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Stability properties of a crack inverse problem in half space
  • darko volkov,
  • Yulong Jiang
darko volkov

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Yulong Jiang
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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