Stability properties of a crack inverse problem in half space
- darko volkov,
- Yulong Jiang
Abstract
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.17 Jul 2020Submitted to Mathematical Methods in the Applied Sciences 24 Jul 2020Submission Checks Completed
24 Jul 2020Assigned to Editor
01 Aug 2020Reviewer(s) Assigned
16 Feb 2021Review(s) Completed, Editorial Evaluation Pending
01 Mar 2021Editorial Decision: Revise Minor
12 Apr 20211st Revision Received
13 Apr 2021Submission Checks Completed
13 Apr 2021Assigned to Editor
13 Apr 2021Editorial Decision: Accept
30 Sep 2021Published in Mathematical Methods in the Applied Sciences volume 44 issue 14 on pages 11498-11513. 10.1002/mma.7509