Stability properties of a crack inverse problem in half space

TL;DR

This paper establishes a Lipschitz stability result for an inverse problem involving detecting a crack in a half space using boundary data, with implications for geological and fluid flow applications.

Contribution

It proves the unique solvability and Lipschitz stability of reconstructing a planar crack's geometry from boundary measurements, even with unknown forcing terms.

Findings

Lipschitz stability for crack reconstruction in half space

Unique solvability of the inverse problem under geometric assumptions

Stability holds despite unknown forcing term and boundary data bounds

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.