Analysis of a Model of Elastic Dislocations in Geophysics
Analysis of a Model of Elastic Dislocations in Geophysics
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DOI：
10.1007/s00205-019-01462-w
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发表时间：
2018-09
影响因子：
2.5
通讯作者：
A. Aspri;E. Beretta;A. Mazzucato;Maarten V. de Hoop
中科院分区：
文献类型：
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作者：
A. Aspri;E. Beretta;A. Mazzucato;Maarten V. de Hoop
我们分析了弹性位错的数学模型及其在地球物理学中的应用，其中弹性位错是指弹性固体内部的开放的、定向的利普希茨表面，在该表面上存在位移的不连续性。我们将地球建模为一个无限的、各向同性的、非均匀的、占据半个空间的弹性介质，并假设拉梅参数仅具有利普希茨连续性。我们研究了通过在地球表面施加无牵引力边界条件、牵引力的连续性和跨断层位移的给定跳跃来确定位移的正向问题的非常弱解的适定性。我们采用合适的加权索博列夫空间进行分析。我们利用正问题的适定性和唯一连续论证来确定逆问题的唯一性，即通过测量地球表面的位移来确定位错表面和位移跳跃。唯一性适用于切向或法向跳跃以及表面上的某些几何条件。
We analyze a mathematical model ofelastic dislocationswith applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lamé parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface of the Earth, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well-posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.