Analysis of a Model of Elastic Dislocations in Geophysics
We analyze a mathematical model of elastic dislocations with 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.
The authors thank C. Amrouche, S. Salsa, and M. Taylor for useful discussion and for suggesting relevant literature. They also thank the anonymous referees for their careful reading of this work and useful suggestions. A. Aspri and A. Mazzucato thank the Departments of Mathematics at NYU-Abu Dhabi and at Politecnico of Milan for their hospitality. A. Mazzucato was partially supported by the US National Science Foundation Grant DMS-1615457. M. V. de Hoop gratefully acknowledges support from the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-1815143, and the corporate members of the Geo-Mathematical Group at Rice University.
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