Inverse Problems for Hyperbolic Equations

双曲方程的反问题

• 批准号: 9709637
• 负责人: Lizabeth Rachele
• 金额: $ 1.8万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 1999-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9709637&HistoricalAwards=false
• 关键词: Inverse; Problems; Hyperbolic; Equations

This proposal concerns two inverse problems for hyperbolic equations. The first problem involves the wave operator P for the Laplace-Beltrami operator for a smooth, Riemannian metric on a bounded n-dimensional domain. The inverse problem is to describe material properties (represented by the coefficients of P) of an object (represented by the n-dimensional domain) given only measurements made at the surface of the object (modeled by a boundary operator, the Dirichlet-to-Neumann map). We consider the continuous dependence of the metric on the Dirichlet-to-Neumann map for P. For this inverse problem (and inverse problems in oil prospection and seismology, for example) it is known, though, that the material properties of the object are not uniquely determined by surface measurements, in general. In particular, a metric is not uniquely determined by the Dirichlet-to-Neumann map for P since the pullback of a metric by a diffeomorphism that fixes the boundary has the same associated Dirichlet-to-Neumann map as the original metric. Sylvester and Uhlmann have shown that unique determination does hold at the boundary, though,and in the case that surface measurements are available for all time, it follows from Belishev and Kurylev that uniqueness holds in the interior, up to the pull-back by a diffeomorphism that fixes the boundary. To study the dependence, then, of material properties of the object on surface measurements, for this inverse problem for which uniqueness does not hold, we propose showing that the metric depends continuously on the Dirichlet-to-Neumann map, up to the pull-back by a diffeomorphism that fixes the boundary. Having shown that continuous dependence holds, one can expect that material properties of the interior of the object (represented by the coefficients of the differential equation) can, in principle, be reconstructed arbitrarily accurately from measurements made only at the surface (that is, from the Dirichlet-to-Neumann map). In t he second problem we consider an inverse problem for the system of operators P for elastodynamics with residual stress. The linearly elastic, nonhomogeneous, isotropic object being studied is represented by a bounded, 3-dimensional region with smooth boundary. The behavior of the object is described in terms of solutions of the system of equations for elastodynamics. Coefficients of these equations represent material properties of the object, and surface measurements are modeled by the associated Dirichlet-to-Neumann map. It is assumed there are no body forces acting on the object after time zero but that events in the past have built up a residual stress in the object, which is modeled by a smooth, symmetric, second-rank tensor that is divergence-free and has zero traction on the boundary. The central question we pose here is to what extent the Dirichlet-to-Neumann map associated with P uniquely determines the density, coefficients of elasticity, and residual stress at the boundary.

这一建议涉及双曲型方程的两个反问题。第一个问题涉及n维有界区域上光滑黎曼度量的Laplace-Beltrami算子的波算子P。逆问题是描述对象(由n维域表示)的材料属性(由n维域表示)，仅给出在对象表面进行的测量(由边界运算符Dirichlet-to-Neumann映射建模)。我们考虑度规对P的Dirichlet-to-Neumann映射的连续依赖性。对于这个反问题(例如，石油勘探和地震学中的反问题)，通常情况下，物体的材料性质并不是由表面测量唯一确定的。特别地，度量不是由P的Dirichlet-to-Neumann映射唯一确定的，因为通过固定边界的微分同胚拉回的度量具有与原始度量相同的关联Dirichlet-to-Neumann映射。然而，西尔维斯特和乌尔曼已经证明，唯一的决定确实存在于边界上，在表面测量始终可用的情况下，从别里舍夫和库里列夫得出的结论是，唯一性存在于内部，直到通过固定边界的微分同胚而拉回。为了研究物体材料性质对表面测量的依赖性，对于这个不存在唯一性的反问题，我们建议证明度规连续地依赖于Dirichlet-to-Neumann映射，直到通过固定边界的微分同胚拉回。在证明了连续依赖成立后，人们可以预期，原则上，物体内部的材料属性(由微分方程的系数表示)可以从仅在表面进行的测量(即，从狄里克莱特到诺伊曼映射)任意准确地重建。在第二个问题中，我们考虑了具有残余应力的弹性动力学算子组P的反问题。所研究的线弹性、非均匀、各向同性物体被表示为具有光滑边界的有界三维区域。物体的行为用弹性动力学方程组的解来描述。这些方程的系数表示对象的材料属性，表面测量由关联的Dirichlet-to-Neumann映射建模。假设在时间零之后没有体力作用在物体上，但过去的事件在物体中建立了残余应力，该残余应力由光滑的对称二阶张量来模拟，该张量是无发散的，在边界上没有牵引力。我们在这里提出的中心问题是，与P相关的Dirichlet-to-Neumann映射在多大程度上唯一地决定了边界上的密度、弹性系数和残余应力。