev inequality due to Kenig et al. [14] and on another hand, only involves standard estimates for CGO solutions [22]. We are also able to prove the uniqueness of a potential for 3) the class of $W^{s, 3/s}$ ($\supsetneqq W^{2, 3/2}$) conductivities with $3/2 < s < 2$ in three dimensions. As far as we know, statement 3) is new.

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摘要：

英文摘要：

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