Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map
Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map
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非平稳Dirichlet-Neumann映射的双曲逆边值问题和时间延拓
DOI：
10.1017/s0308210500001943
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
M. Lassas
中科院分区：
文献类型：
--
作者：
Y. Kurylev;M. Lassas
设M是具有非空边界的紧致黎曼流形。本文考虑二阶双曲型初边值问题utt + but + a（x，D）u = 0在M R+，u中的反问题|MR+ = f，u| t=0 = ut| t=0 = 0。我们的目标是确定（M，g），B和a（x，D）的知识的非平稳狄利克雷-诺依曼映射（双曲响应算子）RT，具有足够大的T 0。响应算子RT是映射，其中是初边值问题解的法向导数。更具体地说，我们展示了以下内容。(i)它是可能的，以确定Rt为任何t 0，如果我们知道RT足够大的T和一些几何条件上的测地线行为（M，g）是满意的。(ii)这样就有可能唯一地确定（M，g）和B以及椭圆算子a（x，D）模广义规范变换.
Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem. More specifically, we show the following. (i) It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.(ii) It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.