Analysis of a Model of Elastic Dislocations in Geophysics

Andrea Aspri, Elena Beretta, Anna L. Mazzucato, Maarten V. De Hoop

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
JournalArchive for Rational Mechanics and Analysis
DOIs
StateAccepted/In press - Jan 1 2019

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Geophysics
Dislocation
Forward Problem
Jump
Earth (planet)
Well-posedness
Uniqueness
Very Weak Solutions
Unique Continuation
Sobolev spaces
Lipschitz Continuity
Model
Weighted Sobolev Spaces
Free Boundary
Inverse problems
Half-space
Lipschitz
Discontinuity
Inverse Problem
Interior

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

Analysis of a Model of Elastic Dislocations in Geophysics. / Aspri, Andrea; Beretta, Elena; Mazzucato, Anna L.; De Hoop, Maarten V.

In: Archive for Rational Mechanics and Analysis, 01.01.2019.

Research output: Contribution to journalArticle

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