论文标题
半空间中裂纹反问题的稳定性特性
Stability properties of a crack inverse problem in half space
论文作者
论文摘要
我们在本文中显示了Lipschitz稳定性在半空间中发生裂纹逆问题的稳定性结果。直接问题是在顶部边界上的neumann条件为零的拉普拉斯方程。强迫术语是整个裂缝的不连续性。该公式可能与弹性介质中的地质断层有关，也可能与半空间中的不可压缩流量减去内壁有关。直接问题在适当的功能空间中很好地构成。我们研究了相关的逆问题，在裂缝上的跳跃未知，更重要的是，裂纹的几何形状和位置尚不清楚。逆问题的数据是在顶部边界的一部分上方的Dirichlet类型。我们证明，在裂纹几何形状的某些假设下，可以解决这个反问题。本文的亮点显示了这个反问题的稳定性结果。假设裂缝是平面，我们表明，尽管底层PDE的强迫术语尚不清楚，但重建包含裂缝的平面是Lipschitz稳定的。这种均匀的稳定性结果是在上面的强迫项界定的，并且在适当的规范中远离零的限制。
We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem is of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data is bounded below away from zero in appropriate norms.
