Determination of second-order elliptic operators in two dimensions from partial Cauchy data
Determination of second-order elliptic operators in two dimensions from partial Cauchy data
复制标题
DOI：
10.1073/pnas.1011681107
复制
发表时间：
2010-12
期刊：
影响因子：
--
通讯作者：
O. Imanuvilov;G. Uhlmann;Masahiro Yamamoto
中科院分区：
文献类型：
--
作者：
O. Imanuvilov;G. Uhlmann;Masahiro Yamamoto
研究了在非空任意相对开子集上测量的Cauchy数据在二维空间中确定一般二阶椭圆算子系数的反边值问题。我们给出了产生相同部分柯西数据的系数集的完整刻画。作为推论，我们证明了各向同性电导率方程、Schrödinger方程、对流-扩散方程、各向异性电导率方程模一组在边界处恒等的微分同态以及磁性Schrödinger方程模规范变换在用部分柯西数据确定系数时的几个唯一性结果。关键的一步是利用Carleman估计构造新的复杂几何光学解。
We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollary we prove several uniqueness results in determining coefficients from partial Cauchy data for the isotropic conductivity equation, the Schrödinger equation, the convection–diffusion equation, the anisotropic conductivity equation modulo a group of diffeomorphisms that are the identity at the boundary, and the magnetic Schrödinger equations modulo gauge transformations. The key step is the construction of novel complex geometrical optics solutions using Carleman estimates.